Abstract. In these notes we prove Harish-Chandra’s formula for the joint probability distribution for a random Hermitian matrix in a co-adjoint orbit O_{A_{0}}.

THE PROBABILITY DISTRIBUTION OF THE DIAGONAL ENTRIES OF A MATRIX WITH SPECIFIED EIGENVALUES

TERENCE TAO

Let A_{0} = diag(a_{1}, a_{2}, …, a_{n}) be a diagonal matrix with real decreasing entries a_{1} > a_{2} > … > a_{n}. Let O_{A_{0}} denote the co-adjoint orbit of A_{0} with respect to the unitary group U(n), i.e.

O_{A_{0}} = {UA_{0}U^{∗} : U ∈ U(n)}.

Note that O_{A_{0}} ⊂ u(n)^{∗}, where u(n)^{∗} is the space of Hermitian matrices.

This co-adjoint orbit comes with a natural U(n)-invariant measure dO_{A_{0}}, which we normalize to have total mass 1. Let d : O_{A_{0}} → R^{n} be the diagonal map defined by d(X) = (x_{11}, x_{22}, …, x_{nn}), where x_{ij} is the ij entry of X. We are interested in the measure d_{∗}(dO_{A_{0}}). Our main result is

Theorem 0.1. [7] For any ξ we have

{ \atop d_{∗}(dO_{A_{0}})}(ξ) = \frac{m!}{(−2πi)^{m}} \frac{1}{∏_{j<k}(a_{k} − a_{j})} \frac{∑_{σ∈S_{n}} sgn(σ)e^{−2πi⟨σ(A_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})} . (1)

The measure d_{∗}(dO_{A_{0}}) is clearly finite, compactly supported, and symmetric with respect to the standard action of the symmetric group S_{n}. