AUTOMORPHIC INDUCTION AND LEOPOLDT TYPE CONJECTURES FOR GL(n)

HARUZO HIDA

Dedicated to Professor Mikio Sato on the occasion of his 70-th birthday

March 3, 1999

1. Introduction

Let F be a number field and G_{0} be a non-split inner form of GL(n)_{/F} split at a rational prime p and ∞. We define G = Res_{F/ℚ}G_{0}. We start with cohomological study of automorphic forms on the reductive algebraic group G_{/ℚ} which is isomorphic to a product of copies of GL(n) over a number field. In particular, under not so restrictive conditions on G, we shall show that the nearly p–ordinary (cuspidal) cohomology group

H = H_{n.ord}^{q}(X_{1}(Np^{∞}), ℚ_{p}/ℤ_{p}) = \underrightarrow{lim}_{r}H_{n.ord}^{q}(X_{1}(Np^{r}), ℚ_{p}/ℤ_{p})

is of co-finite type over Λ = ℤ_{p}[[T (𝔯_{p})]] for a maximal (split) torus T_{/ℤ} of GL(n)_{/ℤ} (Theorem 6.2), where the injective limit is taken with respect to the restriction maps, 𝔯_{p} = 𝔯⊗_{ℤ} ℤ_{p} for the integer ring 𝔯 of F and Λ is the completed group algebra: Λ = \underleftarrow{lim}_{n}ℤ_{p}[T (ℤ/p^{n}ℤ)]. Thus for a suitable q = r (that is the degree, called the bottom degree; see (BD) in 6.3 for the exact formula of r), we will also show the finiteness over Λ of the universal p–adic Hecke algebra h^{n.ord} defined as a subalgebra of End_{Λ}(H) generated by the standard Hecke operators (Corollary 6.3). Actually we prove, under some assumptions, that the Hecke algebra of weight χ ∈ X(T) is obtained, up to finite error, as the specialization of the universal Hecke algebra along the algebra homomorphism χ : Λ → O induced by χ (Control theorem: Theorem 6.5).

Here we do not disregard the torsion part of finite level cohomology groups (although in my earlier works for GL(2), we restricted our study to their torsionfree part), because the Hecke algebra might be too small for non totally real or non CM fields if we kill torsion of each cohomology group.

Although the algebra h^{n.ord} satisfies various good properties as listed above, basic invariants in ring theory, for example, its Krull dimension, are still difficult to determine in general. The structure of h^{n.ord} is known to our satisfaction only for GL(1) and GL(2) over ℚ, for which h^{n.ord} and the Pontryagin dual H^{∗} of H are projective Λ–modules of finite rank. More generally, we know the expected value of the Krull dimension for GL(1) over an arbitrary number field and for all inner forms of GL(2) over totally real fields F, assuming the Leopoldt conjecture for (F, p) (see the description after Conjecture 7.1 for known information). However, we can make a reasonable guess for the dimension of h^{n.ord} for general G. Here we shall make the following conjecture predicting the Krull dimension of h^{n.ord}:

Research of the author is partially supported by an NSF grant.

Conjecture 1.1. Suppose that G_{0} is an inner form of GL(n)_{/F}. Then we have

dim(h^{n.ord} ⊗_{ℤ_{p}} ℚ_{p}) ≤ \begin{cases}m[F : ℚ] + 1 & if n = 2m, \\ m[F : ℚ] + r_{2} + 1 & if n = 2m + 1, \end{cases}

where r_{2} is the number of complex places of F.

It might look odd to have inequality, but depending on the prime-to–p level N, the Hecke algebra may be just trivial. Also the algebra h^{n.ord} might be very small if F is neither a totally real field nor a CM field. In other words, we conjecture the equality in the above formula if N is sufficiently small (so its norm sufficiently large) and if F is either a totally real field or a CM field. This conjecture is a GL(n)– version of the Leopoldt conjecture. In fact, when m = 0 (⇔ n = 1), the conjecture is just the Leopoldt conjecture for p and F. We will check in 7.2 the compatibility of the conjecture with various Langlands functoriality; in particular, it behaves well under automorphic induction. If we admit the automorphic induction functoriality (see [Cl] 1.1) relative to finite extensions L/F′ for real abelian extensions F′/ℚ and Conjecture 1.1 for GL(n)_{/F′}, the above conjecture implies the original Leopoldt conjecture for an arbitrary totally real field F (see Corollary 7.4). Note here that the Leopoldt conjecture is known to be true for abelian extensions F′/ℚ.

The above conjecture was first made in 1994 when I started writing this paper (after having finished an earlier work [H95]) and was also stated in a series of lectures I gave at the Galilée Institute of Université de Paris-Nord in June 1994. Earlier than this work, the lower bound of the Krull dimension of the full and p– ordinary deformation ring of a Galois representation had been computed by Mazur [M], and the formula of Mazur for p–ordinary case is the same as the one presented above for modular two dimensional Galois representations over F = ℚ. As conjectured by Mazur, for modular two dimensional representations, the Hecke algebra is isomorphic to the deformation ring (at least in the p–ordinary cases); so, the identity of the formulas for n = 2 is a natural consequence of Mazur’s conjecture, which is now a theorem of Taylor-Wiles [W] and [TW] (see also [HM] Section 4.3). In the meantime, generalizing Mazur’s perspective, J. Tilouine [Ti] has made a conjecture predicting the Krull dimension of the (nearly p–ordinary) universal deformation ring deforming a fixed p–adic Galois representation having values in a smooth reductive group (over ℤ_{p}). Of course, if the deformation ring for GL(n) is isomorphic to the Hecke algebra, Conjecture 1.1 is a special case of Tilouine’s conjecture. One of the purposes of this paper is to describe evidences for the conjecture from the automorphic side. Related to this, we will prove the following fact, as a special case of Theorem 8.1:

If p splits completely in F and π is nearly p–ordinary, then the Newton polygon of the Hecke polynomial of π at p coincides with the Hodge polygon of the motive (conjecturally) associated to π.

Thus if such a motive exists, the near p–ordinarity of automorphic representations implies the near p–ordinarity of the Galois representation (in the sense of [Ti]) of the motive. In particular, from this, we can deduce that the Newton polygon of any cohomological automorphic representation is located on or above the Hodge polygon. The corresponding fact for motives is a well known result of Mazur and Fontaine. We have said that the above fact is a special case of Theorem 8.1, because we actually prove a precise result without assuming the splitting of p in F/ℚ (see also Remark 8.1).

Since automorphic forms are invariant under the left translation by rational elements in G(ℚ), h^{n.ord} is actually an algebra over Λ = ℤ_{p}[[T (𝔯_{p})/\overline{𝔯^{×}}]] for the integer ring 𝔯 of F. If we admit the Leopoldt conjecture for F and p, we see that

dim(Λ ⊗_{ℤ_{p}} ℚ_{p}) = (n − 1)[F : ℚ] + r_{2} + 1.

Thus dim(Λ ⊗_{ℤ_{p}} ℚ_{p}) is possibly equal to dim(h^{n.ord} ⊗_{ℤ_{p}} ℚ_{p}) if and only if one of the following conditions holds:

(i) F is totally real and n = 2; (ii) n = 1.

Admitting the conjecture, we see

dim(Λ ⊗_{ℤ_{p}} ℚ_{p}) − dim(h^{n.ord} ⊗_{ℤ_{p}} ℚ_{p}) = \begin{cases}(m − 1)[F : ℚ] + r_{2} & if n = 2m, \\ m[F : ℚ] & if n = 2m + 1. \end{cases} (∗)

We write Δ for the right-hand side of the above formula. The left-hand side of (∗) measures the non-abelian (or non-Leopoldt) part of the above conjecture. It seems plausible that if Δ ≤ 1, one can prove (∗) without assuming the Leopoldt conjecture. This has been basically done in [H88] and [H89] when Δ = 0, that is, the two cases (i) and (ii). It is remarkable that the non-abelian part of the conjecture is proven in Case (i) without assuming any particular condition on the structure of the Galois group of F over ℚ, because only known cases of the Leopoldt conjecture assume that F is abelian either over ℚ or an imaginary quadratic field. This is one of the reason why we believe that the above conjecture (or at least its non-abelian part) might be more accessible than the original Leopoldt conjecture. Thus the next target of our investigation would be the case where Δ = 1. We see that Δ = 1 if and only if one of the following conditions is satisfied:

(a) F has only one complex place and n = 2;

(b) 3 ≤ n ≤ 4 and F = ℚ. The formula (∗) has been proven in [H94b] Sections 5 and 6 for all inner forms of GL(2) in Case (a). We shall prove in this paper the conjecture for some inner forms of GL(3)_{/ℚ} and GL(4)_{/ℚ} (see Corollary 6.3 and the explanation after Conjecture 7.1).

The two key ingredients of the proof of the control theorem (Theorem 6.5) are strong multiplicity one theorem valid for cuspidal cohomology on GL(n) and the semi-simplicity (Corollary 8.3) of the (cuspidal) nearly ordinary Hecke algebra. The proof of the semi-simplicity can be generalized to interior cohomology on any reductive group split at p (Corollary A.4). We prove the semi-simplicity in Sections 5 and 8 for GL(n) and in Appendix A for general reductive groups. Since this paper is based on the results obtained in [H95], we have added in Appendix B a list of corrections to [H95].

1.1. Notation. Here is general notation we will use throughout the paper. First of all, we keep the notation introduced in Section 1 throughout the paper. For two algebraic groups G ⊃ H and a polynomial representation ρ of H (that is, a morphism of algebraic groups from H to GL(d) for some d > 0), Ind_{H}^{G} ρ indicates the induced representation in the category of polynomial representations of algebraic groups. Thus, for a representation space V of a polynomial representation ρ : H → GL(V),

Ind_{H}^{G}(ρ) = {f : G → V: polynomial|f(gh) = ρ(h^{−1})f(g) for h ∈ H} .

Then the action of G is given by gf(g′) = f(g^{−1}g′). For compact p–adic groups G ⊃ H and a continuous representation ρ of H on a topological module V, ind_{H}^{G}ρ is the representation of G on

ind_{H}^{G}V = {f : G → V: continuous|f(gh) = ρ(h^{ι})f(g) for h ∈ H} ,

where ι is a suitable involution of G (usually h^{ι} = h^{−1}) specified in the text. Then the action of G is given by gf(g′) = f(g^{ι}g′). For locally compact p–adic groups G ⊃ H and a smooth representation ρ of H acting from the right on a vector space V over a field of characteristic 0 with the discrete topology, Ind_{H}^{G}ρ is the right G–module of locally constant functions f : G → V compactly supported modulo H satisfying f(gh) = f(g)ρ(h) for h ∈ H. The action of G on such functions f is given by f(g′)g = f(gg′).

For an n_{i} × n_{i} matrix A_{i} for i = 1, …, r, we write diag(A_{1}, …, A_{r}) for the n × n matrix (n = n_{1} + ⋯ + n_{r}) whose i–th diagonal block is A_{i} (i = 1, …, r) and all the other off-diagonal blocks are zero. To a partition n = n_{1} + ⋯ + n_{r}, we can associate a standard parabolic subgroup P of GL(n) generated by M_{P} = {diag(A_{1}, …, A_{r})|A_{i} ∈ GL(n_{i})} and upper unipotent matrices. The notion of near-ordinarity depends on a choice of a conjugacy class of a proper parabolic subgroup. In other words, for a fixed standard parabolic subgroup P, we define the nearly ordinary part to be the maximal quotient (or equivalently a maximal subspace) on which the Hecke operator associated to elements in the center of M_{P} is invertible (see 6.2 for the precise definition). Thus for a given open compact subgroup S, we can have different nearly ordinary cohomology groups depending on the choice of the standard parabolic subgroup. If it is necessary to emphasize the dependence on P, we write H_{P −n.ord}^{q} for the nearly p–ordinary part with respect to the parabolic subgroup P. When no confusion is likely, we drop “P ” from the subscript. There is one exception: When P = B, the standard Borel subgroup, we write H_{n.ord}^{q} for H_{B−n.ord}^{q} all the time. Then for the standard parabolic subgroup P, the standard Levi component is given by M_{P} as above. Then P = M_{P} N_{P} for the unipotent radical N_{P} of P. We write M^{∘} for the derived group of M_{P}. Thus M^{∘} = SL(n_{1}) × ⋯ × SL(n_{r}). We define a torus T_{P} by P ∩ SL(n)/M^{∘}N_{P}, which is isomorphic to GL(1)^{r−1} via diag(x_{1}, …, x_{r−1}, x_{r}) ↦ (det(x_{1}), …, det(x_{r−1})). For each algebraic group H, we write Z(H) for its center. In particular, we write T_{M} = Z(M). By the above determinant map, we have T_{P} ≅ T_{M}^{∘} = SL(n) ∩ T_{M}, and often we identify the two tori.

For the adele ring 𝔸 of ℚ and a finite set Σ of places of ℚ, we write 𝔸^{(Σ)} = {x ∈ 𝔸|x_{v} = 1, ∀v ∈ Σ}; in particular 𝔸^{(p∞)} = {x ∈ 𝔸|x_{p} = x_{∞} = 1}.

Contents

1. Introduction 1 1.1. Notation

2. Preliminaries 6

3. Cohomological automorphic representations for GL(n) 7 3.1. Rational representations of GL(n) 3.2. Infinity type of cohomological representations of GL_{n}(ℝ) 3.3. Infinity type of cohomological representations of GL_{n}(ℂ)

4. Local Hecke algebras 10 4.1. Hecke algebras for parabolic subgroups 4.2. Hecke algebras of local congruence subgroups

5. Jacquet modules and nearly p–ordinary part 12 5.1. Jacquet modules for GL(n) 5.2. Jacquet modules and Hecke operators

6. p–Adic Hecke algebras 15 6.1. Hecke algebra as a double coset algebra 6.2. Nearly p–ordinary cohomology groups 6.3. Universal nearly p–ordinary Hecke algebras

7. A conjecture on Krull dimension of h^{n.ord} 28 7.1. Statement of the conjecture 7.2. Compatibility with automorphic functoriality

8. Ordinarity condition via Newton polygon 35 8.1. Newton polygon and Hodge polygon 8.2. Newton polygon is above Hodge polygon

Appendix A. Semi-simplicity of Hecke algebras for reductive groups 39 A.1. Jacquet modules for reductive groups A.2. Double coset algebras A.3. Rational representations of G A.4. Nearly p–ordinary representations A.5. Semi-simplicity of interior cohomology groups

Appendix B. Correction to [H95] 46

References 47

2. Preliminaries

Let F be a number field, that is, a finite extension of ℚ. Let G_{/ℚ} be a reductive algebraic group satisfying the condition ι : G(ℚ_{p}) ≅ GL_{n}(F_{p}) for F_{p} = F ⊗_{ℚ} ℚ_{p}. We fix an isomorphism ι such that it induces for the derived group G^{∘} of G

ι : G^{∘}(ℚ_{p}) ≅ SL_{n}(F_{p}). (SL(p))

From time to time, we write G^{∘}(ℤ_{p}) for ι^{−1}(SL_{n}(𝔯_{p})), where 𝔯_{p} = 𝔯 ⊗_{ℤ} ℤ_{p}. We also suppose, throughout the paper, the strong approximation theorem:

(SA) G^{∘}(ℚ) is dense in G^{∘}(𝔸^{(∞)}).

In Section 6, we will prove the control theorem for the nearly p–ordinary cohomology group and the associated Hecke algebra. The proof is divided into two steps: The first step is the control theory of cohomology groups of arithmetic subgroups of G^{∘}(ℚ) which has been basically taken care of in [H95]. Thus we will resort to a sort of induction process from G^{∘} to G in Section 6 in order to prove the control theorem for the nearly p–ordinary sheaf cohomology groups on modular varieties of G.

Let us explain briefly why we think important this problem of controlling Hecke algebras and cohomology groups. Let Φ be a discrete subgroup of G^{∘}(ℚ) ∩ G^{∘}(ℤ_{p}) dense in G^{∘}(ℤ_{p}). Then for a (ℤ_{p}, Φ)–module L, we study various cohomology groups for Φ, for example, group cohomology H^{q}(Φ, L). In particular, we would like to study their p–adic behavior when one shrinks Φ p–adically along a parabolic subgroup P of GL(n)_{/ℤ}. For example, we take the Borel subgroup B in this preliminary section as P. Shrinking Φ along B means that we take

Φ_{1}(p^{α}) = {γ ∈ Φ|ι(γ) mod p^{α} ∈ N_{B}(𝔯/p^{α}𝔯)}

for the unipotent radical N_{B} of B. We study H^{q}(Φ_{1}(p^{α}), L) and its limit

H^{q}(Φ_{1}(p^{∞}), L) = \underrightarrow{lim}_{α}H^{q}(Φ_{1}(p^{α}), L) and Ĥ^{q}(Φ_{1}(p^{∞}), L) = \underleftarrow{lim}_{α}H^{q}(Φ_{1}(p^{α}), L).

Here the limit is taken with respect to the restriction maps and transfer maps.

We also look at the Γ_{0}–type groups:

Φ_{0}(p^{α}) = {γ ∈ Φ|ι(γ) mod p^{α} ∈ B(𝔯/p^{α}𝔯)} .

Since the maximal split torus T^{∘} in SL(n) normalizes N_{B} and Φ_{1}(p^{α}), we have a natural action of T^{∘}(𝔯_{p}) (𝔯_{p} = r ⊗_{ℤ} ℤ_{p}) on the above cohomology groups. We want to study the T^{∘}(𝔯_{p})–module structure of the cohomology groups.

Why is it important to study the limit cohomology groups and the T^{∘}(𝔯_{p})– module structure? We answer this question by an example: If L is a rational representation of G^{∘}, then we can consider modules L(A) with coefficients in A for various rings, for example ℂ. It is well known after works of Eichler and Shimura on elliptic modular forms, the cohomology groups with coefficients in L(ℂ) is directly related to some specific modular forms (called cohomological modular forms) on G^{∘}. Therefore, by cohomological functoriality, on can put a rational or integral structure on the space of modular forms in question. Moreover, for a p–adic ring O, if one can isolate a part of Ĥ^{q}(Φ_{1}(p^{∞}), L(O)) or H^{q}(Φ_{1}(p^{∞}), L(O) ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}) which is of finite or co-finite type over the completed group ring O[[T^{∘}(𝔯_{p})]], it gives a p–adic deformation (over an explicitly given Spec(O[[T^{∘}(𝔯_{p})]])) of cohomological modular forms through cohomological functoriality. Here we say a module is of co-finite type if its Pontryagin dual is of finite type. In the p–adic situation, the spectrum of p–adic automorphic representations is always continuous. Therefore it is natural to study the limit alongside the individual H^{q}(Φ_{1}(p^{α}), L) of finite level. Through the action of T^{∘}(𝔯_{p}), we may be able to describe the spectrum explicitly. This is a principal reason to study the limit and the action.

There is a general and simple method to isolate such components of finite type over O[[T^{∘}(𝔯_{p})]]. Since G^{∘} ⊂ G, we have Hecke operators acting on cohomology groups which preserve integrality. Let 𝕋 be one of such operators. If L or its Pontryagin dual is of finite type over ℤ_{p}, we have a unique projector e_{𝕋} acting on H^{q}(Φ_{1}(p^{α}), L) such that 𝕋 is an automorphism on e_{𝕋}H^{q}(Φ_{1}(p^{α}), L) and is p– adically nilpotent on (1 − e_{𝕋})H^{q}(Φ_{1}(p^{α}), L) (see [H93] 1.11). Here if L is discrete of co-finite type (that is, its Pontryagin dual is of finite type over ℤ_{p}), we say that 𝕋 is p–adically nilpotent if

(1 − e_{𝕋})H^{q}(Φ_{1}(p^{α}), L) = ⋃_{j≥1}(1 − e_{𝕋})H^{q}(Φ_{1}(p^{α}), L)[𝕋^{j}],

where the bracket “[𝕋^{j}]” indicates the kernel of the operator 𝕋^{j}. More generally, if L is an injective limit of discrete (ℤ_{p}, Φ)–modules of co-finite type, we have well defined projector e_{𝕋}. We call a Φ–module L admissible if L is an injective limit of discrete (ℤ_{p}, Φ)–modules of co-finite type. We will see, for 𝕋 given by Hecke operator at p (associated to our choice of parabolic subgroup P), the image of e_{𝕋} has the required property.

Let L^{∗} denote the Pontryagin dual of L. Then we have by Poincaré duality (see [H93] 1.9), if Φ is co-compact,

H^{q}(Φ_{1}(p^{α}), L)^{∗} ≅ H^{d−q}(Φ_{1}(p^{α}), L^{∗}) and H^{q}(Φ_{1}(p^{∞}), L)^{∗} ≅ Ĥ^{d−q}(Φ_{1}(p^{∞}), L^{∗}),

where d is the (real) dimension of the symmetric space of G^{∘}(ℝ). Therefore, hereafter, we may assume that L is a p–divisible admissible module.

3. Cohomological automorphic representations for GL(n)

In this section, we describe cohomological tempered (modulo center) representations of GL_{n}(ℝ) and GL_{n}(ℂ) following Clozel [Cl] Section 3. This is useful in relating cohomology groups to modular forms.

3.1. Rational representations of GL(n). Let T be the standard diagonal torus of G = GL(n). Let X(T) = Hom_{alg.gp}(T, G_{m}). The standard base of the root system with respect to the upper triangular Borel subgroup B is explicitly given by {α_{i}} for α_{i}(diag(t_{1}, t_{2}, …, t_{n})) = t_{i}t_{i+1}^{−1} for i = 1, …, n − 1. Then the coroot α̌_{i} of α_{i} is explicitly given by

α̌_{i}(t) = diag(1, …, 1, t^{i}, t^{−1}, 1, …, 1).

The fundamental dominant weights {ω_{i}} which form the dual base of {α̌_{i}} are given by

ω_{i}(diag(t_{1}, t_{2}, …, t_{n})) = t_{1} · t_{2} ⋯ t_{i} (1 ≤ i ≤ n − 1).

We simply write ω_{n}(x) = det(x) for x ∈ T. Thus we may identify X(T) with ℤ^{n} so that α = (m_{1}, m_{2}, …, m_{n}) ∈ ℤ^{n} gives the character

α(diag(t_{1}, t_{2}, …, t_{n})) = t_{1}^{m_{1}} · t_{2}^{m_{2}} ⋯ t_{n}^{m_{n}}.

Then the cone of dominant weights in X(T) is given by

C_{n} = {(m_{1}, m_{2}, …, m_{n}) ∈ ℤ^{n}|m_{1} ≥ m_{2} ≥ ⋯ ≥ m_{n}} .

Now we look at the standard parabolic subgroup P of GL(n) corresponding to the partition n = n_{1} + n_{2} + ⋯ + n_{r} of n into r positive integers (r > 0). We write n_{i}^{∗} = n_{1} + n_{2} + ⋯ + n_{i}. We may regard T as a maximal split torus of M = M_{P}. Then the center Z(M) of M is a subtorus of T. The set of dominant weights of M = M_{P} is given by

C(M) = C_{n_{1}} × ⋯ × C_{n_{r}} in ℤ^{n} = X(T).

For each χ ∈ C(M), we have an irreducible representation ρ_{χ} = Ind_{B}^{P} χ^{−1} of M rational over F having χ^{−1} as a highest weight (with respect to the reverse ordering making −C_{n} positive), which we regard as a representation of P through the projection pr : P → M. By definition, C(M) ⊃ C_{n}, and Ind_{P}^{G} ρ_{χ} ≠ 0 if and only if χ ∈ C_{n}. It is known that Ind_{P}^{G} ρ_{χ} ≅ Ind_{B}^{G} χ^{−1} if χ ∈ C_{n}, which is absolutely irreducible and is rational over F. For each ℚ–algebra A, we write L(χ; A) for the space of Ind_{B}^{G} χ^{−1} over A.

3.2. Infinity type of cohomological representations of GL_{n}(ℝ). Here we follow description of cohomological admissible representations of GL_{n}(ℝ) given in [Cl] Section 3 (see also [H95] Section 8). For a given dominant weight χ ∈ X(T), there is at most one irreducible representation π = π(χ) of GL_{n}(ℝ) such that (i) its restriction to SL_{n}(ℝ) is tempered and

H^{q}(𝔤, O_{n}(ℝ); H(π) ⊗ L(χ; ℂ)) ≠ 0, (ii)

where H(π) is the space of smooth vectors of π, and 𝔤 is the complexified Lie algebra of GL_{n}(ℝ). We call such π cohomological (or χ–cohomological).

In [Cl] Section 3, actually the classification of cohomological π′ with non-vanishing:

H^{q}(𝔤, O_{n}(ℝ); H(π′) ⊗ L̃(χ; ℂ)) ≠ 0

is given for the contragredient L̃(χ; ℂ) of L(χ; ℂ). Since

H^{q}(𝔤, O_{n}(ℝ); H(π′) ⊗ L̃(χ; ℂ)) ≅ H^{q}(𝔤, O_{n}(ℝ); H(π′) ⊗ L(χ; ℂ))

(cf. [BW] p.43), the contragredient π′ of π′ associated to L̃(χ; ℂ) is χ–cohomological. Since π is classified in [Cl] using representation of the Weil group at ∞, what we need to keep in mind is that a representation ρ of the Weil group W_{ℂ/ℝ} corresponds to π with non-vanishing H^{q}(𝔤, O_{n}(ℝ); H(π) ⊗ L̃(χ; ℂ)) if and only if ρ^{−1} corresponds to π(χ). Thus what we need to do is to replace ρ(z) in [Cl] by ρ(z^{−1}) for z ∈ ℂ^{×} ⊂ W_{ℂ/ℝ}. In this sense, our classification looks a bit different from the one given in [Cl].

Now regard χ as an element of ℤ^{n}. Then π is classified by a representation ρ of the Weil group W_{ℂ/ℝ} into GL_{n}(ℂ) (and a character χ ∈ X(T)). We have that H^{q}(𝔤, O_{n}(ℝ); H(π ⊗ ε) ⊗ L(χ; ℂ)) ≠ 0 for a character ε of ℝ^{×} with ε^{2} = 1 if and only if for z ∈ ℂ^{×} ⊂ W_{ℂ/ℝ}, writing (see [Cl] p.114-119)

ρ(z^{−1})

= \begin{cases}diag(z^{a_{1}} \overline{z}^{a_{2}}, z^{a_{2}} \overline{z}^{a_{1}}, …, z^{a_{2m−1}} \overline{z}^{a_{2m}}, z^{a_{2m}} \overline{z}^{a_{2m−1}}) & if n = 2m, \\ diag(z^{a_{1}} \overline{z}^{a_{2}}, z^{a_{2}} \overline{z}^{a_{1}}, …, z^{a_{2m−1}} \overline{z}^{a_{2m}}, z^{a_{2m}} \overline{z}^{a_{2m−1}}, (z\overline{z})^{a_{2m+1}}) & if n = 2m + 1, \end{cases}

we have

\begin{matrix}a_{2} > a_{4} > ⋯ > a_{2m} > a_{2m−1} > ⋯ > a_{3} > a_{1} if n = 2m, \\ a_{2} > a_{4} > ⋯ > a_{2m} > a_{2m+1} > a_{2m−1} > ⋯ > a_{3} > a_{1} if n = 2m + 1, \end{matrix} (3.1)

w = a_{2i−1} + a_{2i} (i = 1, …, m) and 2a_{2m+1} (if n = 2m + 1) are independent of i, (3.2)

χ = \begin{cases}(a_{2}, a_{4}, …, a_{2m}, a_{2m−1}, …, a_{1}) + 1 − δ & if n = 2m, \\ (a_{2}, a_{4}, …, a_{2m}, a_{2m+1}, a_{2m−1}, …, a_{3}, , a_{1}) + 1 − δ & if n = 2m + 1, \end{cases} (3.3)

where δ = ω_{1} + ω_{2} + ⋯ + ω_{n} = (n, n − 1, …, 1) and 1 = (1, 1, …, 1).

By [Cl] Lemma 3.14, we have

H^{q}(𝔤, O_{n}(ℝ); H(π(χ)) ⊗ L(χ; ℂ)) ≅ \begin{cases}⋀^{q−m^{2}} ℂ^{m−1} & if n = 2m \\ ⋀^{q−m(m+1)} ℂ^{m} & if n = 2m + 1. \end{cases} (3.4)

Since H^{q}(𝔤, SO_{n}(ℝ); H(π(χ)) ⊗ L(χ; ℂ)) really contributes to H^{q}(Φ, L(χ; ℂ)) for a discrete subgroup Φ of the identity connected component G(ℝ)_{+} of G(ℝ), we deduce from (3.4) combined with [BW] I.5.1 that

(3.5) H^{q}(𝔤, SO_{n}(ℝ); H(π(χ)) ⊗ L(χ; ℂ))

≅ \begin{cases}(⋀^{q−m^{2}} ℂ^{m−1}) ⊗_{ℂ} ℂ [O_{n}(ℝ)/SO_{n}(ℝ)] & if n = 2m \\ (⋀^{q−m(m+1)} ℂ^{m}) ⊗_{ℂ} ℂ [O_{n}(ℝ)/SO_{n}(ℝ)] & if n = 2m + 1. \end{cases}

3.3. Infinity type of cohomological representations of GL_{n}(ℂ). Let G = Res_{ℂ/ℝ}GL(n). Then G(ℝ) = GL_{n}(ℂ), and every irreducible rational representation of G is isomorphic to τ ⊗ (τ′ ∘ c) for a pair (τ, τ′) of rational representations of GL(n)_{/ℂ} for complex conjugation c. Thus x ∈ G(ℝ) acts via τ(x) ⊗ τ′(\overline{x}) for complex conjugation “−”. Thus we can view τ ⊗ (τ′ ∘ c) as an element χ of C_{n} × C_{n}. On the other hand, an admissible irreducible representation π is classified by representations ρ of the Weil group W_{ℂ/ℂ} = ℂ^{×}. Following [Cl] p.112-3, we have, for admissible irreducible π with tempered restriction to SL_{n}(ℂ), H^{q}(𝔤, U_{n}(ℝ); H(π)⊗ L(χ; ℂ)) ≠ 0 if and only if for z ∈ ℂ^{×} = W_{ℂ/ℂ}

ρ_{π}(z^{−1}) = diag(z^{p_{1}} \overline{z}^{q_{1}}, z^{p_{2}} \overline{z}^{q_{2}}, …, z^{p_{n}} \overline{z}^{q_{n}}); (3.6) (3.7) p_{i} > p_{i+1} and q_{i} < q_{i+1} for all i = 0, …, n − 1; (3.8) w = p_{i} + q_{i} is independent of i; χ = (p − δ + 1,^{w}q − δ + 1), (3.9)

where the involution “w” indicates the conjugation by the longest element in the Weyl group of T: ^{w}(a_{1}, …, a_{n}) = (a_{n}, a_{n−1}, …, a_{1}).

Again by [Cl] Lemma 3.14, we have

H^{q}(𝔤, U_{n}(ℝ); H(π(χ)) ⊗ L(χ; ℂ)) ≅ ⋀^{q− \frac{n(n−1)}{2}} ℂ^{n−1}. (3.10)

4. Local Hecke algebras

In this section, we reformulate some results in [H95] Section 2 on the structure of local Hecke algebras made of double cosets of Γ_{0}-type open compact subgroups (with respect to a standard parabolic subgroup) of GL(n). We refer to [H95] for the proof of the results stated in this section.

4.1. Hecke algebras for parabolic subgroups. Let P be a proper standard parabolic subgroup of GL(n) associated with the partition n = n_{1} + n_{2} + ⋯ + n_{r}. Let V be a discrete valuation ring finite flat over ℤ_{ℓ} for a rational prime ℓ. Let ϖ be a prime element of V and we write 𝔪 = Vϖ and k = V/𝔪. We write v for the valuation with v(ϖ) = 1.

Let

(4.1) D = D_{P} = D_{P} (V)

= {diag(a_{1}1_{n_{1}}, …, a_{r}1_{n_{r}}) ∈ M_{n}(V)|V ⊃ a_{1}V ⊃ a_{2}V ⊃ ⋯ ⊃ a_{r}V ≠ 0} .

In this section, we often write P for P (V). We then consider Δ_{∞} = P DP = P (V)DP (V) ⊂ GL_{n}(F) for the field F of fractions of V. Then it is easy to see

(4.2) Δ_{∞} = P DP is a semi-group.

Let N = N(V) be the unipotent radical of P. For a subgroup X of Δ_{∞} with P ▷ X ⊃ N, let R(X; Δ_{∞}) be the space of compactly supported X–bi-invariant functions (with values in ℤ) compactly supported on Δ_{∞}. The space R(X; Δ_{∞}) has a natural structure of algebra under the convolution product with respect to the Haar measure μ with μ(X) = 1. We write XxX ∈ R(X; Δ_{∞}) for the characteristic function of the set given by the double coset XxX. For each s > 0, we put

ξ_{s} = ( \begin{matrix}1_{n−s} & 0 \\ 0 & ϖ1_{s} \end{matrix} ) ∈ Δ_{∞} and let s_{j} = n_{r−j+1} + n_{r−j+2} + ⋯ + n_{r} for j = 1, …, r.

Then we write T_{s_{j}}(ϖ) for Xξ_{s_{j}}X.

Proposition 4.1. We have the following equality in R(X; Δ_{∞}):

XξXζX = XξζX = XζξX = XζXξX and XξX · XζX = XξζX (4.3)

for ξ, ζ ∈ D, and the ring R(P; Δ_{∞}) is isomorphic to the polynomial ring ℤ[T_{1}, …, T_{r}] via T_{s_{j}}(ϖ) ↦ T_{j}.

This is proven in [H95] Section 2. A key to the proof is the following explicit coset decomposition of XξX of ξ ∈ D = D_{P}: Decompose X = ⨆_{η∈Ξ}(ξ^{−1}Xξ ∩ X)η. Then multiplying by ξ^{−1}Xξ from the left, we get

ξ^{−1}XξX = ⨆_{η∈Ξ} ξ^{−1}Xξη ⟺ XξX = ⨆_{η∈Ξ} Xξη = ⨆_{η∈ξN_{P} (V)ξ^{−1}/N_{P} (V)} Xηξ. (4.4)

Write ξ = diag(a_{1}1_{n_{1}}, a_{2}1_{n_{2}}, …, a_{r}1_{n_{r}}). To describe the group ξ^{−1}Xξ ∩ X, we write an n × n matrix A as (A_{ij}) for n_{i} × n_{j} blocks A_{ij}. Then we see

ξ^{−1}Xξ ∩ X = {g = (g_{ij}) ∈ X|g_{ij} ∈ a_{i}^{−1}a_{j}M_{n_{i}×n_{j}}(V) for all j > i} .

Thus we may choose the representative set Ξ to be the set of unipotent matrices in N_{P} (V) such that

Ξ ∋ η ⟼ (η_{ij} mod a_{i}^{−1}a_{j}V)_{j>i} ∈ ⨁_{j>i} M_{n_{i}×n_{j}}(V/a_{i}^{−1}a_{j}V) (4.5)

is an isomorphism.

Let M be the standard Levi-subgroup of P. Then

M(A) = {diag(x_{i})|x_{i} ∈ GL_{n_{i}}(A)}.

We write π : P → M for the natural projection, and we put

P^{∘} = {x ∈ P |π(x) = diag(x_{i}) with x_{i} ∈ SL_{n_{i}}(A) for all i} .

Then P/P^{∘} ≅ T_{M} via det : M → T_{M} given by det(diag(x_{i})) = diag(det(x_{i})1_{n_{i}}).

Corollary 4.2. We have an algebra isomorphism:

R(P^{∘}(V); Δ_{∞}) ≅ ℤ[T_{M}(V)][T_{1}, …, T_{r}]

given by T_{s_{j}}(ϖ) ↦ T_{j} and P^{∘}(V)uP^{∘}(V) ↦ [det(π(u))] for u ∈ P (V), where [t] is the group element t in the group algebra ℤ[T_{M}(V)].

Suppose that P is the standard Borel subgroup B. Then T ≅ B/N, and we may regard a characters λ = (λ_{1}, …, λ_{n}) : T (F) → K^{×} as a character of B. Thus λ(diag(a_{1}, …, a_{n})) = ∏_{j} λ_{j}(a_{j}). Let V be a right K[B(F)]–module with λ– eigensubspace V [λ]. Then we can let T_{s}(ϖ) = Nξ_{s}N = ⨆_{i} Nx_{i} act on eigenvectors v ∈ V [λ] by v|T_{s}(ϖ)(x) = ∑_{i} vx_{i}, and we have

v|T_{s}(ϖ) = (|ϖ|_{v}^{−ts} ∏_{j=t+1}^{n} λ_{j}(ϖ)) v, (4.6)

where t = n − s and |x|_{v} = |V/𝔪|^{−v(x)} is the normalized absolute value of V.

4.2. Hecke algebras of local congruence subgroups. We keep the notation introduced in the previous section. We consider the following subgroups:

\begin{matrix}I_{α} = I_{P,α} = \{g ∈ GL_{n}(V)|g mod ϖ^{α} ∈ P (V/ϖ^{α})\} \\ I_{α}^{∘} = I_{P,α}^{∘} = \{g ∈ GL_{n}(V)|g mod ϖ^{α} ∈ P^{∘}(V/ϖ^{α})\} . \end{matrix} (4.7)

Let C be an open compact subgroup of GL_{n}(V) such that

(P) C contains I_{α}^{∘} and C is contained in I_{α}.

Put Δ_{C} = I_{α}D_{P} I_{α}. Then we have

Proposition 4.3. Suppose (P) . Then we have

(4.8) Δ_{C} is a semi-group; CξC · CζC = CξζC = CζC · CξC for ξ, ζ ∈ D; (4.9) R(C; Δ_{C}) ≅ ℤ[I_{α}/C][T_{1}, T_{2}, …, T_{r}] by T_{s_{j}}(ϖ) ↦ T_{j}, (4.10)

where T_{s_{j}}(ϖ) = Cξ_{s_{j}}C.

5. Jacquet modules and nearly p–ordinary part

We recall the definition of Jacquet modules of admissible representations of GL(n) and study its relation to nearly p–ordinary vectors. The result obtained here will be used to show semi-simplicity of the cohomology groups over their Hecke algebras.

5.1. Jacquet modules for GL(n). Let F be a finite extension of ℚ_{ℓ} with ℓ–adic integer ring V. Let π be an admissible irreducible representation of G = GL_{n}(F) over a field K of characteristic 0. We write V for the representation space of π. Since we mainly deal with cohomology group with right GL_{n}(F)–action, here against usual convention, we suppose that V is a right GL_{n}(F)–module. For a standard parabolic subgroup P of GL(n)_{/V} with unipotent radical N = N_{P}, the Jacquet module V_{P} of V with respect to P is the N(F)–co-invariant space of V, that is, V_{P} = V/V (P) for the subspace V (P) of V generated by

{vπ(n) − v|n ∈ N(F) and v ∈ V } .

Since V (P) is stable under P, for the standard Levi subgroup M of P, M(F) naturally acts on V_{P}. It is known that

(1) V_{P} gives an admissible representation of M(F) if V is admissible (a theorem of Jacquet [BZ] 3.14);

(2) The functor V ↦ V_{P} is exact ([BZ] 2.35);

(3) Let ξ = diag(1_{n_{1}}, ϖ1_{n_{2}}, ϖ^{2}1_{n_{3}}, … ϖ^{r−1}1_{n_{r}}) ∈ D_{P}. Then v ∈ V (P) ⟺ there exists m > 0 such that ∫_{ξ^{M} N(V)ξ^{−M}} vπ(n)dn = 0 for all M ≥ m for a Haar measure dn of N(F) ([BZ] 2.33);

(4) V is super-cuspidal if and only if V (gN_{P} g^{−1}) = V for all g ∈ G and all proper parabolic subgroups P ([BZ] 3.18).

Since M^{∘}(F)T_{M}(F) is of finite index in M(F) for the derived group M^{∘} of M, V_{P} is an admissible representation of M^{∘}(F). Since N(V) is a compact group, for any admissible N(V)–module X, we have a projector

X → X^{N(V)} = H^{0}(N(V), X) given by x ↦ ∫_{N(V)} xπ(n)dn

for the Haar measure “dn” normalized so that ∫_{N(V)} dn = 1. Thus we have a natural exact sequence

0 → V (P)^{N(V)} → V^{N(V)} → V_{P}^{N(V)} → 0. (5.1)

Since each component of the above exact sequence is stable under N(V), it is an exact sequence of R(N; Δ_{∞})–modules.

5.2. Jacquet modules and Hecke operators. We write 𝕋 = 𝕋_{P} for the Hecke operator corresponding to P ξP, where

ξ = diag(1_{n_{1}}, ϖ1_{n_{2}}, ϖ^{2}1_{n_{3}}, … ϖ^{r−1}1_{n_{r}}) ∈ D_{P} .

Then we see from (4.4) and (4.5) that

v|𝕋_{P}^{m} = (∫_{ξ^{m}N(V)ξ^{−m}} xπ(n)dn) π(ξ^{m}) for v ∈ V^{N(V)}. (5.2)

Since N_{P} (F) = ⋃_{m≥1} ξ^{m}N_{P} (V)ξ^{−m}, 𝕋_{P} is nilpotent on V (P)^{N(V)}.

For any open compact subgroup S with N(V) ⊂ S ⊂ I_{P,1}, by (4.5), SξS = ⨆_{x∈Ξ} Sξx if N(V)ξN(V) = ⨆_{x∈Ξ} N(V)ξx. Thus the action of 𝕋_{P} on V^{S} induces the action of the double coset SξS, and the finite dimensional space V^{S} is stable under 𝕋_{P}. Then we can decompose V^{S} = V_{nil}^{S} ⊕ V_{ss}^{S} so that 𝕋_{P} is nilpotent on V_{nil}^{S} and invertible on V_{ss}^{S}. Since V^{N(V)} = ⋃_{S} V^{S} for S running over all open subgroups of I_{P,1} containing N(V), we have the decomposition:

(5.3) V^{N(V)} = V_{P}^{nil} ⊕ V_{P}^{∘}

such that V_{P}^{nil} = ⋃_{S} V_{nil}^{S} = V^{N(V)} ∩ V (P) and V_{P}^{∘} = ⋃_{S} V_{ss}^{S},

where 𝕋_{P} is invertible on V_{P}^{∘} and nilpotent on V_{P}^{nil}.

Proposition 5.1. Let the notation be as above. Let V be an admissible representation of G(F) over a field K of characteristic 0. Then V_{P}^{∘} ≅ V_{P} as R(P (V), Δ_{∞})– modules.

Proof. By the above construction (5.3), the functor V ↦ V_{P}^{∘} is an exact functor, and hence, by taking semi-simplification if necessary, we may assume that V_{P}^{∘} and V_{P} are both semi-simple R(N; Δ_{∞})–modules. By the above splitting, the projection V → V_{P} induces an inclusion V_{P}^{∘} ↪ V_{P} of R(N; Δ_{∞})–modules. Since an algebraic closure \overline{K} of K is faithfully flat over K, the result for V ⊗_{K} \overline{K} implies that of V. Thus we may assume that K is an algebraically closed field. Since ξ ∈ T_{M}(F) = Z(M), the Hecke operator 𝕋_{P} is invertible on the admissible P (F)-module V_{P}. Thus we need to show that any \overline{v} ∈ V_{P} can be lifted to v ∈ V_{P}^{∘}. Pick v ∈ V such that 0 ≠ \overline{v} = (v mod V (P)) ∈ V_{P}. We may assume that \overline{v} is an eigenvector of T_{M}(F)–action, because we have assumed that V_{P} is semi-simple M(F)–module and K is algebraically closed. By a simple computation, if v is fixed by an open subgroup of N(V), vπ(ξ^{−α}) (≡ c\overline{v} mod V (P) with c ≠ 0) is fixed by N(V) for sufficiently large α. Therefore, we may assume that v ∈ V^{N(V)}. Then v|𝕋_{P}^{β} for sufficiently large β gets into V_{P}^{∘}. Then \overline{v} is the image of (v|𝕋_{P}^{β})|𝕋_{P}^{−β} in V_{P}^{∘}, where we have taken the inverse 𝕋_{P}^{−β} inside V_{P}^{∘} on which 𝕋_{P} is invertible. This shows the desired surjectivity.

By the above proposition, we get the following isomorphism of R(P; Δ_{∞})–modules:

V^{N_{P} (V)} = V_{P}^{∘} ⊕ V_{P}^{nil} ≅ V_{P}^{N_{P} (V)} ⊕ V (P)^{N_{P} (V)}. (5.4)

When P = B, then M = T_{M} = T is commutative, and hence, V_{B} is finite dimensional. We quote the following theorem of Kazhdan [BZ] 5.21 and 7.12:

Proposition 5.2. Suppose that either V is absolutely irreducible or is a subquotient of Ind_{B}^{V}λ for a continuous character λ : T (F) → K^{×}. Then we have

dim_{K} V_{B}^{∘} ≤ dim_{K} V_{B} ≤ n! for N = N_{B}.

Proposition 5.3. Suppose that π ⊂ Ind_{P}^{G}ρ for an admissible irreducible representation ρ of M_{P} (F). Let B be the standard Borel subgroup of GL(n). Then the operator 𝕋_{B} defined above for the Borel subgroup B is nilpotent if ρ is super-cuspidal and P ≠ B.

Proof. We suppose the contrary to the assertion: V_{B}^{∘} ≅ V_{B} is non-trivial. Then by [BZ] 3.19, 3.27 and 3.13 (or more precisely, [BW] XI.2), there exists a continuous character λ : T (F) → K^{×} of T (F) such that

(5.5) π can be embedded isomorphically into Ind_{B}^{G}λ.

Thus π cannot be a factor of Ind_{P}^{G}ρ for a super-cuspidal ρ of M_{P} for P ≠ B. This shows the result.

Of course, taking the dual of the above statement (5.5), we can realize π as a quotient of Ind_{B}^{G}λ′ for λ′ : T (F) → K^{×} possibly different from λ.

For a continuous character λ : T (F) → K^{×} (with respect to the discrete topology of K^{×}), writing λ(diag(t_{1}, …, t_{n})) = ∏_{j=1}^{n} λ_{j}(t_{j}), we define a new character λ̃ by

λ̃(diag(t_{1}, …, t_{n})) = ∏_{j=1}^{n}(λ_{j}(t_{j})|t_{j}|_{𝔭_{σ}}^{j−1}). Then we write I_{B}^{G}(λ) = Ind_{B(F)}^{G(F)}(λ̃).

For a Haar measure dμ_{N} on the unipotent radical N(F) of B(F), we have the modulus character δ_{B} given by

∫_{N} ϕ(x)dμ_{N}(x) = δ_{B}(b) ∫_{N} ϕ(b^{−1}xb)dμ_{N}(x).

The character is explicitly given by

\sqrt{δ_{B}}(x) = | det(x)|_{v}^{(n−1)/2} ∏_{j=1}^{n} |t_{j}|^{1−j}

for x = diag(t_{1}, …, t_{n}). Thus we see λ̃(x) = δ_{B}(x)^{−1/2}| det(x)|_{v}^{(n−1)/2}λ(x). Since δ_{B} is the modulus character of the conjugation action of T on the Haar measure μ_{N}, I_{B}^{G}(λ) is the induction in the category of algebraic admissible representations as in [Cl] Definition 1.9 performed for right B–module (not left ones in [Cl]).

Proposition 5.4. Let W be the Weyl group of T = T_{B}. Let λ : T (F) → K^{×} be a continuous character (under the discrete topology on K), and put V = V (I_{B}^{G}(λ)). If λ^{w} for w ∈ W are all distinct (where λ^{w}(t) = λ(wtw^{−1})), then the Jacquet module V_{B} is a semisimple T (F)–module. Moreover V_{B} ≅ ⨁_{w∈W} λ̃^{w} as T (F)–modules; in particular, dim_{K} V_{B} = n!.

Proof. The Jacquet module functor V ↦ V_{B} is exact; thus (X_{B})^{ss} = (X^{ss})_{B}, where the superscript “ss” indicates the semi-simplification of a representation X. Since (I_{B}^{G}λ)^{ss} = | |_{v}^{(n−1)/2} ⊗ (Ind_{B}^{G}(δ_{B}^{−1/2}λ))^{ss} ≅ | |_{v}^{(n−1)/2} ⊗ (Ind_{B}^{G}(δ_{B}^{−1/2}λ^{w}))^{ss} = (I_{B}^{G}λ^{w})^{ss} (see [BZ1] 2.9), λ̃^{w} is a subquotient of V_{B} by Frobenius reciprocity ([BZ] 3.13). Thus we have a T (F)–linear surjection of (V_{B})^{ss} onto ⨁_{w∈W} λ̃^{w}, because λ̃^{w} are all distinct. Since dim_{K} V_{B} ≤ n! and |W | = n!, we know that V_{B} ≅ (V_{B})^{ss} = ⨁_{w} λ̃^{w} and dim_{K} V_{B} = n!.

By (4.6), we get immediately from the above proposition the following assertion:

Corollary 5.5. Let the notation be as in Proposition 5.4. Suppose that λ^{w} (w ∈ W) are all distinct. Then for each λ̃^{w}–eigenvector v ∈ V_{B} ≅ V_{B}^{∘},

v|T_{s}(ϖ) = (|ϖ|_{v}^{s(s−1)/2} ∏_{j=n−s+1}^{n} λ_{j}^{w}(ϖ)) v,

where λ_{j}^{w}(ϖ) = λ^{w}(diag(1, …, 1, ϖ^{j}, 1, …, 1)). In particular, V_{B}^{∘} is a semisimple R(N_{B}; Δ_{∞})–module of dimension n!.

6. p–Adic Hecke algebras

In this section, we first define the nearly p–ordinary universal Hecke algebra and prove its finiteness. We shall further prove the control theorem for the universal nearly p–ordinary Hecke algebra (of level p^{∞} for G) from the strong multiplicity one theorem for GL(n) and its inner twists.

6.1. Hecke algebra as a double coset algebra. We consider an algebraic group G_{/ℚ} = Res_{F/ℚ}G_{0} for a number field F as in the introduction. We write 𝔩 (resp. v) for prime ideals (resp. places) of F and ℓ for primes of ℤ. We write G_{ℓ} = G(ℚ_{ℓ}) and G_{v} = G_{0}(F_{v}). Then G(𝔸) is the restricted product of local groups G_{v} over all places v of F. Since G_{0} is an inner form of GL(n)_{/F}, we have

(GL(𝔩)) i_{𝔩} : G_{𝔩} ≅ GL_{n}(F_{𝔩}) for almost all prime ideals 𝔩.

Thus there is a finite set of primes Σ for which (GL(𝔩)) fails to hold. We fix an isomorphism i_{𝔩} for every 𝔩 ∉ Σ. We put

𝔯^{(Σ)} = ∏_{𝔩∉Σ} 𝔯_{𝔩} ⊂ 𝔯̂ = ∏_{𝔩} 𝔯_{𝔩}.

We fix a rational prime p outside Σ and a standard (proper) parabolic subgroup P of GL(n) associated with the partition n = n_{1} + ⋯ + n_{r}. Let U be an open compact subgroup of G(𝔸) of the form U = ∏_{𝔩} U_{𝔩} for the projection U_{𝔩} of U in G_{𝔩}. We suppose the following conditions on U:

(G_{B,𝔩}) If U_{𝔩} ≠ GL_{n}(𝔯_{𝔩}) and 𝔩 ∉ Σ ∪ {𝔭|p}, then I_{B,α} ⊃ U_{𝔩} ⊃ I_{B,α}^{∘}, (G_{P,p}) I_{P,α} ⊃ U_{𝔭} ⊃ I_{P,α}^{∘} (α > 0) for all 𝔭|p

for the subgroups I_{B,1} in GL_{n}(𝔯_{𝔩}) and I_{P,α}^{∘} and I_{P,α} in GL_{n}(𝔯_{𝔭}) defined in (4.7).

We write D_{𝔩} (𝔩 ∉ Σ ∪ {𝔭|p}) for D_{B}(𝔯_{𝔩}) in (4.1). We put D_{𝔭} = D_{P} (𝔯_{𝔭}). Then by elementary divisor theory, if U_{𝔩} = GL_{n}(𝔯_{𝔩}) and 𝔩 ∉ Σ ∪ {𝔭|p}, then Δ_{𝔩} = U_{𝔩}D_{𝔩}U_{𝔩} = M_{n}(𝔯_{𝔩}) ∩ GL_{n}(F_{𝔩}). If U_{𝔩} ≠ GL_{n}(𝔯_{𝔩}) (𝔩 ∉ Σ), as seen in 4.1,

Δ_{𝔩} = U_{𝔩}D_{𝔩}U_{𝔩} if 𝔩 ∤ p, and Δ_{𝔭} = I_{P,α}D_{𝔭}I_{P,α} if 𝔭|p

are semi-groups. Anyway Δ = Δ_{U} = (∏_{𝔩∈Σ} U_{𝔩}) × (∏_{𝔩∉Σ} Δ_{𝔩}) is a sub-semi-group

in G(𝔸^{(∞)}).

We consider the Hecke algebra R(U; Δ) made of compactly supported (locally constant) U–bi-invariant functions supported on Δ with values in ℤ. Then by definition, we have

(6.1) R(U; Δ_{U}) ≅ ⨂_{l∉Σ} R(U_{𝔩}; Δ_{𝔩})

≅ (⨂_{𝔭|p} ℤ[I_{P,α}/U_{𝔭}][T_{s_{1}}(𝔭), …, T_{s_{r}}(𝔭)]) ⊗ ( ⨂_{𝔩∉Σ∪{𝔭|p}} ℤ[T_{1}(𝔩), …, T_{n}(𝔩)]) ,

for T_{s}(𝔩) which is the characteristic function of U_{𝔩}ξ_{s}U_{𝔩} for ξ_{s} = ( \begin{matrix}1_{t} & 0 \\ 0 & ϖ_{𝔩}1_{s} \end{matrix} ) ∈ D_{𝔩}. Here s_{j} is as in Proposition 4.1, and the above result follows from the proposition combined with [Sh] Theorem 3.20 when U_{𝔩} = G_{0}(𝔯_{𝔩}).

We define

R = R_{P} = (⨂_{𝔭|p} ℤ[T_{M}(𝔯_{𝔭})][T_{s_{1}}(𝔭), …, T_{s_{r}}(𝔭)]) ⊗ ( ⨂_{𝔩∉Σ∪{𝔭|p}} ℤ[T_{1}(𝔩), …, T_{n}(𝔩)]) .

Since we have a natural projection P/P^{∘}(𝔯_{𝔭}) ≅ T_{M}(𝔯_{𝔭}) → I_{P,α}/U_{𝔭} for all U satisfying (G_{P,p}), the ring R(U; Δ_{U}) is a homomorphic image of R.

6.2. Nearly p–ordinary cohomology groups. Let G(ℝ)_{+} be the identity connected component of G(ℝ). Let

X = X(U) = G(ℚ)\G(𝔸)/UZ(ℝ)_{+}C_{∞+},

where C_{∞+} is a maximal compact subgroup of G(ℝ)_{+} and Z(ℝ)_{+} is the center of G(ℝ)_{+}. We now consider the following condition:

G(ℝ) ≅ GL_{n}(F ⊗_{ℚ} ℝ). (GL(∞))

Then if U is sufficiently small, X is a Riemannian manifold. For the moment, we suppose that

(TF) X(U) is smooth.

Let L be a left Δ_{p}^{−1}–module for Δ_{p} = ∏_{𝔭|p} Δ_{𝔭} on which the central elements in 𝔯_{U}^{×} = 𝔯^{×} ∩ (U · G(ℝ)_{+}) act trivially. Then we consider the covering

X = X (U) = G(ℚ)\(G(𝔸) × L)/UZ(ℝ)_{+}C_{∞+},

where the action is given by γ(x, ℓ)u = (γxu, u_{p}^{−1}ℓ). We study the sheaf \underline{L} = \underline{L}_{U} of locally constant sections of X (U) over X(U). We have the cohomology groups H^{q}(X(U), \underline{L}_{U}) with coefficients in \underline{L}. Since U satisfies (G_{B,𝔩}) and (G_{P,p}), we may regard L as a right Δ–module through the projection of Δ_{U} into Δ_{p}, that is, for δ ∈ Δ and ℓ ∈ L, ℓδ = δ_{p}^{−1}ℓ.

For each double coset UxU as an element of R(U; Δ), we can define the action of [UxU] on H^{q}(X(U), \underline{L}) as follows: We look at the following morphism [x] : X (V) → X (V^{x}) for V = xUx^{−1} ∩ U and V^{x} = x^{−1}V x given by [x](y, ℓ) = (yx, x_{p}^{−1}ℓ) = (yx, ℓx_{p}). Then [x] induces a morphism

[x] : H^{q}(X(V), \underline{L}) → H^{q}(X(V^{x}), \underline{L}),

and we define

[UxU] = T r_{X(V^{x})/X(U)} ∘ [x] ∘ res_{X(V)/X(U)}. (6.2)

Thus under (G_{P,p}) and (G_{B,𝔩}), the (double coset) Hecke algebra R_{P} acts on the cohomology group H^{q}(X(U), \underline{L}).

Since G(ℚ) is isomorphic to the multiplicative group of a central simple algebra D over F, choosing a maximal order ℜ of D, we can extend G to a group scheme over ℤ. Hereafter we suppose the following two conditions:

(D1) G(A) = (ℜ ⊗_{ℤ} A)^{×} for a maximal order ℜ of a division algebra over F; (D2) The algebra ℜ ⊗_{𝔯} F_{𝔩} is isomorphic either to M_{n}(F_{𝔩}) or a division algebra

for all primes 𝔩. By (D1), it is known that X(U) is compact. The set Σ is made of primes 𝔩 at which G_{𝔩} is the multiplicative group of a division algebra central over F_{𝔩}.

We are going to specify the open compact subgroup U and the Δ^{−1}–module L. As for U, we fix an open subgroup U of G(ℤ̂) (containing the center Z(ℤ) = 𝔯^{×}) satisfying (G_{B,𝔩}) with U_{p} = G(ℤ_{p}). Recall that P is a standard parabolic subgroup of GL(n) associated to a partition n = n_{1} + ⋯ + n_{r}. Let M be the standard Levi-subgroup of P. Thus

M(A) = {diag(x_{1}, …, x_{r})|x_{i} ∈ GL_{n_{i}}(A)} .

Let M′ be a factor of M given by

M′(A) = {diag(x_{1}, …, x_{r−1})|x_{i} ∈ GL_{n_{i}}(A)} ,

and put

M^{∘} = {x = diag(x_{1}, …, x_{r}) ∈ M|x_{i} ∈ SL(s_{i}) for i = 1, …, r} . (6.3)

Let π : P → M be the projection, and write p for the product of all prime ideals 𝔭|p in F. Then we define

U_{0,P} (p^{α}) = {u ∈ U|u_{p} mod p^{α} ∈ P (𝔯/p^{α})}
U_{1,P} (p^{α}) = {u ∈ U_{0,P} (p^{α})|π(u_{p} mod p^{α}) ∈ M^{∘}(𝔯/p^{α})} .

Let S be an open subgroup of U such that U_{0,P} (p^{α}) ⊃ S ⊃ U_{1,P} (p^{α}) (α > 0). We now specify the Δ_{S}^{−1}–modules we are going to study. Recall that N = N_{P} denotes the unipotent radical of P. We put S_{p}^{∘} = S_{p} ∩ SL_{n}(𝔯_{p}), where S_{p} is the p–component of S. By abusing the notation, we write I_{P,α} for the product of I_{P,α} with respect to V = 𝔯_{𝔭} over all p–adic places 𝔭 of F. Then Y_{S} = S_{p}^{∘}/N(𝔯_{p}) is an open subspace of the space Y_{α} = (I_{P,α} ∩SL_{n}(𝔯_{p}))/N(𝔯_{p}) we studied in [H95]. There are two group actions on the space Y_{S}. Firstly, since P (𝔯_{p}) ∩ SL_{n}(𝔯_{p}) normalizes N(𝔯_{p}), the quotient M(𝔯_{p}) ∩ SL_{n}(𝔯_{p}) ≅ (P (𝔯_{p}) ∩ SL_{n}(𝔯_{p}))/N(𝔯_{p}) naturally acts on Y_{S} from the right. Secondly, for each choice ϖ = (ϖ_{𝔭}) ∈ 𝔯_{p} with prime element ϖ_{𝔭} ∈ 𝔯_{𝔭}, we have a natural left action of the semi-group

Δ_{S,p} = S_{p} (∏_{𝔭|p} D_{P} (𝔯_{p})) S_{p}

on Y_{S} as described in [H95] Section 3.

We recall briefly the action of Δ_{S} on Y_{S}. Since Δ_{S,p} = S_{p}^{∘}T (𝔯_{p})D(𝔯_{p})S_{p}^{∘} and S_{p}^{∘} acts on Y_{S}, we only need to define the action of T (𝔯_{p})D(𝔯_{p}) for D(𝔯_{p}) = ∏_{𝔭|p} D(𝔯_{𝔭}). First note here that Y_{S} ⊂ (GL(n)/N)(F_{p}) (F_{p} = F ⊗_{ℚ} ℚ_{p}). Thus there is a natural left action of SL_{n}(F_{p}) and right action of M(F_{p}) = (P/N)(F_{p}) on Y = (GL(n)/N)(F_{p}). Then for each x ∈ T (𝔯_{p})D(𝔯_{p}), decompose S_{p}x_{p}S_{p} = ⨆_{u∈xN(𝔯_{p})x^{−1}/N(𝔯_{p})} S_{p}ux_{p} (see (4.4) and (4.5)). For each class yN in the homogeneous space Y_{S} ⊂ SL_{n}(F_{p})/N(F_{p}) (with y ∈ S_{p}^{∘}), x_{p}y = y_{u}ux_{p} for a unique u ∈ xN(𝔯_{p})x^{−1}/N(𝔯_{p}). Then the action of x: y ↦ y_{u} = x_{p}yx_{p}^{−1} mod N(F_{p}) preserves Y_{S} in GL_{n}(F_{p})/N(F_{p}). This action induces a left action of (Δ_{S})^{−1} on continuous functions ϕ on Y_{S} by xϕ(y) = ϕ(x_{p}^{−1} ∘ y).

Now we explicitly write down the Δ^{−1}–module L. Let ρ be an absolutely irreducible rational representation of \overline{M} = Res_{𝔯_{p}/ℤ_{p}}M_{/𝔯_{p}} ⊂ G_{/ℤ_{p}} into GL(m) defined over O for a valuation ring O ⊂ \overline{ℚ}_{p} containing σ(𝔯) for all embeddings σ : F ↪ \overline{ℚ}_{p}. Let K be the field of fractions of O. Let V (O) ≅ O^{m} be the representation space of the contragredient ρ̃ of ρ, which is again a polynomial representation. For each O–module X, we define an M–module V (X) by V (O) ⊗_{O} X. Regard ρ as a representation of \overline{P} = Res_{𝔯_{p}/ℤ_{p}}P_{/𝔯_{p}} ⊂ G_{/ℤ_{p}} pulling it back to P by the isomorphism P/N_{P} ≅ M. Then we suppose that the polynomial representation L(ρ; K) = IndG ; \overline{P} ρ̃ is non-trivial, which is then absolutely irreducible G(ℚ_{p})–module. Physically we have L(ρ; K) = Ind_{P ∩G^{∘}}^{G^{∘}} ρ̃|_{P ∩G^{∘}}:

(6.4) L(ρ; K)

= {ϕ : (G^{∘}/N_{P})(ℚ_{p}) → V (K)|ϕ : polynomial, ϕ(yx) = ρ̃(x^{−1})ϕ(y)} for all x ∈ (M ∩ SL(n))(F_{p}). We often identify two induced modules IndG ; \overline{P} ρ̃ and Ind_{P ∩G^{∘}}^{G^{∘}} ρ̃|_{P ∩G^{∘}}.

We are going to modify the action of Δ_{p}^{−1} ⊂ G(ℚ_{p}) so that it preserves an O– lattice L(ρ; O) ⊂ L(ρ; K). Note that Y_{S} ⊂ (G^{∘}/N_{P})(ℚ_{p}). We then define L(ρ; O) by the subspace of L(ρ; K) made of all functions having values in V (O) on Y_{S}. If ϕ ∈ L(ρ; O), then for s ∈ S_{p}, we may write s = s^{∘}t with s^{∘} ∈ S_{p}^{∘} and t ∈ M(𝔯_{p}). Then ϕ(s) = ϕ(s^{∘}t) = ρ̃(t^{−1})ϕ(s^{∘}) ∈ V (O), and hence L(ρ; O) is an Δ_{S}^{−1}–module by the pull-back action of the action of Δ_{S} on Y_{S}: bϕ(y) = ϕ(b^{−1}∘y) for ϕ ∈ L(ρ; O) and b ∈ Δ_{S}^{−1}. This action coincides with the action induced from L(ρ; K) = IndG ; \overline{P} ρ̃ on S_{p}^{∘} but differs by a scalar factor for general elements in Δ_{S}^{−1}.

We also consider a far bigger Δ_{S}^{−1}–module C_{ρ}^{S} made of continuous functions ϕ on Y_{S} with values in V (K/O) satisfying ϕ(yx) = ρ̃(x^{−1})ϕ(y) for all x ∈ M^{∘}(𝔯_{p}), where M^{∘} = M_{P}^{∘} = SL(n_{1}) × ⋯ × SL(n_{r}) ⊂ M. Thus

C_{ρ}^{S} = ind_{P^{∘}(𝔯_{p})}^{S_{p}^{∘}}V (K/O) as S_{p}^{∘}–module.

The module C_{ρ}^{S} has a natural action of Δ_{S}^{−1} given by bϕ(y) = ϕ(b_{p}^{−1} ∘ y). In particular, we write C_{ρ} for C_{ρ}^{S} when S = U. Similarly for each topological O– module X, we may consider the following Δ_{S}^{−1}–modules:

C(Y_{S}, ρ; X) = {ϕ : Y_{S} → V (X) : continuous|ϕ(yx) = ρ̃(x^{−1})ϕ(y) ∀x ∈ M^{∘}(𝔯_{p})} ,

where we let b ∈ Δ_{S}^{−1} act on ϕ by bϕ(y) = ϕ(b_{p}^{−1}y).

In [H95], the symbols L(ρ; A) and C_{ρ}^{S} are used to denote induced module from ρ in place of ρ̃ here. However results obtained there are valid by replacing ρ by ρ̃; so, hereafter we quote results proven in [H95] without giving any warning about the change of ρ to its contragredient ρ̃.

We equip C(Y_{S}, ρ; X) with a T_{P} (𝔯_{p})–module structure in the following way: First note that T_{P} ≅ M_{P}^{∘} \(M_{P} ∩ SL(n)). Then take γ ∈ (M_{P} ∩ SL(n))(𝔯_{p}). For the class [γ] ∈ T_{P} (𝔯_{p}) and ϕ ∈ C(Y_{S}, ρ; X), we define

[γ] · ϕ(y) = ρ̃(γ)ϕ(yγ). (6.5)

By our definition of C(Y_{S}, ρ; X), for γ^{∘} ∈ M^{∘}(𝔯_{p}),

[γγ^{∘}] · ϕ(y) = ρ̃(γγ^{∘})ϕ(yγγ^{∘}) = ρ̃(γ)ρ̃(γ^{∘})ρ̃(γ^{∘})^{−1}ϕ(yγ) = [γ] · ϕ(y).

Thus the action is well defined.

Since Y_{S} is Zariski dense in the algebraic variety SL(n)/N, the induced polynomial representation L(ρ; K), realized on the space of polynomial functions on SL(n)/N with values in V (K), by restriction to Y_{S}, is sent isomorphically into C(Y_{S}, ρ; K). That is, we have an embedding L(ρ; K) ⊂ C(Y_{S}, ρ; K) compatible with the T_{P} (𝔯_{p})–action. The action of Δ_{S}^{−1} on Y_{S} induces a new modified Δ_{S}^{−1}– action on L(ρ; K), which coincides with the original one on S_{p}^{∘}. For φ = Ind_{P}^{GL(n)} ρ̃, noting the fact: φ|_{SL(n)} = Ind_{P ∩SL(n)}^{SL(n)} ρ̃|_{P ∩SL(n)} and ξ ∘ y = ξyξ^{−1} mod N(F_{p}) for ξ ∈ D_{P} (𝔯_{p}), we have the new action of ξ on ϕ ∈ L(ρ; K) given by

ξ^{−1}ϕ(y) = ϕ(ξyξ^{−1}) = ρ̃(ξ)ϕ(ξy) = ω(ξ)ρ(ξ^{−1})ϕ(y),

where ω is the central character of ρ̃, which is equal to χ^{−1}|_{T_{M}} if ρ̃ = Ind_{B}^{P} χ^{−1}.

Let χ be a rational character of Res_{𝔯_{p}/ℤ_{p}}T_{M}. By the determinant map: P/P^{∘} ≅ T_{M} as in 1.1, we regard χ a character of \overline{P}, and we define ρ ⊗ χ by ρ ⊗ χ(x)v = χ(x)ρ(x)v. We suppose that χ is dominant with respect to ρ (that is, L(ρ⊗χ; K) ≠ 0). Then as already remarked, we can realize L(ρ⊗χ; K) in C(Y_{S}, ρ; K) uniquely. In particular, the action of T_{P} on C(Y_{S}, ρ; K) preserves L(ρ ⊗ χ; K) and hence induces an action of T_{P} (𝔯_{p}) on L(ρ ⊗ χ), which is actually given by χ.

We are now going to twist the module L(ρ; A) by a finite order character of T_{M}. Let ε be a character of T_{M}(𝔯/p^{α}) → O^{×} and S = U_{0,P} (p^{α}). We extend ε to

T_{M}(F_{p}) = T_{M}(𝔯_{p}) × ∏_{i=1}^{r} (∏_{𝔭|p} ϖ_{𝔭}^{ℤ})

by just putting

ε((1, …, 1, ϖ_{𝔭}^{i}, 1, …, 1)) = 1

for all 𝔭|p. For each x ∈ Δ_{S}, we write x_{ii} for the i–th n_{i} × n_{i} diagonal block. Then regarding det(x) = (det(x_{ii}))_{0≤i≤r} ∈ G_{m}^{r}(F_{p}) as an element of T_{M}(F_{p}), we consider ε to be a character ε : Δ_{S} → O^{×}. Thus we may think of L(ρ ⊗ χε; K) which is equal to L(ρ ⊗ χ; K) as a K–vector space but the Δ_{S}^{−1}–action is twisted as bv = ε(b^{−1})b ∘ v, where “b ∘ v” indicates the action of b ∈ Δ_{S}^{−1} on L(ρ ⊗ χ; K).

Define a function ε_{Y} : Y_{S} → O^{×} by ε_{Y} (y) = ε(det(y)) for det : P ∩ SL(n) → T_{M} which is well defined because det(y) for y ∈ S_{p}^{∘} ⊂ Δ_{S} depends only on the coset yN(𝔯_{p}). Then first realizing L(ρ ⊗ χ; K) in C(Y_{S}, ρ; K) and then multiplying functions in L(ρ ⊗ χ; K) by ε_{Y}, we get the space L(ρ ⊗ χε; K) realized inside C(Y_{S}, ρ; K). We then define

L(ρ ⊗ χε; O) = L(ρ ⊗ χε; K) ∩ C(Y_{S}, ρ; O),

where we take α as small as possible so that ε factors through T_{M}(𝔯/p^{α}). Then we write simply

L(ρ ⊗ χε) = L(ρ ⊗ χε; O) ⊗_{O} K/O.

By definition, L(ρ ⊗ χε) is a p–divisible subspace of C_{ρ}^{S} on which T_{M}(𝔯_{p}) acts by χε.

We call a continuous homomorphism χ of T_{M}(𝔯_{p}) into \overline{ℚ}_{p}^{×} arithmetic if χ coincides with a rational character in X(Res_{F/ℚ}T_{M}) on a small p–adic neighborhood of the identity in T_{M}(𝔯_{p}). Thus we have shown that L(ρ ⊗ χ) can be realized in a unique way in C_{ρ}^{S} for a suitable α (S = U_{0,P} (p^{α})). Then we have well defined cohomology groups:

H^{q}(X(S), \underline{C}_{ρ}^{S}) and H^{q}(X(S), \underline{L}(ρ ⊗ χ))

on which the double coset algebra R(S; Δ_{S}) naturally acts.

We define the p–adic Hecke algebra h_{ρ⊗χ,q}(S; O), for arithmetic χ dominant with respect to ρ, to be the O–subalgebra of End_{O}(H^{q}(X(S), L(ρ ⊗ χ))) generated by the operators in R(S, Δ_{S}). Let 𝕋_{j}(𝔭) be the operator induced by T_{s_{j}}(ϖ_{𝔭}) under the action on L(ρ ⊗ χ) of Δ_{S}^{−1} described above. Note here that we have modified the original action of G_{𝔭} on L(ρ ⊗ χ; K) so that it is minimal among the actions preserving integrality. We write T_{j}(𝔭) for the operator induced by T_{s_{j}}(ϖ_{𝔭}) using the original action of G_{𝔭}. The two operators 𝕋_{j}(𝔭) and T_{j}(𝔭) differ by a constant as specified in (6.6) below.

We can now define the nearly p–ordinary part h_{ρ⊗χ,q}^{n.ord}(S; O) to be the largest direct summand of h_{ρ⊗χ,q}(S; O) on which the image of ∏_{1≤j≤r} ∏_{𝔭|p} 𝕋_{j}(𝔭) is a unit. We write e_{P} for the projector of h_{ρ⊗χ,q}(S; O) onto h_{ρ⊗χ,q}^{n.ord}(S; O). Then

e_{P} = lim_{n→∞} ( ∏_{1≤j≤r} ∏_{𝔭|p} 𝕋_{j}(𝔭))^{n!}

is an element in h_{ρ⊗χ,q}(S; O) (see Section 2 and [H93] 1.11). We then define the nearly ordinary cohomology group H_{P −n.ord}^{q}(X(S), L(ρ ⊗ χ; A)) (with respect to P) to be e_{P} H^{q}(X(S), L(ρ ⊗ χ; A)) as long as we have a well defined e_{P} acting on it. The notion of near ordinarity depends on the choice of the (conjugacy class of) parabolic subgroup P. Thus for a given S, we may have

H_{P −n.ord}^{q}(X(S), L(ρ ⊗ χ; A)) ≠ H_{Q−n.ord}^{q}(X(S), L(ρ ⊗ χ; A))

for two non-conjugate parabolic subgroups P and Q. See Appendix A for a little more intrinsic description of the dependence on P for general reductive groups. Therefore, if confusion is likely, we use the symbol H_{P −n.ord}^{q} to indicate P –nearly ordinary cohomology groups.

Since the action of Δ_{S}^{−1} on L(ρ′; X) for ρ′ = ρ ⊗ χ′ is modified, if one uses the original action of GL(n) on GL(n)/N to define the Hecke operator T_{j}(𝔭) = T (ξ_{s_{j}}) on the cohomology group, we have the following relation

𝕋_{j}(𝔭) = ω(ξ_{s_{j}})T_{j}(𝔭) on H^{q}(X(S), \underline{L}(ρ ⊗ χ)), (6.6)

where ξ_{s} = ( \begin{matrix}1_{n−s} & 0 \\ 0 & ϖ1_{s} \end{matrix} ) in T (F_{𝔭}) and ω is the algebraic character in X(T_{M}) which gives the central character of ρ′ on an open neighborhood of the identity of T_{M}(𝔯_{p}). If ρ̃ = ρ_{χ} = Ind_{B}^{P} χ^{−1}, then ω = (χχ′)^{−1}. The operator T_{j}(𝔭) is induced by the action of ξ_{s_{j}}^{−1} under ρ′ = Ind_{B}^{G}(χχ′)^{−1} and 𝕋_{j}(𝔭) is induced by the action on Y_{S} of ξ_{s_{j}}^{−1} ∈ Δ_{S}^{−1}. The modified action is given by ω(ξ_{s_{j}}^{−1})^{−1}ρ′(ξ_{s_{j}}^{−1}) = ω(ξ_{s_{j}})ρ′(ξ_{s_{j}}^{−1}), and hence we get (6.6).

Since we have a natural map: H^{q}(X(S), \underline{L}(ρ; K)) → H^{q}(X(S), \underline{L}(ρ)) with finite cokernel (L(ρ) = L(ρ; O) ⊗_{O} K/O), an eigenspace H_{P −n.ord}^{q}(X(S), \underline{L}(ρ; K))[λ] of Hecke operators T_{j}(𝔩) (for all 𝔩) with eigenvalue λ(T_{j}(𝔩)) gives rise to an eigenspace H_{P −n.ord}^{q}(X(S), \underline{L}(ρ))[λ] in the discrete module H^{q}(X(S), \underline{L}(ρ)) with the same eigenvalues. On the other hand, fixing embeddings i_{∞} : \overline{ℚ} ↪ ℂ and i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p}, we pull back K to ℂ: K_{0} = i_{∞}(i_{p}^{−1}(K)). Then we have

H^{q}(X(S), \underline{L}(ρ; ℂ)) = H^{q}(X(S), \underline{L}(ρ; K_{0})) ⊗_{K_{0}} ℂ,

and hence the system of eigenvalues λ is given by an automorphic representation π of G(𝔸). We call such π nearly p–ordinary with respect to P. By definition, e_{P} H^{q}(X(S), \underline{L}(ρ))[λ] ≠ 0 if and only if i_{p}(λ(𝕋_{j}(𝔭))) is a p–adic unit in K. Thus π is nearly p–ordinary if and only if for a common eigenvector v ∈ H^{0}(S, V (π)) of T_{j}(𝔭) (for all 𝔭|p), we have

(6.7) |λ(T_{j}(𝔭))|_{p} = |ω^{−1}(ξ_{s_{j}})|_{p} for all j = 1, …, r and all 𝔭|p.

In my earlier work [H95] (especially in Theorem 5.1 of the paper), we studied the action of T_{M}^{∘}(𝔯_{p}) = (T_{M} ∩ SL(n))(𝔯_{p}) on C_{ρ}^{S} induced by the above action of T_{P} (𝔯_{p}) via the isogeny: T_{M}^{∘} ↪ M ↠ T_{P}. Actually the use of the action of T_{P} in place of T_{M} makes the result more transparent, and we obtain, in exactly the same manner as in the proof of [H95] Theorem 5.1 an isomorphism of R_{P} –modules:

ι_{χ} : H_{n.ord}^{q}(Γ^{∘}(S_{1,P} (p^{∞})), L(ρ ⊗ χ)) ≅ H_{n.ord}^{q}(Γ^{∘}(S_{1,P} (p)), C_{ρ}^{S}) (6.8)

satisfying ι_{χ}(χ(z)zc) = zι_{χ}(c) for c ∈ H_{n.ord}^{q}(Γ^{∘}(S_{1,P} (p^{∞})), L(ρ ⊗ χ)) under the action of T_{P} given in (6.5), where Γ^{∘}(S) = (S · G^{∘}(ℝ)) ∩ G^{∘}(ℚ). If we assume the vanishing of cohomology degree less than r: H_{n.ord}^{q}(Γ^{∘}(S_{1,P} (p^{α})), L(ρ ⊗ χ; K)) = 0 for 0 ≤ q < r, we have an isogeny (cf. [H95] Theorem 5.2)

ι_{χ} : H_{n.ord}^{r}(Γ^{∘}(S_{1,P} (p^{α})), L(ρ ⊗ χ)) → H_{n.ord}^{r}(Γ^{∘}(S_{1,P} (p)), C_{ρ}^{S})[χ], (6.9)

where “[χ]” indicates the χ–eigenspace in the cohomology group under the action of T_{P} (𝔯_{p}).

6.3. Universal nearly p–ordinary Hecke algebras. Now we fix the following decomposition given by the approximation theorem:

G(𝔸) = ⨆_{t∈Ξ} G(ℚ)tU_{0,P} (p^{α})G(ℝ)_{+} and G(𝔸) = ⨆_{t∈Ξ} ⨆_{s∈Ξ(S)} G(ℚ)tsSG(ℝ)_{+}, (App)

where G(ℝ)_{+} is the identity connected component of the real Lie group G(ℝ) and Ξ (resp. Ξ(S)) is a finite subset of G(𝔸^{(p∞)}) (resp. U_{0,P} (p^{α})_{p}). Let T_{M} = Z(M) and T_{M}^{∘} = T_{M} ∩ SL(n). Since the reduced norm map ν : G → Res_{F/ℚ}G_{m} induces the isomorphism:

ν : G(ℚ)\G(𝔸)/SG(ℝ)_{+} ≅ F^{×}\F_{𝔸}^{×}/ν(S)F_{ℝ+}^{×},

we may choose the finite sets Ξ and Ξ(S) so that the following four conditions are satisfied:

(i) Ξ is independent of α; (ii) t_{p} = t_{∞} = 1 for t ∈ Ξ; (iii) Ξ(S) ⊂ U_{0,P} (p^{α})_{p} if S ⊃ U_{1,P} (p^{α}); (iv) Ξ(S) ⊂ Ξ(S′) if S′ ⊂ S.

Let T_{M}^{∘} = T_{M} ∩SL(n). Since U_{0,P} (p^{α})/U_{1,P} (p^{α}) = T_{P} (𝔯/p^{α}) ≅^{det} T_{M}^{∘}(𝔯/p^{α}) and by strong approximation theorem (SA) in Section 2, we may approximate each element in T_{M}^{∘}(𝔯_{p}) ⊂ U_{0,P} (p^{α}) sufficiently closely by an element in G(ℚ) ∩ tU_{0,P} (p^{α})t^{−1}, ν induces an isomorphism: Ξ(U_{1,P} (p^{α})) onto a quotient C of T_{M}(𝔯/p^{α})/T_{M}^{∘}(𝔯/p^{α}) ≅ (𝔯/p^{α})^{×}. The quotient C is the image of 𝔯_{p}^{×} in Cl(ν(S)) = F^{×}\F_{𝔸}^{×}/ν(S)F_{ℝ+}^{×}. We write T_{M}^{∘}(α) for the subgroup of T_{M}(𝔯/p^{α}) containing T_{M}^{∘}(𝔯/p^{α}) such that T_{M}^{∘}(α)\T_{M}^{∘}(𝔯/p^{α}) ≅ C. This group depends on the choice of U. We then define

T_{M}^{∘}(∞) = \underleftarrow{lim}_{α}T_{M}^{∘}(α).

Thus T_{M}^{∘}(∞) contains T_{M}^{∘}(𝔯_{p}) as a subgroup, and T_{M}^{∘}(∞)/T_{M}^{∘}(𝔯_{p}) is isomorphic to the closure of ν(𝔯_{U}^{×}) in 𝔯_{p}^{×}, where 𝔯_{U}^{×} = 𝔯^{×} ∩ (U · G(ℝ)).

For each congruence subgroup Γ ⊂ G(ℚ)_{+} for G(ℚ)_{+} = G(ℚ)∩G(ℝ)_{+}, we define \overline{Γ} for its image in G(ℚ)/Z(ℚ) for the center Z of G and Γ^{∘} = Γ ∩ G^{∘}(ℚ). Then the natural map Γ ↠ \overline{Γ} induces a homomorphism ι : Γ^{∘} → \overline{Γ} with finite kernel and cokernel killed by n. We put X(Γ) = \overline{Γ}\G(ℝ)_{+}/C_{∞+}Z(ℝ). The space X(Γ) is compact if D is a division algebra and is a smooth Riemannian manifold if \overline{Γ} is torsion-free.

We define Γ_{t}(S) = G(ℚ) ∩ tS · G(ℝ)_{+}t^{−1} in G(𝔸) for t ∈ Ξ,

\overline{Γ}_{t}(S) = Γ_{t}(S)/F^{×} ∩ Γ_{t}(S) ⊂ P GL_{n}(F)

and Γ_{t}^{∘}(S) = G^{∘}(ℚ) ∩ tS · G(ℝ)_{+}t^{−1}, which is a congruence subgroup of the derived group G^{∘}(ℚ) of G(ℚ). Since

Γ_{t}(U_{0,P} (p^{α}))/Γ_{t}(U_{1,P} (p^{α})) ≅ T_{M}^{∘}(α),

the cohomology group H^{q}(\overline{Γ}_{t}(U_{1,P} (p^{α})), L) is naturally a T_{M}^{∘}(α)–module. By definition X(S) ≅ ⨆_{t∈Ξ} ⨆_{s∈Ξ(S)} X(Γ_{t}(S)) via gtsu ↦ u_{∞} for u ∈ S · G(ℝ)_{+}, and

H^{q}(X(S), \underline{L}) ≅ ⨁_{t∈Ξ} ⨁_{s∈Ξ(S)} H^{q}(\overline{Γ}_{t}(S), L); (6.10)

H^{q}(X_{1,P} (p^{α}), \underline{L}) = ⨁_{t∈Ξ} ind_{T_{M}^{∘} (α)}^{T_{M} (𝔯/p^{α})}H^{q}(\overline{Γ}_{t}(U_{1,P} (p^{α})), L), (6.11)

where we have written L = L(ρ ⊗ χ) for arithmetic χ dominant with respect to ρ and X_{1,P} (p^{α}) = X(U_{1,P} (p^{α})). Here the involution “ι” in the definition in 1.1 of the induction “ind_{T_{M}^{∘} (α)}^{T_{M} (𝔯/p^{α})}” is the identity map. Thus by (6.11), H^{q}(X(U_{1,P} (p^{α})), L) is a T_{M}(𝔯/p^{α})–module. Even if X(S) is not smooth, we can think of the right hand side of (6.10), which also has a natural action of the double coset algebra R_{P}. Abusing notation, we hereafter write H^{q}(X(S), \underline{L}) for ⨁_{s∈Ξ(S)} H^{q}(\overline{Γ}_{t}(S), L) if (TF) is not satisfied by S.

We consider

H^{q}(X_{1,P} (p^{∞}), \underline{L}) = \underrightarrow{lim}_{α}H^{q}(X_{1,P} (p^{α}), \underline{L}). (6.12)

Since the restriction map res : H^{q}(X_{1,P} (p^{α}), \underline{L}) → H^{q}(X_{1,P} (p^{β}), \underline{L}) for β > α is a morphism of R_{P} –modules, it is compatible with e_{P}. Thus e_{P} acts on H^{q}(X_{1,P} (p^{α}), \underline{L}) for α = 1, 2, …, ∞, and we have, writing H_{n.ord}^{q}(X_{1,P} (p^{α}), \underline{L}) for e_{P} H^{q}(X_{1,P} (p^{α}), \underline{L}),

(6.13) H_{n.ord}^{q}(X_{1,P} (p^{∞}), \underline{L}) = \underrightarrow{lim}_{α}H_{n.ord}^{q}(X_{1,P} (p^{α}), \underline{L})

= ⨁_{t∈Ξ} ind_{T_{M}^{∘} (∞)}^{T_{M} (𝔯_{p})}H_{n.ord}^{q}(Γ_{t}(U_{1,P} (p^{∞})), \underline{L}),

where T_{M}^{∘}(∞) = \underleftarrow{lim}_{α}T_{M}^{∘}(α) and

H^{q}(\overline{Γ}_{t}(U_{1,P} (p^{∞})), \underline{L}) = \underrightarrow{lim}_{α}H^{q}(\overline{Γ}_{t}(U_{1,P} (p^{α})), \underline{L}).

We define the p–adic Hecke algebra h_{ρ⊗χ,q}^{n.ord}(U_{1,P} (p^{∞}); O) of level p^{∞} by the subalgebra of End_{Λ}(H_{n.ord}^{q}(X_{1,P} (p^{∞}), \underline{L}(ρ ⊗ χ))) generated over Λ = O[[T_{M}(𝔯_{p})]] by operators in R_{P}. By definition, h_{ρ⊗χ,q}^{n.ord}(U_{1,P} (p^{α}); O) is an O[[T_{M}(𝔯_{p})]]–algebra.

Although in [H95] Section 5, results are formulated using I_{P,α} and Y_{α}, the result proven there is valid without modification replacing I_{P,α} and Y_{α} by S_{p} and Y_{S}. All arguments, as easily checked, go through without modification. In particular e_{P} is well defined on H^{q}(Γ_{t}(S), \underline{C}_{ρ}), and by (6.8), we get a canonical isomorphism

ι_{χ} : H_{n.ord}^{q}(Γ_{t}^{∘}(U_{1,P} (p^{∞})), L(ρ ⊗ χ)) ≅ H_{n.ord}^{q}(Γ_{t}^{∘}(U_{0,P} (p)), C_{ρ}) (6.14)

such that ι_{χ}(χ(z)zc) = zι_{χ}(c) for c ∈ H^{q}(Γ_{t}^{∘}(U_{1,P} (p^{∞})), L(ρ ⊗ χ)) and z ∈ T_{M}^{∘}(𝔯_{p}).

Let 𝔯_{+}^{×} = 𝔯^{×} ∩ Z(𝔸^{(∞)}) · G(ℝ)_{+}. The quotient

H = 𝔯_{+}^{×}Γ_{t}(U_{1,P} (p^{α}))/𝔯_{+}^{×}Γ_{t}^{∘}(U_{1,P} (p^{α}))

is independent of α ≫ 0 and t and is a subgroup of a finite group 𝔯_{+}^{×}/(𝔯_{+}^{×})^{n}. The finite group H is canonically isomorphic to \overline{Γ}_{t}(U_{1,P} (p^{α}))/\overline{Γ}_{t}^{∘}(U_{1,P} (p^{α})). Similarly, the kernel H′ = Ker(Γ_{t}^{∘}(U_{1,P} (p^{α})) → \overline{Γ}_{t}(U_{1,P} (p^{α}))) is finite, of exponent n and independent of α ≫ 0 and t.

Recall that L(ρ; O) is a representation of S_{p} (not just that of S_{p}^{∘}). We assume that

(1) the central elements in 𝔯_{U}^{×} = (𝔯_{+}^{×} ∩ U · G(ℝ)_{+}) act trivially on L(ρ; O);
(2) χ(𝔯_{U}^{×}) = 1.

Thus we may regard L(ρ; O) as a \overline{Γ}_{t}(U)–module. We assume that p ∤ n. Then by the Hochschild-Serre spectral sequence applied to the exact sequence:

1 → H′ → Γ_{t}^{∘}(U_{1,P} (p^{α})) → \overline{Γ}_{t}^{∘}(U_{1,P} (p^{α})) → 1,

we have

ι_{χ} : H_{n.ord}^{q}(\overline{Γ}_{t}^{∘}(U_{1,P} (p^{∞})), L(ρ ⊗ χ)) ≅ H_{n.ord}^{q}(Γ_{t}^{∘}(U_{1,P} (p^{∞})), L(ρ ⊗ χ)), (6.15)

because H^{q}(H′, X) = 0 if q > 0 and X is a ℤ_{p}–module. Thus

H_{n.ord}^{q}(\overline{Γ}_{t}^{∘}(U_{1,P} (p^{∞})), L(ρ ⊗ χ))

is independent of χ by (6.14).

Again by Hochschild-Serre spectral sequence applied to the exact sequence:

1 → \overline{Γ}_{t}^{∘}(U_{1,P} (p^{α})) → \overline{Γ}_{t}(U_{1,P} (p^{α})) → H → 1,

we have

ι_{χ} : H_{n.ord}^{q}(\overline{Γ}_{t}(U_{1,P} (p^{∞})), L(ρ ⊗ χ)) ≅ H^{0}(H, H_{n.ord}^{q}(Γ_{t}^{∘}(U_{1,P} (p^{∞})), L(ρ ⊗ χ))). (6.16)

Thus H_{n.ord}^{q}(\overline{Γ}_{t}(U_{1,P} (p^{∞})), L(ρ ⊗ χ)) is again independent of χ.

The space C_{ρ} is naturally a T_{P} (𝔯_{p})–module as described above. This induces the action of T_{P} on the right hand side of (6.14). We have a natural isomorphism induced by determinant 1.1: T_{P} ≅ T_{M}^{∘}. Thus we can view C_{ρ} as a T_{M}^{∘}(𝔯_{p})–module. The unit group ν(𝔯_{U}^{×}) acts trivially on H_{n.ord}^{q}(\overline{Γ}_{t}(U_{1,P} (p^{∞})), L(ρ ⊗ χ)). Since

T_{M}^{∘}(∞) = \overline{T_{M}^{∘}(𝔯_{p})(ν(𝔯_{U}^{×}))}

in T_{M}(𝔯_{p}), we can extend the action of T_{M}^{∘}(𝔯_{p}) to T_{M}^{∘}(∞) making ν(𝔯_{U}^{×}) act on it trivially. Then we have a canonical isomorphism

ι_{χ} : H_{n.ord}^{q}(X_{1,P} (p^{∞}), L(ρ ⊗ χ)) ≅ ⨁_{t∈Ξ} ind_{T_{M}^{∘} (∞)}^{T_{M} (𝔯_{p})}H_{n.ord}^{q}(\overline{Γ}_{t}(U_{1,P} (p^{∞})), L(ρ)), (6.17)

satisfying ι_{χ}(χ(z)zc) = zι_{χ}(c) for z ∈ T_{M}(𝔯_{p}). This shows

Theorem 6.1. Suppose (D1-2) , (SL(p)) and that p ∤ n. Then we have an algebra isomorphism

ι_{χ} : h_{ρ⊗χ,q}^{n.ord}(U_{1,P} (p^{∞}); O) ≅ h_{ρ,q}^{n.ord}(U_{1,P} (p^{∞}); O)

taking T_{j}(𝔩) to T_{j}(𝔩) (1 ≤ j ≤ n) for all 𝔩 ∉ Σ∩{𝔭|p} and 𝕋_{j}(𝔭) to 𝕋_{j}(𝔭) (1 ≤ j ≤ r). In particular, if P = B, the Hecke algebra h_{χ,q}^{n.ord}(U_{1,P} (p^{∞}); O) is independent of χ.

Proof. By our construction, the isomorphism in (6.17) is equivariant under Hecke operators listed in the theorem (or more precisely, it is an isomorphism of R_{P} – modules). Since the Hecke algebra h_{ρ⊗χ,q}^{n.ord}(U_{1,P} (p^{∞}); O) acts faithfully on the cohomology group H_{n.ord}^{q}(X_{1,P} (p^{∞}), L(ρ⊗χ)) and generated by these Hecke operators, we get the identity of the algebras by (6.17).

Let r_{1} (resp. r_{2}) for the number of real (resp. complex) places of F. We write h_{q}^{n.ord}(U_{1,P} (p^{∞}); O) for h_{χ,q}^{n.ord}(U_{1,P} (p^{∞}); O) if P = B. Hereafter we assume (GL(∞)) and that q is equal to the following number r given by

r = \begin{cases}r_{1}m^{2} + r_{2}m(n − 1) & if n = 2m, \\ r_{1}m(m + 1) + r_{2}mn & if n = 2m + 1. \end{cases} (BD)

The above number r gives the bottom degree of the cuspidal cohomology group of GL(n) under (GL(∞)). Here the word “bottom degree” means that

H_{cusp}^{j}(X(S), L(ρ; ℂ)) = 0 if j < r, and

H_{cusp}^{j}(X(S), L(ρ; ℂ)) = 0 for all j if H_{cusp}^{r}(X(S), L(ρ; ℂ)) = 0.

Then the above explicit value of the bottom degree follows from (3.5) and (3.10).

Under this choice of q, we write h_{ρ}^{n.ord} for h_{ρ,r}^{n.ord}. When P = B, we simply write h^{n.ord} for h_{χ}^{n.ord} which is independent of χ ∈ X(Res_{F/ℚ}T). We write W = W_{P,ρ} for

⨁_{t∈Ξ} ind_{T_{M}^{∘} (𝔯_{p})}^{T_{M} (𝔯_{p})}H_{n.ord}^{r}(\overline{Γ}_{t}(U_{0,P} (p)), C_{ρ}).

Let W = W_{P} be the Pontryagin dual module of W_{P,ρ}. We put W (K) = W ⊗_{O} K. Then using (6.16), we can deduce from (6.9) the following result:

Theorem 6.2. Suppose (D1-2), (GL(∞)), (SL(p)) and that p ∤ n. Let χ be an arithmetic character of T_{M} dominant with respect to ρ such that χχ_{0}^{−1} factors through T_{M}(𝔯/p^{α}) for χ_{0} ∈ X(T_{M}). Then for the ideal P = Ker(χ) in O[[T_{M}(𝔯_{p})]], we have

W (K)/PW (K) ≅ H_{n.ord}^{r}(X(U_{0,P} (p^{α})), L(ρ ⊗ χ; K)).

Proof. We have (from [H95] Theorems 5.1 and 5.2) a Hecke equivariant isogeny

H_{n.ord}^{r}(Γ_{t}^{∘}(U_{0,P} (p^{α})), L(ρ ⊗ χ)) → H_{n.ord}^{r}(Γ_{t}^{∘}(U_{1,P} (p^{∞})), L(ρ))[χ].

We have [χ] in place of [ω_{χ}] in the theorems in [H95], because present normalization of the action of T_{M} uses the T_{P} –action given in (6.5), which is different from the one used in [H95]. Since we have an exact sequence:

1 → H′ → Γ_{t}^{∘}(p^{α}) → \overline{Γ}_{t}(p^{α}) → H → 1

with finite group H and H′ independent of exponent α ≫ 0 and of t, we get an isogeny of T_{M}^{∘}(𝔯_{p})–modules:

H_{n.ord}^{r}(\overline{Γ}_{t}(U_{0,P} (p^{α})), L(ρ ⊗ χ)) → H_{n.ord}^{r}(\overline{Γ}_{t}(U_{1,P} (p^{∞})), L(ρ))[χ].

By Shapiro’s lemma applied to the right-hand-side of (6.17), we have

W_{P,ρ}[χ] ≅ ⨁_{t∈Ξ} H_{n.ord}^{r}(\overline{Γ}_{t}(U_{1,P} (p^{∞})), L(ρ))[χ].

This gives rise to the Hecke equivariant isogeny:

H_{n.ord}^{r}(X_{0,P} (p^{α}), \underline{L}(ρ ⊗ χ)) ≅ ⨁_{t∈Ξ} H_{n.ord}^{r}(\overline{Γ}_{t}(U_{0,P} (p^{α})), L(ρ ⊗ χ)) → W_{P,ρ}[χ].

The desired isomorphism is induced from the Pontryagin dual map of the above isogeny, after tensoring K.

By the above proposition, it is easy to see the Pontryagin dual module W_{P,ρ} of W_{P,ρ} is of finite type over O[[Γ_{T_{M}}]] for the maximal torsion-free subgroup Γ_{T_{M}} of T_{M}(𝔯_{p})/\overline{𝔯^{×}}, which is canonically isomorphic to the maximal torsion-free subgroup of T_{M}(𝔯_{p})/\overline{ν(𝔯_{U}^{×})}. We simply write h for h_{ρ}^{n.ord}(U_{1,P} (p^{∞}); O). Then the Hecke algebras h and h(K) = h ⊗_{O} K act faithfully on W and W (K), respectively. In particular, h is finite over O[[Γ_{T_{M}}]]. Moreover if n > 2 or F has at least one complex place (that is, r_{2} > 0), we can find by (3.3) and (3.9) a dominant character χ ∈ X(Res_{F/ℚ}T) such that there is no cuspidal cohomological automorphic representation whose infinity type is χ (see the following Section 7 for more details). Then for P = B, W (K)/PW (K) = 0 (P = Ker(χ)), and hence W (K) is a torsion O[[Γ_{T}]]–module for the maximal torsion-free subgroup Γ_{T} of T (𝔯_{p}). Note that Λ = O[[Γ_{T}]] is isomorphic to the power series ring O[[T_{1}, …, T_{d}]] for d = r[F : ℚ] + 1 + δ − r_{1} − r_{2}, where δ = δ_{p} is the defect of the Leopoldt conjecture given by dim_{ℚ}(𝔯^{×} ⊗_{ℤ} ℚ) − dim_{ℚ_{p}}(\overline{𝔯^{×}} ⊗_{ℤ_{p}} ℚ_{p}) for the p–adic closure \overline{𝔯^{×}} in 𝔯_{p}. Then the Hecke algebra has dimension less than d + 1 if n > 2 or r_{2} > 0. We thus have Corollary 6.3. The Hecke algebra h_{ρ}^{n.ord}(U_{1,P} (p^{∞}); O) (of bottom degree) is finite over O[[Γ_{T_{M}}]]. In particular,

dim(h_{ρ}^{n.ord}(U_{1,P} (p^{∞}); O)) ≤ dim(O[[Γ_{T_{M}}]])

= rank(Res_{F/ℚ}T_{M}) + 2 + δ_{p} − r_{1} − r_{2} = (r − 1)[F : ℚ] + 2 + δ_{p} + r_{2},

where r_{1} (resp. r_{2}) is the number of real (resp. complex) places of F and P is associated to the partition of n into r parts. If n > 2 or F has at least one complex place, then the above inequality is strict for P = B.

We now assume

(JL) The global Jacquet-Langlands correspondence compatible with the local correspondence holds for G and Res_{F/ℚ}GL(n).

The local Jacquet-Langlands correspondence is known by [DKV] and [R]. The existence of global correspondence is known under a certain ramification condition (see [AC]) Theorem B and [Cl1] Theorem 3.3). In particular, (JL) holds under the conditions (D1-2) (as long as Σ ≠ ∅). Under this assumption, the Hecke module H^{i}(\overline{Γ}, L(ρ; ℂ)) can be embedded into H_{sq}^{i}(\overline{Γ}′, L(ρ; ℂ)) for a suitable congruence subgroup Γ′ of GL_{n}(F) ([Cl] Section 3.5), where the latter cohomology group is the square integrable cohomology groups. Then from the strong multiplicity one theorem (e.g. [Cl] Theorem 1.1 and [Cl1] Theorem 3.3) valid for Res_{F/ℚ}GL(n), we know the strong multiplicity one theorem for G.

As an application of the strong multiplicity one, we have

Proposition 6.4. Suppose that U = ℜ̂^{×} for a maximal order ℜ of D satisfying (D1-2), (SL(p)) and (GL(∞)), where ℜ̂ = ℜ ⊗_{ℤ} ℤ̂. Suppose that the strong multiplicity one theorem holds for cuspidal automorphic representations of G(𝔸). Then we have

(1) The commutative K–algebra h_{χ}^{n.ord}(U_{0,B}(p^{α}); K) is semi-simple;

(2) The cohomology group

H_{n.ord}^{r}(X(U_{0,B}(p^{α})), L(χ; K))

is free of finite rank over the Hecke algebra h_{χ}^{n.ord}(U_{0,B}(p^{α}); K).

Proof. Since K is a finite extension of ℚ_{p}, we can embed K into ℂ (algebraically). We fix such an embedding. We have a natural action of C = C_{∞}/C_{∞+} ≅ {±1}^{r_{1}} on H^{r}(X(S), L(ρ ⊗ χ; ℂ)) which commutes with Hecke operators. We fix a character ε : C → {±1} and consider the ε–eigenspace: H^{r}(X(S), L(ρ ⊗ χ; ℂ))[ε]. Then this space is isomorphic to a space S of cusp forms on G(𝔸) invariant under the group S with a fixed infinity type (see (3.5) and (3.10) and [H95] Section 8 for the description of S). We will see later in Corollary 8.3 from p–near ordinarity that if π is an automorphic representation of G(𝔸) intervening in H_{n.ord}^{r}(X(U_{0,B}(p)), L(ρ ⊗ χ; K))[ε], the local component π_{𝔭} of π at 𝔭|p can be embedded into I_{B(F_{𝔭})}^{G_{𝔭}}λ for a character λ : T (F_{𝔭}) → K^{×} with distinct λ^{w} for all elements w in the Weyl group W of T. By Corollary 5.5, we now know that V (π_{𝔭})^{∘} is a semi-simple R(N_{B}(𝔯_{𝔭}), Δ_{∞})–module, and each character of R(N_{B}(𝔯_{𝔭}), Δ_{∞}) appears on V (π_{𝔭})^{∘} at most multiplicity 1. Actually we will see that e_{B}V (π_{𝔭})^{∘} is one dimensional (Corollary 8.3). For 𝔩 ∈ Σ, since U_{0,B}(p^{α})_{𝔩} is the maximal compact subgroup of D_{𝔩}^{×} for a division algebra D_{𝔩/F_{𝔩}}, the 𝔩–component π_{𝔩} of π is a one dimensional representation. If 𝔩 ∉ Σ∪{𝔭|p}, U_{0,B}(p^{α})_{𝔩} is a maximal compact subgroup of GL_{n}(F_{𝔩}) and hence π_{𝔩} is spherical, and H^{0}(U_{0,B}(p^{α})_{𝔩}, V (π_{𝔩})) is one dimensional. Thus by the strong multiplicity one theorem, S is a semi-simple module over R_{B} and each character of R_{B} has multiplicity at most one. In other words, H_{n.ord}^{r}(X(U_{0,B}(p^{α})), L(ρ ⊗ χ; ℂ))[ε] is free of rank 1 over h_{ρ⊗χ}^{n.ord}(U_{0,P} (p^{α}); K) ⊗_{K} ℂ (fixing an embedding K ↪ ℂ), which is semi-simple. This shows the desired result by a descent from ℂ to K, since ℂ is a faithfully flat K–module.

Actually under (GL(∞)) and (D1-2) with Σ ≠ ∅, the strong multiplicity one theorem is known (see [AC] Theorem B and [Cl1] Theorem 3.3); so, the above proposition holds.

For simplicity, we write h for h_{id,r}^{n.ord}(U_{1,B}(p^{∞}); O). Let P be the kernel of the algebra homomorphism χ : O[[T (𝔯_{p})]] → O) induced by χ. We study the localization h_{P}. By definition, h acts faithfully on ⨁_{ε} H_{n.ord}^{r}(X_{1,B}(p^{∞}), K/O)[ε]. Thus for an ideal 𝔞_{ε} of h, h/𝔞_{ε} acts faithfully on H_{n.ord}^{r}(X_{1,B}(p^{∞}), K/O)[ε] for a given ε : C → {±1}. Write H_{ε} for the Pontryagin dual module of H_{n.ord}^{r}(X_{1,B}(p^{∞}), K/O)[ε]. Note that Ph_{P} ⊃ 𝔞_{ε,P} for all ε by Proposition 6.4. Since

h_{P}/Ph_{P} ≅ \frac{(h/𝔞_{ε}h)_{P}}{P(h/𝔞_{ε}h)_{P}} ↠ (H_{ε}/PH_{ε}) ⊗ K ≅ H_{n.ord}^{r}(X_{0,B}(p^{α}), L(χ; K))[ε]

by Theorem 6.2 and Proposition 6.4, choosing \overline{x} ∈ H_{n.ord}^{r}(X_{0,B}(p^{α}), L(χ; K)) which is the image of 1 ∈ h/𝔞_{ε}, we can define an h–linear map h → H_{ε} by T ↦ T x for x in H_{ε} such that x mod P = \overline{x}. Then by Theorem 6.2, the induced map h_{P} → H_{ε,P} is surjective (by Nakayama’s lemma). Since the action is faithful, we have (h/𝔞_{ε})_{P} ≅ W (K)[ε]_{P}, and we get

Theorem 6.5. Suppose that U = ℜ̂^{×} for a maximal order ℜ of D satisfying (D1-2) (GL(∞)) and (SL(p)). For each dominant arithmetic character χ of T (𝔯_{p}), writing P for Ker(χ : O[[T (𝔯_{p})]] → O), we have a natural algebra isomorphism:

(h^{n.ord}(U_{1,B}(p^{∞}); O)/Ph^{n.ord}(U_{1,B}(p^{∞}); O)) ⊗_{O} K ≅ h_{χ}^{n.ord}(U_{0,B}(p^{α}); K),

which takes T_{j}(𝔩) to T_{j}(𝔩) for all 𝔩 outside Σ.

7. A conjecture on Krull dimension of h^{n.ord}

In this section, we restate the conjecture in the introduction and give supporting arguments in terms of infinity types and functoriality.

7.1. Statement of the conjecture. We have seen in Corollary 6.3 that dim h^{n.ord} is finite for h^{n.ord} = h^{n.ord}(U_{1,B}(p^{∞}); O). The following conjecture giving the upper bound of the dimension is a slightly stronger version of the conjecture in the introduction:

Conjecture 7.1. Let h^{n.ord} = h^{n.ord}(U_{1,B}(p^{∞}); O) for ρ = id. Then we have

dim(h^{n.ord}) ≤ r(n, F) + 1

where

r(n, F) = \begin{cases}m[F : ℚ] + 1, & if n = 2m, \\ m[F : ℚ] + r_{2} + 1, & if n = 2m + 1. \end{cases}

The conjecture is equivalent to the Leopoldt conjecture if n = 1. As for n = 2, the conjecture is known ([H94a] and [H95]), if we assume the following two conditions:

(1) The Leopoldt conjecture holds for F and p;

(2) F has at most one complex place: r_{2} ≤ 1. As we will explain more, later in this subsection, the conjecture holds under (D1-2), (SL(p)) and (GL(∞)) when 3 ≤ n ≤ 4 and F = ℚ, because we know from Corollary 6.3 that dim(h^{n.ord}) ≤ n = r(n; ℚ) + 1 if 3 ≤ n ≤ 4.

We shall give a heuristic argument for the conjecture in terms of the infinity type of cohomological automorphic representation. We will see some other supporting evidences for this conjecture in the following subsection and also study some implication of the validity of the conjecture.

Let π be a cuspidal cohomological automorphic representation of G(𝔸). For each embedding σ : F ↪ ℂ, we write π_{σ} for the local component at the infinite place [σ] induced by the embedding. Thus π_{σ} = π_{cσ} for complex conjugation c. In Section 3, we have associated to π_{σ}, its infinity type χ_{σ} ∈ X(T) for real embedding σ and a pair (χ_{σ}, χ_{cσ}) ∈ X(T)^{2} for complex σ. Actually χ_{σ} is an element of positive weights C_{n}.

We fix an open compact subgroup U of G(𝔸^{(∞)}) so that U ⊃ G(ℤ_{p}). Let X (U) be the set of dominant arithmetic characters χ : T (𝔯_{p}) → K^{×} of cohomological automorphic representations occurring in H^{r}(X(U_{1,B}(p^{α})), \underline{L}(χ; K)) for some α > 0, where r is the bottom degree defined in (BD). We write X_{∞}(U) for the set of infinity types χ_{∞} for χ ∈ X (U). Choosing a complete representative subset ∞ ⊂ I for infinite places of F, we may identify the set X_{∞}(U) with a subset of X(T)[I] = X(Res_{F/ℚ}T) by the result of Section 3, where I is the set of embedding of F into ℂ and X(T)[I] is the set of formal linear combinations of elements in I with coefficients in the abelian group X(T). In other words, decomposing ∞ = ∞(ℝ) ⨆ ∞(ℂ) for real embeddings ∞(ℝ) and complex embeddings ∞(ℂ), we have associated to each π the sum

χ_{∞}(π) = ∑_{σ∈∞(ℝ)} χ_{σ}σ + ∑_{τ∈∞(ℂ)}(χ_{τ} τ + χ_{cτ} cτ).

Let W be the Weyl group of T in GL(n) and w be the longest element of W. Then t ↦ wt^{−1}w is an automorphism of T preserving the positivity with respect to B. For each character χ : T (𝔯_{p}) → O^{×}, we write χ^{w}(t) = χ(wtw^{−1}). For the determinant composed with the norm character, we write ν : T (F) ⟶^{det} F^{×} ⟶^{norm} ℚ^{×}. Then by the description in Section 3, we get

X_{∞}(U) ⊂ X = C_{n}[I] ⋂ {χ ∈ X(T)[I]|χ^{w} + χ^{c} = [χ]ν with [χ] ∈ ℤ} , (7.1) where complex conjugation c acts on the values of χ so that χ(t)^{c} = c(χ(t)), identifying X(T) = ℤ^{n}, ν corresponds to ∑_{σ} 1σ with 1 = (1, 1, …, 1), and χ^{c} = ∑_{σ∈I} χ_{σ}cσ. Although in Section 3, the property characterizing χ is stated using χ̃ = χ + δ − ν for δ = ∑_{σ}(n, n − 1, …, 1)σ, the description of X does not change since δ^{w} + δ^{c} = (n + 1)ν.

The set X_{∞}(U) may not coincide with X. For example, if n = 1 and F contains no CM fields, X_{∞}(U) = ℤν. However X has rank r_{2}+1 when n = 1. It is interesting to study when X ≠ ⋃_{U} X_{∞}(U) happens for general n > 1.

Let Γ_{T} be the maximal p–profinite subgroup of T (𝔯_{p})/\overline{𝔯^{×}}. Then Λ = O[[Γ_{T}]] is isomorphic to the power series ring of d variables for d = [F : ℚ]n + 1 + δ_{p} − r_{1} − r_{2}. Thus Spec(Λ)(\overline{ℚ}_{p}) is the product of d copies of the open unit disk in \overline{ℚ}_{p}. Suppose that F is either totally real or a CM field. By Langlands functoriality, for sufficiently large α, we expect (as we will see later) to be able to create a non-trivial nearly p–ordinary π of level p^{α} with χ = χ(π) for any given χ ∈ X. If this is the case, by Theorem 6.5, the support on Spec(O[[T (𝔯_{p})]] ⊗_{ℤ} ℚ) of h^{n.ord} ⊗_{ℤ} ℚ contains the closure X of X in Spec(Λ⊗_{ℤ}ℚ). Since the dimension of the closure X is equal to the ℤ_{p}–rank of the p–adic closure of X, which is the rank of the ℤ–module generated by X. Thus if either n > 2 or r_{2} > 0, the linear span of X is smaller than X(T)[I], and hence we can find χ ∈ C_{n}[I] which is never an infinity type of a cohomological automorphic representation. This fact is used to show that the Hecke algebra is a torsion Λ–module in Corollary 6.3 under the condition that n > 2 or r_{2} > 0. In particular, if 3 ≤ n ≤ 4 and F = ℚ, it is enough to conclude dim h^{n.ord} ≤ 3; so, the conjecture holds in this case. Anyway it is easy to see that the rank of X is equal to the upper bound of the dimension in the conjecture, and this gives a reason for the conjecture.

Remark 7.1. It is a part of a much general conjecture of Langlands that for each cuspidal cohomological automorphic representation π of G(𝔸), there exists a pure motive M defined over F with coefficients in a number field E such that L(s, π) = L(s, M_{/F}). Since the infinity type of π determines the gamma factor of L(s, π), we can describe the Hodge type of M in terms of χ_{∞}(π). The outcome is as follows: Write χ(π)_{σ} + δ − 1 = (m_{σ,1}, …, m_{σ,n}). Then the Hodge type of M at the infinite place σ is given by

(m_{σ,1}, m_{cσ,n}), (m_{σ,2}, m_{cσ,n−1}), …, (m_{σ,j}, m_{cσ,n−j+1}), …. (7.2)

Since M is supposed to be pure, the Hodge numbers (p_{i}, q_{i}) of M has to satisfy p_{i} + q_{i} = w for the weight w of M. Thus the description of the infinity type of cohomological representations given above is a consequence of the purity of M, admitting Langlands’ conjecture.

Fix a complex embedding i_{∞} : \overline{ℚ} ↪ ℂ. Now suppose that F is either a CM field or a totally real field. Thus we have well defined complex conjugation ρ acting on F such that c ∘ i_{∞} ∘ σ = i_{∞} ∘ σ ∘ ρ for all σ ∈ I. If F is totally real, ρ is just the identity map of F. Then for T^{∘} = SL(n) ∩ T, we consider the automorphism ι : T^{∘}(𝔯_{p}) → T^{∘}(𝔯_{p}) given by ι(t) = wρ(t)^{−1}w^{−1} for the longest element w of the Weyl group of T. Let Γ^{∘} be the torsion-free part of T^{∘}(𝔯_{p}) and we write Γ_{j}^{∘} = (Γ^{∘})^{p^{j}}. The involution ι : T^{∘}(𝔯_{p}) → T^{∘}(𝔯_{p}) induces an involution ι of Λ_{j} = O[[Γ_{j}^{∘}]] into itself. Then Conjecture 7.1 follows from the Leopoldt conjecture for F and p combined with the following vanishing property of the nearly p–ordinary cohomology groups:

Conjecture 7.2. Suppose that F is either totally real or a CM field. Then for any given open compact subgroup U ⊂ G(𝔸^{(∞)}) with U ⊃ G(ℤ_{p}), there exists j ≥ 0 such that H_{n.ord}^{r}(\overline{Γ}(U_{1,B}(p^{∞})), K/O) is annihilated by the ideal Λ_{j}(ι − 1)Λ_{j}, where r is the bottom degree for G.

For any dominant arithmetic character χ : T^{∘}(𝔯_{p}) → O^{×} we can find j > 0 such that χ|_{Γ_{j}^{∘}} is induced by a dominant rational character χ_{0} in X(T^{∘})[I]. By the definition of ι, the character χ_{0} is in X if and only if the O–algebra homomorphism χ : Λ_{j} → O induced by χ factors through Λ_{j}/Λ_{j}(ι − 1)Λ_{j}. Thus if Conjecture 7.2 holds, then the Pontryagin dual module W^{∘} of H_{n.ord}^{r}(\overline{Γ}(U_{1,B}(p^{∞})), K/O) is supported by Spec(Λ_{j}/Λ_{j}(ι − 1)Λ_{j}), which has relative dimension r(n, F) − (r_{2} + 1) over O. It is then clear from the induction process (6.17) that the Pontryagin dual module W of H_{n.ord}^{r}(X_{1,B}(p^{∞})), K/O) is supported by the spectrum of

(Λ_{j}/Λ_{j}(ι − 1)Λ_{j})[[Cl_{F} (p^{∞})]],

whose relative dimension over O is r(n, F) + δ_{p} for the defect of the Leopoldt conjecture δ_{p} of F.

When F is neither a CM field nor totally real, the situation is rather murky, and even conjecturally, there is no clear-cut description of the annihilator of the cohomology group H_{n.ord}^{r}(\overline{Γ}(U_{1,B}(p^{∞})), K/O) or the Hecke algebra h^{n.ord}(p^{∞}; O) except for CM components. The annihilator is computed for CM components of the Hecke algebra in [H94b] Section 5 when r_{2} = 1 and D is a quaternion algebra positive definite over all real places of F. In this case, the annihilator is directly related to the units of the quadratic extension of F (containing a CM field). We hope to come back this question in a forthcoming paper.

7.2. Compatibility with automorphic functoriality. In the previous subsection, we described how the conjectural dimension of the universal nearly p–ordinary Hecke algebra is deduced by the linear rank of the module of infinity types of cuspidal cohomological automorphic representations. Here we show that the conjectural dimension formula is compatible with some of the Langlands functorialities: tensor products, base-change and automorphic induction.

Since we have a (conjectural) motivic interpretation of cohomological automorphic representations, it would be easier to deal with the set of Hodge weight (m_{σ,1}, …, m_{σ,n}) of motives rather than X. We therefore define for the standard diagonal torus T = T_{n} ⊂ GL(n), under the notation of Remark 7.1,

H_{n} = H_{n/F} = {χ ∈ X(T_{n})[I]|χ^{c} + χ^{w} = [χ]ν} .

Then we associate to each rank n pure motive M a Hodge weight

χ(M) = ∑_{σ}(m_{σ,1}, …, m_{σ,n})σ ∈ H_{n}

if the Hodge type of M is given by (7.2). Then X gives rise to a spanning cone of H_{n} via χ ↦ χ + δ − ν. We call χ = ∑_{σ}(m_{σ,1}, …, m_{σ,n})σ ∈ H_{n} regular if m_{σ,i} ≠ m_{σ,j} for all i ≠ j and σ. We call the motive regular if χ(M) is regular. Let H_{n}^{+} be the set of regular elements in H_{n}. Thus our reason for the conjecture is that regular elements in H_{n} span the full module H_{n} over ℤ, and its rank should give the dimension of h^{n.ord} over K.

Let F/F′ be a finite field extension. We write I_{F} = \{σ : F ↪ \overline{ℚ}\} to distinguish I_{F} and I_{F′}. Starting from rank n regular pure motive M_{/F}, we consider the restriction of scalar Res_{F/F′}M, then

χ(Res_{F/F′}(M)) = ∑_{σ∈I_{F′}} ( ⨁_{τ:Res(τ)=σ} χ_{τ} (M)) σ.

This induces a linear map Res_{F/F′} : H_{n/F} → H_{n[F:F′]/F′}, and the map is obviously injective. Then comparing the rank of the source and the target, we get the following result

Proposition 7.3. For a finite extension F/F′, we have r(n; F) ≤ r(n[F : F′], F′), and the equality holds if and only if one of the following conditions is satisfied:

(1) n is even;

(2) F is totally imaginary;

(3) n[F : F′] is odd, and there is at most one real place of F over each real place of F′.

Proof. Let H = {χ ∈ H_{n}|[χ] = 0}. Then r(n, F) = rank H + 1. When n is even, the assertion is obvious; so, we may assume that n is odd. We write r_{1}(σ) (resp. r_{2}(σ)) for the number of real (resp. complex) places of F over a place σ of F′. Then the contribution of each σ to rank H is given by

mr_{1}(σ) + (2m + 1)r_{2}(σ) = m[F : F′] + r_{2}(σ).

Thus r(n, F) = m[F : ℚ] + r_{2}(F) + 1. Similarly we can compute

r(n[F : F′], F′) = \begin{cases}m[F : ℚ] + ℓ[F′ : ℚ] + 1 & if [F : F′] = 2ℓ, \\ m[F : ℚ] + ℓ[F′ : ℚ] + r_{2}(F′) + 1 & if [F : F′] = 2ℓ + 1. \end{cases}

From this, we conclude r(n; F) ≤ r(n[F : F′], F′) and the equality holds if and only if F satisfies (2) or (3).

There is some hope that assuming the Leopoldt conjecture for the base field, we might be able to prove Conjecture 7.1 over the given field. Anyway this is the case where n ≤ 2 and r_{2} ≤ 1. As we will see in 8.1, as long as χ(Res_{F/F′}(M)) remains regular, automorphic induction for F/F′ preserves near p–ordinarity if all the prime factors of p in F′ split in F. The Hodge type χ(Res_{F/F′}(M)) of the restriction of scalar remains regular for χ(M_{/F}) in a full dimensional cone inside H_{n/F} if F is either a CM field or a totally real field. If F is neither a CM field nor a totally real field, preserving regularity requirement imposes a strong restriction. Anyway, by the above proposition, if automorphic induction exists for GL(n), we can reduce the general (totally real) case of the conjecture to the special case where the Leopoldt conjecture is valid for F′:

Corollary 7.4. Let L be a number field, and we choose a real abelian extension F′/ℚ so that every prime factor of p of F′ splits in F = LF′. Then the Leopoldt conjecture for the number field L follows from Conjecture 7.1 for F′ and p under the following conditions:

(1) L is totally real;

(2) Automorphic induction exists for cuspidal cohomological automorphic representations from GL(2)_{/F} to GL(2[F : F′])_{/F′}.

Proof. It is easy to see that the Leopoldt conjecture for F and p implies that of L and p; so, we may assume that L = F. Let G be the algebraic group Res_{F′/ℚ}(D′)^{×} for a division algebra D′_{/L} satisfying (D1-2) with dim_{F′} D′ = 4[L : F′]^{2}. We write H = Res_{L/ℚ}D^{×} for a quarternion algebra D_{/L} satisfying the following conditions:

(a) D is unramified at every finite place;

(b) D is either totally definite or D ⊗_{ℚ} ℝ ≅ M_{2}(ℝ) × ℍ × ⋯ × ℍ.

Then we look at the universal nearly p–ordinary Hecke algebra h = h^{n.ord}(Np^{∞}; O) for H defined in [H89]. Let Γ be the maximum torsion-free subgroup of T_{H}(ℤ_{p})/\overline{𝔯^{×}} for the p–adic closure \overline{𝔯^{×}} in 𝔯_{p}^{×}. Then rank_{ℤ_{p}} Γ = [L : ℚ] + 1 + δ for the defect of the Leopoldt conjecture δ for (L, p).

Let ℙ be a minimal prime ideal of h and put 𝕀 = h/ℙ. We call an O–algebra homomorphism P : 𝕀 → O arithmetic if P induces a dominant arithmetic character χ_{P} : T_{H}(𝔯_{p}) → O^{×}. The projection h → 𝕀 induces, for each arithmetic O–algebra homomorphism P : 𝕀 → O, an O–algebra homomorphism h/ Ker(χ_{P})h → O, which in turn gives rise to the Hecke eigenvalue system of an automorphic representation π(P) of H(𝔸). For a place 𝔩 of F prime to p, it is known that if π(P)_{𝔩} is special (resp. super-cuspidal) for one point P, then π(P)_{𝔩} is special (resp. super-cuspidal) for all P, because the corresponding Galois representation restricted to the inertia group I_{𝔩} is rigid if 𝔩 ∤ p. For 𝔭|p, π(P)_{𝔭} is always principal except possibly when χ_{P} = 0, for which it could be special. For a given cohomological automorphic representation π of H(𝔸) of cohomological weight 0, if π_{𝔭} for all 𝔭|p is special, then π is automatically p–ordinary. Computing limit multiplicity by the trace formula, we can find π such that π is of cohomological weight 0 and π_{𝔩} is special for a given finite set Σ ∪ {𝔭|p} of primes. Then we can find 𝕀 as above so that π(P) = π for a prime ideal P of 𝕀.

Now we choose Σ so that

(1) every 𝔩 ∈ Σ is totally split in L/F′;

(2) If 𝔩 is a prime ideal of F′ with non-split D′_{𝔩}, Σ contains all places of F over 𝔩.

Then the automorphic representation π(P) has automorphic induction Π(P) in cuspidal (cohomological) automorphic representations of G, because Π(P)_{𝔩} for 𝔩 ∈ Σ is a Steinberg representation and hence square integrable ([B] Section 5).

Let Λ_{j} = O[[Γ^{p^{j}}]]. We choose a regular weight χ ∈ X (which is a square in X(T_{H})) for a p–split torus T_{H} of H so that χ′ = Res_{F/F′}χ is still regular. Then h ⊗_{Λ_{j},χ} K is the Hecke algebra of finite level. We find an open compact subgroup U ⊂ G(𝔸^{(∞)}) and have an algebra homomorphism Π_{j} : h_{χ′}^{n.ord}(U_{1,B}(p^{α(j)}), K) → 𝕀⊗_{Λ_{j},χ}K for suitable level α(j) depending on j such that P ∘Π_{j} corresponds to Π(P) for P factoring through 𝕀 ⊗_{Λ_{j},χ} K. The group U is independent of j if we choose it sufficiently small. Let h′ = h^{n.ord}(U_{1,B}(p^{∞}); O). Since we have a natural O– algebra homomorphism h′ → h_{χ′}^{n.ord}(U_{1,B}(p^{α(j)}), K) induced by the isomorphism in Theorem 6.2, we may pull back Π_{j} to a unique algebra homomorphism Π_{j} : h′ → 𝕀 ⊗_{Λ_{j},χ} K (still denoted by Π_{j}). We have a Galois representation ρ : Gal(\overline{ℚ}/L) → GL_{2}(𝕀) such that T r(ρ(F rob_{𝔩})) = T_{1}(𝔩) for almost all primes 𝔩 of L. Thus Π_{j} actually has values in 𝕀 ⊗_{Λ_{j},χ} O, because the image is generated by the coefficients of the characteristic polynomial of ind_{F}^{F′}ρ(F rob_{𝔩}). Taking the projective limit of Π_{j} with respect to j, we get Π : h′ → 𝕀. Since 𝕀 is generated by Hecke operators T_{1}(𝔩) for almost all primes 𝔩, and image of the map Π is generated by T r(ind_{L}^{F′}ρ(F rob_{𝔩})). From this, it is clear that 𝕀 is an h′–module of finite type. It is known by [H89] that dim 𝕀 = [L : ℚ] + 2 + δ. This combined with Conjecture 7.1 shows

[L : ℚ] + 2 + δ = dim 𝕀 ≤ dim h = [F′ : ℚ][L : F′] + 2.

Thus δ = 0 and the Leopoldt conjecture for L follows.

Remark 7.2. If Conjecture 7.1 is valid for the split GL(n)_{/F′} (instead of G associated to division algebra D′), we can apply the same argument as in the proof of the corollary to GL(1)_{/L} for a CM extension L/F′ and GL([L : F′])_{/F′} in place of H_{/L} and G_{/F′}. The result is the same, that is, the Leopoldt conjecture for (L, p) follows from Conjecture 7.1 for GL([L : F′])_{/F′}. The difficulty of applying the conjecture for G associated to a division algebra is that the automorphic induction image of an arithmetic Hecke character of L has to be super-cuspidal at ramified places of D′, which is impossible if the place split in L/F′ (which is not always but often the case). Anyway the Leopoldt conjecture for a CM field L is equivalent to that for its maximal totally real field; so, this case is basically covered by Corollary 7.4. Anyway, this remark shows that the function (n, F) ↦ r(n, F) is the minimal assignment of dimensions so that Proposition 7.3 holds for the given r(1, F) predicted by the Leopoldt conjecture.

We look into the base-change functoriality from Res_{F′/ℚ}GL(n) to Res_{F/ℚ}GL(n) for a finite extension F/F′. Thus the functoriality is induced by π(M_{/F′}) ↦ π(M_{/F}) for pure regular motive M_{/F′} of rank n. The corresponding linear map of the infinity types of π (or Hodge types of M) is the the inflation map Inf_{F/F′} : H_{n/F′} → H_{n/F} given by Inf_{F/F′}(χ)_{σ} = χ_{σ|_{F′}}. This is a representation theoretic dual of Res_{F/F′}. Since H_{n/F} is naturally a Galois module, we see easily that Im(Inf_{F/F′}) = H^{0}(Gal(\overline{ℚ}/F′), H_{n/F′}), and Inf_{F′/F} is obviously injective. We see that r(2, F) − r(1, F) = r_{1} + r_{2}. Thus we have

Proposition 7.5. We have r(n, F) ≥ r(n, F′) for a finite extension F/F′. If F is a CM field with maximal real subfield F′, r(2, F) − r(1, F) = r(2, F′) − r(1, F′). Thus under the Leopoldt conjecture for F′ and p, Conjecture 7.1 for (F, n = 2, p) implies that of (F′, n = 2, p).

We consider the linear map ⊞ : H_{n/F} ⊕ H_{ℓ/F} → H_{nℓ/F} given by (m_{i}) ⊕ (n_{j}) ↦ (m_{i} − n_{j})_{0≤i≤n,0≤j≤ℓ} fixing an order of the indices (i, j). This is associated to

χ(M) ⊕ χ(N) ↦ χ(M ⊗ Ň)

for the dual motive Ň of N. It is easy to see that Ker(⊞) = (∑_{σ} ℤ(1_{n}, 1_{ℓ})σ) ∩

(H_{n} ⊕ H_{ℓ}), where 1_{n} = \overbrace{(1, 1, …, 1)}^{n}. Take x = ∑_{σ} x_{σ}(1_{n}, 1_{ℓ})σ ∈ Ker(⊞). Then we have x^{c} +x^{w} = ∑_{σ}(x_{σ} +x_{cσ})(1_{n}, 1_{ℓ})σ = [x]ν. This shows 2x_{σ} = [x] for real place σ and x_{σ} + x_{cσ} = [x] for complex places σ. Therefore we have rank_{ℤ} Ker(⊞) = r_{2} + 1, and

Proposition 7.6. Suppose that n ≥ ℓ ≥ 1. We have

rank_{ℤ} Im(⊞) = r(n; F) + r(ℓ : F) − r_{2} − 1 ≤ r(ℓn; F),

and the equality holds only when either n = ℓ = 2 and r_{2} = 0 or ℓ = 1.

Iterating the above process, we can think of χ_{1} ⊞ ⋯ ⊞ χ_{k} ∈ H_{n_{1}...n_{k}/F} for (χ_{1}, … χ_{k}) ∈ H_{N_{1}/F} ⊕ ⋯ ⊕ H_{n_{k}/F}. This map corresponds to the tensor product of k–motives: (M_{1}, …, M_{k}) ↦ M_{1} ⊗ M_{2} ⊗ ⋯ ⊗ M_{k}, and by induction on k, we get:

rank_{ℤ}(Im((χ_{1}, … χ_{k}) ↦ χ_{1} ⊞ ⋯ ⊞ χ_{k})) ≤ r(n_{1} × n_{2} × ⋯ × n_{k}; F).

Thus our conjecture is compatible with tensor functoriality.

8. Ordinarity condition via Newton polygon

In this section, we study what admissible representations of GL_{n}(F_{𝔭}) give rise to the p–component of a P –nearly ordinary automorphic representation. We fix an algebraic closure \overline{ℚ}_{p} of ℚ_{p}. Then we fix two embeddings i_{∞} : \overline{ℚ} ↪ ℂ and i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p}. For each field embedding σ : F ↪ \overline{ℚ}, we write ∞_{σ} for the infinite place associated to i_{∞} ∘ σ : F → ℂ and 𝔭_{σ} for the p–adic place associated to i_{p} ∘ σ.

8.1. Newton polygon and Hodge polygon. We fix one embedding σ : F ↪ \overline{ℚ} and an irreducible cohomological automorphic representation π of G(𝔸) of infinity type χ. We recall the (modified) induced module I_{B}^{G}(λ) = Ind_{B(F_{𝔭σ})}^{G_{𝔭σ}}λ̃ defined in

Section 5 for a continuous character λ : T (F_{𝔭_{σ}}) → \overline{ℚ}^{×} (with respect to the discrete topology on \overline{ℚ}): Writing λ(diag(t_{1}, …, t_{n})) = ∏_{j=1}^{n} λ_{j}(t_{j}), the modified character λ̃ is given by

λ̃(diag(t_{1}, …, t_{n})) = ∏_{j=1}^{n} λ_{j}(t_{j})|t_{j}|_{𝔭_{σ}}^{j−1},

where we normalize the 𝔭_{σ}–adic absolute value so that |ϖ|_{𝔭_{σ}} = |𝔯_{𝔭_{σ}}/ϖ𝔯_{𝔭_{σ}}|^{−1} for a prime element ϖ ∈ 𝔯_{𝔭_{σ}}. We assume that

(1) 𝔭_{σ} ∉ Σ and π_{𝔭_{σ}} ↪ I_{B}^{G}(λ) for a continuous character λ : T (F_{𝔭_{σ}}) → \overline{ℚ}^{×};

(2) π_{∞} is cohomological and associated to a dominant character χ ∈ X as in

3.2 and 3.3. When π is nearly p–ordinary with respect to B, we know that the condition (1) is satisfied by (5.5). We write [𝔭_{σ}] for the set of embedding τ : F ↪ \overline{ℚ} such that 𝔭_{τ} = 𝔭_{σ}.

We define the (reciprocal) Hecke polynomial of π by

H_{𝔭_{σ}}(T) = ∏_{i=1}^{n}(1 − λ_{i}(ϖ)T) = ∑_{s=0}^{n}(−1)^{s}|ϖ|_{𝔭_{σ}}^{−s(s−1)/2}a_{s}T^{s}. (8.1)

By [BZ1] 2.9, if π_{𝔭_{σ}} ↪ I_{B}^{G}(λ′) for another character λ′, then λ′ = λ^{w} for an element w ∈ W; so, H_{𝔭_{σ}}(T) is well defined independent of the choice of λ. When π_{𝔭_{σ}} is unramified (and hence π_{𝔭_{σ}} ≅ I_{B}^{G}(λ)), this polynomial is the Hecke polynomial of the Hecke operators T_{j}(𝔭) (without modification, see (6.6)) as in [Sh] Theorem 3.21. However, when π_{𝔭_{σ}} is ramified, the above polynomial differs from the one constructed in [Sh] by these Hecke operators.

Let V = V (π_{𝔭_{σ}}), and write V_{B}^{∘} for the semi-simple part for 𝕋_{B} = ∏_{j=1}^{n} T_{j}(ϖ) as in Section 5. Then as seen in Section 5, V_{B}^{∘} is canonically isomorphic to the Jacquet module V_{B}, which is a semi-simple T –module. Write V_{B}^{∘}[λ̃] ⊂ V_{B}^{∘} for the space corresponding to the λ̃–eigensubspace V_{B}[λ̃] ⊂ V_{B}. Then V_{B}^{∘}[λ̃] is an eigenspace of T_{s}(ϖ), and its eigenvalue is given by |ϖ|_{𝔭_{σ}}^{s(s−1)/2} ∏_{j=1}^{s} λ_{n−s+j}(ϖ) (see Corollary 5.5).

Let P be a standard parabolic subgroup associated with a partition n = n_{1} + n_{2} + ⋯ + n_{r} of n into r–parts. Define a tuple S = (0 = s_{0} ≤ s_{1} ≤ s_{2} ≤ ⋯ ≤ s_{r}) of r + 1 integers by n_{r−i} = s_{i+1} − s_{i} for all i = 0, ⋯, r. We take an embedding σ : F ↪ \overline{ℚ} and write e = e_{σ} for the ramification index of F_{𝔭_{σ}}/ℚ_{p}. Then for the cohomological weight χ = ∑_{τ} (m_{τ,1}, m_{τ,2}, …, m_{τ,n})τ ∈ X ⊂ X(T)[I] of π (thus m_{τ,1} ≥ m_{τ,2} ≥ ⋯ ≥ m_{τ,n}), we define the Hodge P –polygon Δ_{P} = Δ_{P,σ} by the convex hull of the following vertices:

{(s_{σ,i}, e_{σ}^{−1} ∑_{τ∈[𝔭_{σ}]} ∑_{j=n−s_{i}+1}^{n}(m_{τ,j} + n − j))|i = 0, 1, …, r} . (8.2)

In particular, the slope μ_{σ,i} = μ_{i} of the i-th edge of the polygon Δ_{P,σ} is given by

μ_{σ,i} = \frac{∑_{j=n−s_{i}+1}^{n−s_{i−1}} ∑_{τ∈[𝔭_{σ}]}(m_{τ,j} + n − j)}{e_{σ}n_{r−i+1}}. (8.3)

On the other hand, we define the Newton P –polygon Δ_{P,σ}^{N} of π by the convex hull of the vertices over S (in the horizontal x–axis) of the Newton polygon of H_{𝔭_{σ}}(T), which is above or on the Newton polygon of H_{𝔭_{σ}}(T) and coincides with the Newton polygon only when P = B.

8.2. Newton polygon is above Hodge polygon. We keep the notation introduced in the previous subsection. Let S_{σ} = (s_{0} = s_{σ,0}, …, s_{r} = s_{σ,r}) be the maximal tuple of integers in increasing order such that

|i_{p}(a_{s})|_{p} = |i_{p} ( ∏_{τ∈[𝔭_{σ}]} χ_{τ} (ξ_{s}))|_{p}

for the p–adic absolute value | |_{p} of \overline{ℚ}_{p}, where ξ_{s} = diag(1, …, 1, \overbrace{ϖ, …, ϖ}^{s}) ∈ GL_{n}(F_{𝔭_{σ}}). In particular, we have s_{0} = 0 and s_{r} = n because a_{0} = 1.

From the data S_{σ}, we would like to determine the parabolic subgroup P which is minimal among parabolic subgroups Q so that π is Q–ordinary. Let n = n′_{1} + n′_{2} + ⋯ + n′_{r′} be the partition associated to Q. Define a tuple S_{Q} = (s′_{i})_{i=0,...,r′} by n′_{r′−i} = s′_{i+1} − s′_{i} for all i = 0, …, r′ − 1 with s′_{0} = 0. Since a_{s} is the sum of eigenvalues of T_{s}(ϖ) on V_{B}^{∘} ≅ V_{B} (by Corollary 5.5), we conclude from (6.6) and (6.7) that

(8.4) π is Q–ordinary ⟺ S_{Q} ⊂ S_{σ}.

Here we remark that ω in (6.6) and (6.7) is given by χ^{−1} by our definition of the algebraic induction (6.4). Then we have

Theorem 8.1. Let the assumption and the notation be as above. Then we have

(1) If the cohomological automorphic representation π of infinity type χ ∈ X is nearly Q–ordinary at 𝔭_{σ}|p for a standard parabolic subgroup Q, then the Newton Q–polygon of H_{𝔭_{σ}}(T) coincides with the Hodge Q-polygon Δ_{Q,σ} defined above;

(2) If π_{𝔭_{σ}} ≅ I_{B}^{G}(λ), the parabolic subgroup P determined by S_{σ} is the smallest standard parabolic subgroup for which π is nearly P –ordinary at 𝔭_{σ};

(3) For every irreducible cohomological automorphic representation π with π_{𝔭_{σ}} embedded into I_{B}^{G}(λ) and having infinity type χ, the Newton polygon of the Hecke polynomial H_{𝔭_{σ}}(T) is on or above the Hodge B–polygon Δ_{B,σ}.

Proof. We have already proven the assertion (1) and (2). By the definition of the idempotent e_{B}, we know that

|i_{p}(a_{s})|_{p} ≤ |i_{p} ( ∏_{τ∈[𝔭_{σ}]} χ_{τ} (ξ_{s}))|_{p} .

This shows that Δ_{B} is under or on the Newton polygon of H_{𝔭_{σ}}(T).

Remark 8.1. A general conjecture (due to Langlands) is that for each cohomological cuspidal automorphic representation π, there should exist a pure motive M_{/F} of rank n such that L(s, π) = L(s, M). Since we can write down the Γ–factor of L(s, π) explicitly by the infinity type of π, we can predict the Hodge type of M ×_{F,i_{∞}σ} ℂ, which would be given by

((m_{σ,1}, m_{σ,n}), (m_{σ,2}, m_{σ,n−1}), …, (m_{σ,j}, m_{σ,n−j+1}), …) . (8.5)

Thus if p totally splits in F, Δ_{B} is exactly the Hodge polygon of M at the infinite place ∞_{σ}. If π is unramified at 𝔭_{σ}, we expect that M is crystalline at 𝔭_{σ}. Then the crystalline realization of M at 𝔭_{σ} has the crystalline Frobenius map Φ. Although Φ is not a linear map, its power ϕ = Φ^{[𝔯/𝔭_{σ}:𝔽_{p}]} has a well defined characteristic polynomial H_{cris,σ}(T) of degree n. A standard conjecture is that the crystalline characteristic polynomial H_{cris,σ}(T) should coincide with the characteristic polynomial H_{𝔭_{σ}}(T) of the Frobenius at 𝔭_{σ} of an 𝔩–adic étale realization of M (𝔩 ∤ p). Thus by a well known result of Fontaine and Mazur, if p completely splits in F, the Newton polygon of H_{𝔭_{σ}}(T) = H_{cris,σ}(T) is on or above the Hodge polygon Δ_{B}. This gives a philosophical explanation of the above theorem 8.1. The above theorem is hence a bit stronger than this geometric fact, because (i) Δ_{B} is really above the Hodge polygon if p does not completely split in F, and (ii) the theorem also gives an information even when π ramifies at 𝔭_{σ} (which should corresponds to a non-crystalline motive).

Since near p–ordinarity of a cuspidal cohomological automorphic representation (with respect to the Borel subgroup) implies that π_{𝔭_{σ}} for all 𝔭_{σ} is a subrepresentation of an induced representation, as a direct consequence of the above theorem, we see

Corollary 8.2. For cuspidal cohomological automorphic representations, we write tensor product functoriality as π(M) ⊠ π(M′) = π(M ⊗ M′) from GL(m) × GL(ℓ) to GL(nℓ), automorphic induction functoriality as π(M) ↦ π(Res_{F/F′}M) from Res_{F/ℚ}GL(n) to Res_{F′/ℚ}GL([F : F′]n) and base-change functoriality as π(M_{/F′}) ↦ π(M_{/F}) for a finite extension F/F′. Then, as long as the functorial image remains cohomological (that is, the infinity type of the image is regular), automorphic induction preserves near p–ordinarity (with respect to B) provided that all prime factors of p in F′ split in F, and tensor product and base-change preserve near p–ordinarity (with respect to B) unconditionally.

The following corollary guarantees semi-simplicity of the nearly p–ordinary Hecke algebra:

Corollary 8.3. Let N be the unipotent radical of B. Suppose that π is nearly ordinary at 𝔭_{σ} with respect to B. Then e_{B}H^{0}(N(𝔯_{𝔭_{σ}}), V (π_{𝔭_{σ}})) is a one dimensional space on which B(F_{𝔭_{σ}}) acts by a character λ such that |λ|_{p} = |∏_{τ∈[𝔭_{σ}]} χ_{τ} ν_{τ}^{−1}δ_{τ} |_{p},

where δ = ∑_{σ}(n, n − 1, …, 1)σ and ν = ∑_{σ} 1_{n}σ.

Proof. By near B–ordinarity, we have from the argument in Section 5 that π_{𝔭_{σ}} is a subquotient of I_{B}^{G}(λ) for a character λ : T (F_{𝔭_{σ}}) → \overline{ℚ}^{×}. By near B–ordinarity, we know from the theorem that we can choose λ so that |λ|_{p} = |∏_{τ∈[𝔭_{σ}]} χ′_{τ} |_{p} for

χ′ = χν^{−1}δ. Writing χ′_{τ} = (m_{τ,1}, …, m_{τ,n}), we have m_{τ,1} > m_{τ,2} > ⋯ > m_{τ,n} for all τ ∈ [𝔭_{σ}]. Thus if w ≠ 1,

∏_{s=1}^{n} |λ^{w}(ξ_{s})|_{p} < | ∏_{τ∈[𝔭_{σ}]} ∏_{s=1}^{n} χ′_{τ} (ξ_{s})|_{p} .

In particular, λ^{w} for w ∈ W are all distinct. Then by Proposition 5.4, we know that e_{B}V_{B} ⊂ ⨁_{w∈W} λ̃^{w} as B(F_{𝔭_{σ}})–modules for V = V (π_{𝔭_{σ}}), and as we have already seen in Corollary 5.5 and (4.6), for λ̃^{w}–eigenvector v ∈ V_{B}, we have

v|𝕋_{s}(ϖ) = ( ∏_{τ∈[𝔭_{σ}]} χ_{τ} (ξ_{s}))^{−1} |ϖ|_{𝔭_{σ}}^{−ts}λ̃^{w}(ξ_{s})v = (|λ(ξ_{s})|_{p}^{−1}λ^{w}(ξ_{s})) v.

Thus from |λ^{−1}λ^{w}(ξ_{s})|_{p} < 1 for at least one s if w ≠ 1, we conclude that e_{B} kills the λ̃^{w}–eigenspace if w ≠ 1. Therefore e_{B}V_{B} is at most 1–dimensional and actually is equal to the λ̃–eigenspace under the above isomorphism. Since V_{B}^{∘} ≅ V_{B} as R(B, Δ_{∞})–modules (Proposition 5.1), this shows the desired assertion.

Appendix A. Semi-simplicity of Hecke algebras for reductive groups

We can generalize from cuspidal cohomology of GL(n) to interior cohomology of general split reductive groups the argument which proves semi-simplicity of the nearly ordinary Hecke algebra of p–power level, which we describe here.

Let G_{0/F} be a connected reductive group over a number field F, and we put G = Res_{F/ℚ}G_{0}. If G_{0} is split over F_{p} = F ⊗_{ℚ} ℚ_{p}, we shall prove semi-simplicity of the commutative Hecke algebra acting on the nearly ordinary cohomology group H_{n.ord}^{q}(X(U), L) ⊂ H_{!}^{q}(X(U), L) for a modular variety X(U) associated to an arbitrary p–power level open compact subgroup U of G(𝔸^{(∞)}). Here the locally constant sheaf L on X(U) is associated to a rational representation of G, and H_{!}^{q} indicates the image of the compactly supported cohomology group in the cohomology group without the support condition.

There could be two ways of proving the semi-simplicity. The first one is a bit farfetched: We interpolate p–adically the cohomology group H_{n.ord}^{q}(X(U), L) varying rational representations L, getting a space V, which is a module of finite type over the Iwasawa algebra Λ = O[[T (ℤ_{p})]] of the torus T = Res_{F/ℚ}T_{0} for an F_{p}–split torus T_{0} ⊂ G_{0}. Then one proceed to prove that V_{χ} = V ⊗_{Λ,χ} O for each dominant weight χ is isomorphic to H_{n.ord}^{q}(X(U), L) for the rational representation of highest weight χ^{−1} and the maximal compact subgroup U ⊂ G(𝔸^{∞}). If we find densely populated χ (in Spec(Λ)(O)) with the above specialization property, the Hecke algebra h of V (over Λ) specializes along χ to the semi-simple one of level 1, and hence h must be reduced. Then by specializing h to p–power level Hecke algebra, we get the desired semi-simplicity for almost all such specializations. This method has been exploited for GSp(2g) in [TU] and probably works well for reductive groups G yielding Shimura varieties. However this method is ineffective to prove semisimplicity for all χ and all p–power level. Also this method is probably not feasible for general G whose modular variety does not have complex structure, because in such cases, h is a torsion Λ–module.

We should emphasize here semi-simplicity (or unramifiedness) at arithmetic primes of the universal nearly ordinary Hecke algebra h is important in constructing p–adic L–functions on the spectrum Spec(h) and relating its values with complex L–values of automorphic L–functions.

In earlier works of the author, the semi-simplicity of such Hecke algebras for GL(2)_{/F} is proven using the theory of old and new forms. A key point of this method is to prove one-dimensionality of nearly ordinary vectors in each irreducible (cohomological) automorphic representation, carefully analyzing old vectors. The purpose of this paper is to prove directly the semi-simplicity for all arithmetic characters χ : T (ℤ_{p}) → \overline{ℚ}_{p}^{×} and all reductive G split at p. We prove via the theory of Jacquet modules of local automorphic representations ([BZ] and [BZ1]) that the nearly ordinary vector is unique up to constant multiple if the representation is irreducible. This is a generalization of the argument in Section 5 to general reductive groups. Since the automorphic representation occurring in the cohomology group is unramified outside p, the one-dimensionality gives rise to the semi-simplicity. Our final result is Corollary A.4.

A.1. Jacquet modules for reductive groups. Let G be a split connected reductive group over a finite extension F of ℚ_{p}. Let π be an admissible representation of G(F) on a vector space V over a field K of characteristic different from p. We suppose that G acts on V from the right. Fix a Borel subgroup B with split torus T = B/N for the unipotent radical N. A parabolic subgroup P ⊃ B is called standard. We fix a standard parabolic subgroup P with unipotent radical N_{P}.

Since the characteristic of K is different from p, we have a Haar measure of N_{P} (F). We then define

V (P) = V (P, π) = {v − vπ(n) ∈ V (π)|v ∈ V (π), n ∈ N_{P} (F)} ,

and put V_{P} = V_{P} (π) = V/V (P), which is called the Jacquet module. We take a sufficiently large open compact subgroup U_{w} ⊂ N(F) for each w = v − vπ(n) ∈ V (P) so that n ∈ U_{w}. Then we see that ∫_{U} vπ(u)du = 0 for every open subgroup U of N(F) containing U_{w}. Write U_{0} = N(V) for the p–adic integer ring V ⊂ F, and choose the Haar measure du so that ∫_{U_{0}} du = 1. We also choose an increasing sequence of open compact subgroups U_{i} indexed by 0 ≤ i ∈ ℤ such that ⋃_{i} U_{i} = N(F). Then the map

v ↦ lim_{i→∞}(U_{i} : U_{0})^{−1} ∫_{U_{i}} vπ(u)du

gives rise to a section of V ↠ V_{P}. Thus V ≅ V_{P} ⊕ V (P) canonically, and the association V ↦ V_{P} is an exact functor from the category of admissible representations of G(F) into the category of admissible T (F)–modules.

Proposition A.1. Suppose K has characteristic 0 and that π is absolutely irreducible. Let W be the Weyl group of T in G. Then we have dim_{K} V_{B} ≤ |W | for each Borel subgroup B.

Proof. By extending scalar, we may assume that K is algebraically closed. If V_{B} = 0, there is nothing to prove. Thus we suppose that V_{B} ≠ 0. Then we can find a character λ : T (F) → K^{×} which gives the representation of T = B/N_{B} on a simple factor of V_{B}. Then π is a factor of Ind_{B(F)}^{G(F)} λ by Frobenius reciprocity [BZ1] 2.3. By [BZ1] 2.8, the length of the composition series of V_{B} as T (F)–modules is then bounded by |W |. Since T is abelian, this shows that dim_{K} V_{B} ≤ |W |.

For each character λ : T (F) → K^{×} and w ∈ W, we write λ^{w} for another character of T given by λ^{w}(t) = λ(wtw^{−1}).

Corollary A.2. Suppose that V_{B}[λ̃] ≠ 0, where λ̃ = δ_{B}^{1/2}λ for the right module character δ_{B} on B. Then π ⊂ Ind_{B(F)}^{G(F)} λ̃. If λ^{w}(t) for w ∈ W are all distinct,

V_{B} ⊂ ⨁_{w∈W} λ̃^{w} as T (F)–modules.

The proof is exactly the same as that of Proposition 5.4; so, we leave it to the reader.

A.2. Double coset algebras. Let

D = {x ∈ T (F)|xN_{B}(V)x^{−1} ⊃ N_{B}(V)}

be the expanding sub-semigroup of T (F). When G = GL(n), this semi-group is exactly equal to the one D_{B} defined in (4.1). Let Σ be the set of simple positive roots of T with respect to B. Then for each subset Θ ⊂ Σ, we have a parabolic subgroup P_{Θ} = BW_{Θ}B for the subgroup W_{Θ} of the Weyl group W generated by reflections associated to Σ−Θ. Traditionally the above standard parabolic subgroup is denoted by P_{Σ−Θ} (e.g. [J] II.1.8), but we use the symbol P_{Θ} for that. Write L_{Θ} for the Levi subgroup of P_{Θ} and write Z_{Θ} for the identity connected component of the center of L_{Θ}. We consider the orthogonal complement

X_{∗}(Θ) = {ξ ∈ X_{∗}(T)|θ(ξ) = 0 ∀θ ∈ Σ − Θ}

in X_{∗}(T) = Hom_{alg−gp}(G_{m}, T). Then X_{∗}(Θ) = X_{∗}(Z_{Θ}) and rank_{ℤ} X_{∗}(Θ) = |Θ| + rank Z for the center Z ⊂ G. For each α ∈ Σ, we therefore find a unique generator ξ_{α} ∈ X_{∗}({α}) modulo X_{∗}(Z) so that ξ_{α}(ϖ) ∈ D for a prime element ϖ ∈ V. Then D is generated by ξ_{α}(ϖ) for α ∈ Σ and T (V)Z(F). We write D_{0} for the sub-semigroup of D generated by {ξ_{α}(ϖ)|α ∈ Σ} and Z(F). We write simply P_{α} = P_{{α}} and L_{α} = L_{{α}}.

We consider Δ = Δ_{B} = B(V)DB(V), which is a sub-semigroup of G(F). For simplicity, we write B for B(V) and N for N_{B}(V). Since D commutes with T, for ξ ∈ D,

B = ⨆_{u∈N/ξ^{−1}Nξ}(ξ^{−1}Bξ ∩ B)u and N = ⨆_{u∈N/ξ^{−1}Nξ}(ξ^{−1}Nξ ∩ N)u.

This shows

(A.1) NξN = ⨆_{u∈N/ξ^{−1}Nξ} Nξu = ⨆_{u∈ξNξ^{−1}/N} Nuξ and

BξB = ⨆_{u∈N/ξ^{−1}Nξ} Bξu = ⨆_{u∈ξNξ^{−1}/N} Buξ.

In particular, NξN/N ≅ BξB/B.

We now consider the double coset algebra R = R(N, Δ) spanned over ℤ by NxN for x ∈ D. We let R act on v ∈ V^{N} = H^{0}(N(V), V) by

v|[NξN] = ∑_{u∈N/ξ^{−1}Nξ} vπ(ξu) = ∫_{ξNξ^{−1}} vπ(ξ)du. (A.2)

Let {ξ_{α}(ϖ)}_{α∈Σ} be the generators of D modulo center. Then for ξ = ∏_{α} ξ_{α}, we have

N(F) = ∪_{j=0}^{∞}ξ^{j}Nξ^{−j}.

Thus writing T (ξ) = NξN as an operator on V, we see easily from (A.1) that T (ξ^{j}) = T (ξ)^{j} and T (ξ) is nilpotent on V (B)^{N} by (A.2).

Let ^{t}N be the opposite unipotent subgroup of N. We put U_{1}(r) = ^{t}N(r)T (r)N, where X(r) is the kernel of the reduction map: X(V) → X(V/ϖ^{r}V) for an algebraic group X_{/V}. Then U_{1}(r) is a decreasing sequence of open compact subgroups of G(F) with ∩_{r}U_{1}(r) = N and U_{1}(r)ξU_{1}(r) = ⨆_{u∈ξNξ^{−1}/N} U_{1}(r)uξ for ξ ∈ D. Thus V^{N} = ⋃_{r} H^{0}(U_{1}(r), V), and the finite dimensional space H^{0}(U_{1}(r), V) is stable under T (ξ). Thus we can decompose T (ξ) = s+n for commuting sum of a unique nilpotent operator n and a unique semi-simple operator s first on each H^{0}(U_{1}(r), V) and then over the union V^{N} = ⋃_{r} H^{0}(U_{1}(r), V). Thus we find V^{N} = V (B)^{N} ⊕ s(V^{N}) and the projection V ↠ V_{B} is injective on s(V^{N}).

Since vπ(ξ^{−j}) is N–invariant for a sufficiently large j, the projection V^{N} → V_{B} is surjective. Thus s(V^{N}) ≅ V_{B}, we get a R–linear isomorphism

V^{N} ≅ V (B)^{N} ⊕ V_{B}.

Let λ : T (F) → K^{×} be a character intervening in the T (F)–module V_{B}. Since T is abelian, we can always find a subspace V_{B}[λ] on which T (F) acts via the character λ. We write Ad_{N} for the adjoint representation of Res_{F/ℚ_{p}}(T) on the Lie algebra of Res_{F/ℚ}(N). Then det(Ad_{N}(x)) = ρ^{2}(x) for the half sum of positive roots ρ with respect to Res_{F/ℚ_{p}}(B). Since N acts trivially on V_{B}, we have

v|[NxN] = [N : x^{−1}Nx]λ(x)v = | det(Ad_{N}(x))|_{p}λ(x)v = |ρ^{2}(x)|_{p}λ(x)v, (A.3)

where “| |_{p}” is the standard p–adic absolute value such that |p|_{p}^{−1} = p and Ad is the adjoint representation of Res_{F/ℚ_{p}}(T) on the Lie algebra of Res_{F/ℚ_{p}}(N).

A.3. Rational representations of G. We now suppose that F is a number field and G_{0} is a reductive group defined over F split at F_{𝔭} for all primes 𝔭|p. We then consider G = Res_{F/ℚ}G_{0}. We suppose that G_{0} is actually defined over O_{p} = O_{F} ⊗_{ℤ} ℤ_{p} as a smooth (relatively connected) split group scheme over O_{p}, where O_{F} is the integer ring of F. Then G is a generic fibre of Res_{O_{p}/ℤ_{p}}G_{0/O_{p}}. We sometimes write G(ℤ_{p}) and G_{0}(O_{𝔭}) for ℤ_{p}–points (resp. O_{𝔭}–points) of these schemes defined over p–adic integers. We fix a split Borel subgroup B_{0} ⊂ G_{0/O_{p}} with unipotent radical N_{0}, and write B_{𝔭} = B_{0}(F_{𝔭}) whose unipotent radical N_{𝔭} is given by N_{0}(F_{𝔭}). We define B = Res_{O_{p}/ℤ_{p}}B_{0}, N = Res_{O_{p}/ℤ_{p}}N_{0} and T = B/N = Res_{O_{p}/ℤ_{p}}T_{0} for T_{0} = B_{0}/N_{0}. Let G(ℤ̂) ⊂ G(𝔸^{∞}) denote a maximal compact subgroup (by abusing notation) as maximal as possible (this means that we assume a local component of G(ℤ̂) to be hyperspecial if one need to assume it in order to assure that the spherical representation has only one fixed vector under the maximal compact; see [T] 3.8 and [Ca] III). We assume that the p–component of G(ℤ̂) is given by G(ℤ_{p}).

We write D_{𝔭} (resp. D_{0,𝔭}) for D (resp. D_{0}) with respect to B_{𝔭} and the split torus T_{𝔭} = B_{𝔭}/N_{𝔭}. Let p be the Jacobson radical of O_{p} = O_{F} ⊗_{ℤ} ℤ_{p} for the integer ring O_{F} of F. Thus p = ∏_{𝔭|p} 𝔭O_{p}. Define

U_{0}(r) = {u ∈ G(ℤ_{p})|u mod p^{r} ∈ B_{0}(O/p^{r}O)} (A.4)

U_{1}(r) = {u ∈ G(ℤ_{p})|u mod p^{r} ∈ N_{0}(O/p^{r}O)} .

We fix a subgroup S such that U_{0}(r) ⊃ S ⊃ Z(ℤ_{p})U_{1}(r) for r > 0. Then

(A.5) Δ = Δ_{p} = U_{0}(r)(∏_{𝔭|p} D_{0,𝔭})U_{0}(r) = U_{0}(r)(∏_{𝔭|p} D_{𝔭})U_{0}(r) is a semi-group.

We consider for a topological module A, the space of continuous functions: C(A) = C_{S}(A) = {ϕ : S/Z(ℤ_{p})N(ℤ_{p}) → A}. We would like to make C(A) a left Δ_{p}^{−1}–module for the opposite semi-group Δ_{p}^{−1}. For that, we first define a left action of Δ_{p} on Y_{S} = S/Z(ℤ_{p})N(ℤ_{p}). Since S acts on Y_{S} = S/Z(ℤ_{p})N(ℤ_{p}) from left and U_{0}(r) = ST (ℤ_{p}), we only need to define a left action of D_{0} = ∏_{𝔭|p} D_{0,𝔭}. Pick y ∈ S and consider yN(ℤ_{p}). Then for d ∈ D_{0}, dyN(ℤ_{p})d^{−1} = dyd^{−1}dN(ℤ_{p})d^{−1} ⊂ dyd^{−1}N(ℚ_{p}) and dyd^{−1}N(ℚ_{p}) is well defined in G(ℚ_{p})/Z(ℤ_{p})N(ℚ_{p}). Writing ^{t}N for the opposite unipotent subgroup of G, we have the Iwahori decomposition U_{0}(r) = U′T (ℤ_{p})N(ℤ_{p}) and S = U′T′N(ℤ_{p}) for open subgroups U′ ⊂ ^{t}N(ℤ_{p}) and T′ ⊂ T (ℤ_{p}). Then we see dU′d^{−1} ⊂ U′ and dN(ℤ_{p})d^{−1} ⊃ N(ℤ_{p}) by the definition of D. This shows that the coset dyd^{−1}N(F) ∩ S is well defined single coset of N(ℤ_{p}), which we designate to be the image of the action of d ∈ D_{0}. This action extends to that of the semi-group Δ by an obvious way. We now let Δ_{p}^{−1} act on C_{S}(A) by dϕ(y) = ϕ(d^{−1}y). In this way, C_{S}(A) becomes a Δ_{p}^{−1}–module.

We now fix a finite extension K over ℚ_{p} which contains all conjugates of F in \overline{ℚ}_{p}. Let O be the p–adic integer ring of K. We now assume that A is either an O–module of finite or co-finite type or a vector space over K. We put the p–adic topology on module of finite type and vector spaces over K and the discrete topology on module of co-finite type. We consider the quotient scheme G/N defined over K, and its structure sheaf O_{G/N}. We consider the algebraic induction module:

L(χ; K) = {(ϕ : G/N → K) ∈ H^{0}(G/ZN, O_{G/N})|ϕ(yt) = χ(t)ϕ(y) ∀t ∈ T } , (A.6)

where χ ∈ X(T) = Hom_{alg−gp}(T, G_{m}). We let G act on L(χ; K) by gϕ(y) = ϕ(g^{−1}y). Then L(χ; K) = Ind_{B}^{G} χ^{−1}, which is the induction in the category of scheme theoretic representations (that is, polynomial representations). We write this representation as ρ_{χ} : G → GL(L(χ; K)).

Suppose that χ is trivial on the center for the moment. We restrict functions in L(χ; K) to Y_{S} = S/Z(ℤ_{p})N(ℤ_{p}) and get an embedding L(χ; K) ↪ C_{S}(K). The image is stable under the action of Δ_{p}^{−1} and for ξ ∈ D^{−1} and u ∈ N(ℤ_{p}),

(uξ)ϕ(y) = ϕ(ξ^{−1}u^{−1}y) = χ(ξ)ρ((uξ))ϕ(y) for ϕ ∈ L(χ; K).

When we call L(χ; K) a Δ_{p}^{−1}–module, we take the action induced by C_{S}(A). When we call L(χ; K) a G–module, we take the action given by ρ_{χ}. The two actions differ by scalar. We then define L_{S}(χ; O) = C_{S}(O) ∩ L(χ; K), which is a Δ_{p}^{−1}–module (but not a G(ℚ_{p})–module). By definition, the action of Δ_{p}^{−1} factors through Δ_{p}^{−1}/Z(Δ_{p}^{−1}) for the center Z(Δ_{p}^{−1}) of Δ_{p}^{−1}. We then define L_{S}(χ; K/O) = L_{S}(χ; O) ⊗_{O} K/O. For each character ε : T (O/p^{r}) → O^{×}, we may regard ε as a character of U_{0}(r) because U_{0}(r)/U_{1}(r) ≅ T_{0}(O/p^{r}). Then we define a Δ_{p}^{−1}–module by

(A.7) L(χε; O) = εL(χ; O) ⊂ C_{U_{0}(r)}(O) and L(χε; K/O) = L(χε; O) ⊗_{O} K/O ⊂ C_{U_{0}(r)}(K/O).

Let χ : T (ℤ_{p}) → O^{×} be a general character which may not be trivial on Z but induces an algebraic character χ_{0} on an open neighborhood of the identity. We can always find an algebraic character ψ ∈ X(G) = Hom_{alg−gp}(G, G_{m}) such that ψ|_{Z} = χ_{0}^{h} for some positive integer h. We take a character ψ_{0} : G(ℚ_{p}) → O^{×} such that ψ_{0}^{h} = ψ and ψ_{0} = χ on an open neighborhood of the identity in Z(ℤ_{p}). Then we have a unique subspace in C_{S}(K) ⊗ ψ_{0} isomorphic to L(χ; K) as S_{p}– modules, which we again denote by L(χ; K). We fix such a ψ_{0} and define C_{S}(ψ, A) by C(A) ⊗ ψ_{0}. The choice of ψ_{0} does not matter for our purpose. We then define L(χ; O) = L(χ; K) ∩ C_{S}(ψ; O) and L(χ; K/O) = L(χ; O) ⊗ K/O. These are well defined Δ_{p}^{−1}–modules. The action of Δ_{p}^{−1} may depend on the choice of ψ_{0}, but the difference is only a unit-scalar multiple (in O).

A.4. Nearly p–ordinary representations. Let U be an open subgroup of G(ℤ̂). We consider the associated modular variety:

X(U) = G(ℚ)\G(𝔸)/UC_{∞+}Z(𝔸),

where C_{∞+} is the identity connected component of the maximal compact subgroup of the Lie group G(ℝ) and Z ⊂ G is the center of G. We fix a character χ : T (ℤ_{p}) → O^{×}, which is algebraic on a neighborhood of the identity. We assume that ε = χχ_{0}^{−1} factors through T_{0}(O/p^{r}) for χ_{0} ∈ X(T). We call χ dominant if L(χ; K) ≠ 0. Then we fix a p–adic Hecke character ψ̂ : Z(𝔸)/Z(ℚ)Z(ℝ)_{+} → O^{×} such that ψ̂ coincides with χ on T (ℤ_{p}) ∩ Z(ℚ_{p}). This condition tells us that ψ̂ is a p–adic avatar of a complex Hecke character ψ : Z(𝔸)/Z(ℚ) → ℂ^{×}. This means that i_{∞}^{−1}(ψ(x)) = i_{p}^{−1}(ψ̂(x)) for all x ∈ Z(𝔸) with x_{p} = x_{∞} = 1 and the ∞–type of ψ is given by χ_{0}|_{Z}. We then define a right action, if U_{p} ⊂ U_{0}(r) of uz ∈ UC_{∞+}Z(𝔸) (u ∈ UC_{∞+} and z ∈ Z(𝔸)) on L(χ; A) by ϕ|uz = ψ̂(z)ε(u_{p})ρ_{χ}(u_{p}^{−1})ϕ. The action is well defined since χ = εχ_{0} = ψ̂ on Z(ℚ_{p})∩U. We write this right Z(𝔸)U–module by L(χ, ψ̂; A).

We define the covering space X (U) of X(U) by

X (U) = G(ℚ)\(G(𝔸) × L(χ, ψ̂; A))/UC_{∞+}Z(𝔸), (A.8)

where γ(x, ϕ)u = (γxu, ϕ|u) for γ ∈ G(ℚ) and u ∈ UC_{∞+}Z(𝔸). We use the same symbol L(χ, ψ̂; A) for the sheaf of locally constant sections of X (U) over X(U).

We consider the limit, shrinking U,

L(A) = L^{q}(χ, ψ̂; A) = \underrightarrow{lim}_{U}H_{!}^{q}(X(U), L(χ, ψ̂; A)). (A.9)

On the space L(K) the group G(𝔸^{∞}) acts from the right via a smooth representation, which is completely reducible. The complete reducibility follows from the fact that the interior cohomology H_{!}^{q} is embedded in the square-integrable cohomology and hence the representation over ℂ is unitary ([Cl] 3.17-18). Here we regard L(χ, ψ̂; K) as a G(ℚ_{p})–module via the representation ρ_{χ}. Thus in particular, we have an action on H^{0}(U, L^{q}(χ, ψ̂; K)) = L^{q}(χ, ψ̂; K)^{U} of the double coset algebra R_{U} = R(U^{(p)} ×B(ℤ_{p}), G(𝔸^{p∞})×Δ_{p}) of double cosets UxU with x ∈ G(𝔸^{p∞})×Δ_{p}, where U = U_{p} × U^{(p)} and we have assumed that U_{p} ⊂ U_{0}(1).

As already described, we have a unique co-character ξ_{α} : G_{m} → T for each α ∈ Σ such that ξ_{α}(ϖ) (α ∈ Σ and parameters ϖ of F_{𝔭}) gives a minimum set of generators of D_{𝔭} modulo Z(F_{p}). We consider the double coset T (α) = Uξ_{α}(p)U.

The double coset acts on L(χ, ψ̂; A) for A = O and K/O via the Δ_{p}^{−1}–module structure on L(χ, ψ̂; A). The corresponding operator will be written as 𝕋(α). The operator 𝕋(α) is determined (up to unit multiples in O) independently of the choice of ξ_{α}, because Z(Δ_{p}^{−1}) acts on L(χ, ψ̂; A) by a character ψ_{0} with values in O^{×}, and ξ_{α}(ϖ) is unique modulo center. We then make a product 𝕋 = ∏_{α∈Σ} 𝕋(α).

Similarly, the double coset T (α) acts on L^{q}(χ, ψ̂; K)^{N(ℤ_{p})} through the action of G(𝔸^{∞}). The corresponding operator will be written by the same symbol T (α). Then we put T = ∏_{α∈Σ} T (α). Since the action of G(𝔸^{∞}) is defined using the G–action ρ_{χ} on L(χ; K). Then the two operators 𝕋(α) and T (α) are related on the image of L^{q}(χ, ψ̂; O)^{N(ℤ_{p})} by

(A.10) 𝕋(α) = χ(ξ_{α}(p))^{−1}T (α) up to p–adic units if χ|_{Z} ≠ 1.

The limit e = lim_{n→∞} 𝕋^{n!} exists on H^{q}(X(U), L(χ, ψ̂; A)) for A = O, K and K/O (see Section 2). Thus the limit e exists on L^{q}(χ, ψ̂; A)^{N(ℤ_{p})} for A = O, K/O and K. It is easy to see, if U_{p} ⊃ N(ℤ_{p}),

H^{0}(U, eL^{q}(χ, ψ̂; K)^{N(ℤ_{p})}) = e (H^{q}(X(U), L(χ, ψ̂; K))) . (A.11)

We write L_{n.ord}^{q}(χ, ψ̂; A) for eL^{q}(χ, ψ̂; A)^{N(ℤ_{p})}. An irreducible representation π of G(𝔸^{∞}), which is a subquotient of L^{q}(χ, ψ̂; K), is called nearly ordinary of p–type χ if e (V (π)^{N(ℤ_{p})}) ≠ 0 for the representation space V (π) of π. For a subset Θ ⊂ Σ, we can think of 𝕋(Θ) = ∏_{α∈Θ} 𝕋(α) and e_{Θ} = lim_{n→∞} 𝕋_{Θ}^{n!}. If e_{Θ}V (π) ≠ 0, we call π nearly Θ–ordinary or nearly P_{Θ}–ordinary.

A.5. Semi-simplicity of interior cohomology groups. Recall that 2ρ is the sum of positive root of T with respect to B. If π is a local component of a nearly p–ordinary representation of p–type χ, then for its p–component π_{p} (acting on V), its Jacquet module V_{B} ≠ 0, and hence, by Corollary A.2 and (A.3), we find a character λ : T (ℚ_{p}) → K^{×} such that V_{B}[λ̃] ≠ 0 and |ρ^{−2}λ̃(x)|_{p} = |χ_{0}(x)|_{p} (because ||ρ(x)|_{p}|_{p} = |ρ(x)^{−1}|_{p}), where “| |_{p}” is the p–adic absolute value on K normalized so that |p| = \frac{1}{p}.

By definition, the right modulus function δ_{B} is given by

∫_{N(ℚ_{p})} ϕ(u)du = δ_{B}(b) ∫_{N(ℚ_{p})} ϕ(b^{−1}xb)du.

This shows that

δ_{B} = |ρ^{2}|_{p}^{−1}, (A.12)

where ρ^{2} = det ∘Ad_{N} is the sum of positive roots, and ρ is a sum of fundamental weights with respect to B. This shows

|λ|_{p} = |χ_{0}ρ|_{p}. (A.13)

Note that χ_{0} is non-negative with respect to B because χ_{0} is dominant. Since χ_{0} ≥ 0, χ_{0}ρ > 0, that is, χ_{0}ρ is in the interior of the Weyl chamber of B. This shows that if w ≠ 1,

|λ^{w}(d)|_{p} < |λ(d)|_{p} for all d ∈ D, (A.14)

because W acts simply transitively on Weyl chambers and each element in the interior of the chamber of λ has the maximum p–adic absolute value on D in its conjugates under W. In particular, we get

Theorem A.3. Let π be an irreducible nearly ordinary representation of p–type χ. Then there exists a character λ : T (ℚ_{p}) → K^{×} such that λ̃ ↪ V_{B}(π_{p}) ↪ ⊕_{w∈W} λ̃^{w} and |λ|_{p} = |ρχ|_{p}, where ρ is the sum of fundamental weight with respect to B and | |_{p} is the absolute value on K. Moreover eH^{0}(N(ℤ_{p}), V (π_{p})) is one dimensional, on which T (ξ) = UξU for ξ ∈ D acts by scalar |ρ(ξ)|_{p}λ(ξ).

Now suppose that U = U_{p} × G(ℤ̂^{p}) with U_{1}(r) ⊂ U_{p} ⊂ U_{0}(r) for r > 0. For prime ideals 𝔩 ∤ p of O_{F}, we consider T_{𝔩}(α) = Uξ_{α}(ϖ_{𝔩})U for the prime element ϖ_{𝔩} in O_{𝔩}. Then we define the Hecke algebra h_{q}^{n.ord}(χ, ψ̂; K) by the K–subalgebra of End_{K}(eH_{!}^{q}(X(U), L(χ, ψ̂; K))) generated by the operators T (α) for all α ∈ Σ and Hecke operators associated to double cosets UξU with ξ_{p} = 1. Since

eH_{!}^{q}(X(U), L(χ, ψ̂; K)) = H^{0}(U, eL^{q}(χ, ψ̂; K)^{N(ℤ_{p})}),

we get the following semi-simplicity of the Hecke algebra from the fact that the spherical irreducible representation of G_{0}(F_{𝔩}) has a unique vector fixed by the maximal compact subgroup:

Corollary A.4. Let the notation and the assumption be as above. Then we have

(1) The Hecke module eH_{!}^{q}(X(U), L(χ, ψ̂; K)) has a base made of common

eigenvectors of h_{q}^{n.ord}(χ, ψ̂; K) if K is algebraically closed;
(2) The Hecke algebra h_{q}^{n.ord}(χ, ψ̂; K) is semi-simple.

Appendix B. Correction to [H95]

Since we quoted often results proven in [H95], we here list some of serious misprints and corrections to the result in the paper. One serious mistake is the sign of the character χ in Section 7 (basically, we need to change χ by χ^{−1} in Lemma 7.2 and its proof). Also in Lemma 7.2, ω_{i} (i = 1, …, n) have to be dominant characters with respect to (G_{1},^{t}B) for the lower triangular Borel subgroup ^{t}B instead of the upper trianguler one. In the list below, P.5 L.5b indicates the fifth line from the bottom of the page 5.

Page, line  Statement in [H95]  Correction

P. 453, L. 5

P. 454, L. 6b H_{P −n.ord}^{q}  H_{P −n.ord}^{r} for r = r(χ)
P. 454, L. 6b H_{P −n.ord}^{r}(Φ_{0,P} (p), C_{P,ρ}[ω_{χ}])  H_{P −n.ord}^{r}(Φ_{0,P} (p), C_{P,ρ})[ω_{χ}]

P. 463, L. 11

P. 466, L. 9b χ(π(ξ))^{−1}ρ_{χ}(ξ^{−1})  χ(π(ξ))ρ_{χ}(ξ^{−1})
P. 466 L. 3b |χ(π(d))|_{p} ≥ |η(π(d))|_{p}  |χ(π(d))|_{p} ≤ |η(π(d))|_{p}
P. 467 L. 1b ω_{i}(diag(t_{1}, …, t_{n})) = ∏_{1≤j≤i} t_{j}  ω_{i}(diag(t_{1}, …, t_{n})) = ∏_{1≤j≤i} t_{j}^{−1}

P. 468 L. 2 χ(diag(t_{1}, …, t_{n})) = ∏_{1≤i≤n−1} t_{j}^{j_{i}} χ(diag(t_{1}, …, t_{n})) = ∏_{1≤i≤n−1} t_{j}^{−j_{i}}
P. 468 (∗∗) (ωwu)  (ξwu)
P. 468 (∗∗) χ(π(ξ))  χ(π(ξ))^{−1}
P. 468 L. 11 χ(π(d))  χ(π(d))^{−1}
P. 468 L. 16 |λ(ξ_{s})|_{p} < |χ(π(ξ))|_{p}  |λ(ξ_{s})|_{p} < |χ(π(ξ))^{−1}|_{p}

References

[AC] J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Ann. of Math. Studies, 120, Princeton University Press, 1989

[BZ] J. Bernstein and A. V. Zelevinsky, Representations of the group GL(n, F), where F is a nonarchimedean local field, Uspeksi Mat. Nauk 31 (1976), 5–70 (English translation: Russian Math. Surveys 31 (1976), 1–68)

[BZ1] J. Bernstein and A. V. Zelevinsky, Induced representations of reductive p–adic groups, Ann. Sci. Éc. Norm. Sup. 4-th series 10 (1978), 441–472

[B] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Inventiones Math. 35 (1976), 233–259

[BW] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. of Math. Studies 94, Princeton University Press, Princeton, 1980

[Ca] P. Cartier, Representations of 𝔭–adic groups: a survey, Proc. Symp. Pure Math. 33 (1979), part 1, 111–155

[Cl] L. Clozel, Motifs et formes automorphes: Applications du principe de fonctorialité, Perspective Math. 10 (1990), 77–159

[Cl1] L. Clozel, Représentations galoisienne associées aux représentations automorphes autoduales de GL(n), Publ. IHES 73 (1991), 97–145

[DKV] P. Deligne, D. Kazhdan and M.-F. Vignéras, Représentations des algèbres centrales simples p–adiques, in “Représentations des Groupes réductifs sur un Corps Local,” Hermann, Paris, 1984, 33–117

[H88] H. Hida, On p–adic Hecke algebras for GL_{2} over totally real fields, Ann. of Math., 128 (1988), 295–384 [H89] H. Hida, On nearly ordinary Hecke algebras for GL(2) over totally real fields, Adv.

Studies in Pure Math. 17 (1989), 139–169

[H93] H. Hida, p–Ordinary cohomology groups for SL(2) over number fields, Duke Math. J. 69 (1993), 259–314

[H94a] H. Hida, On the critical values of L–functions of GL(2) and GL(2) × GL(2), Duke Math. J. 74 (1994), 431–529

[H94b] H. Hida, p–Adic ordinary Hecke algebras for GL(2), Ann. l’institut Fourier 44 (1994), 1289–1322

[H95] H. Hida, Control theorems of p–ordinary cohomology groups for SL(n), Bull. Soc. Math. France 123 (1995), 425–475

[HM] H. Hida and Y. Maeda, Non-abelian base change for totally real fields, Olga Taussky Todd memorial issue, Pacific Journal of Math. (1997), 189–217

[J] J. C. Jantzen, Representations of algebraic groups, Academic Press, 1987

[M] B. Mazur, Deforming Galois representations, in “Galois group over ℚ”, MSRI publications 16, (1989), 385–437

[R] J. D. Rogawski, Representations of GL(n) and division algebras over a p–adic fields, Duke Math. J. 50 (1983), 161–196

[Sh] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan 11 (1971), Iwanami-Shoten and Princeton University Press

[TW] R. Taylor and A. Wiles, Ring theoretic properties of certain Hecke modules, Ann. of Math. 142 (1995), 553–572

[Ti] J. Tilouine, Deformations of Galois representations and Hecke algebras, Publ. Mehta Res. Inst., Narosa, New Delhi, 1996

[TU] J. Tilouine and E. Urban, Several variable p–adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, To appear, Ann. Sci. Ec. Norm. Sup

[T] J. Tits, Reductive groups over local fields, Proc. Symp. Pure Math. 33 (1979), part 1, 29–69

[W] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 142 (1995), 443–551

(H. Hida) Department of Mathematics, UCLA, Los Angeles, Ca 90095-1555, U.S.A.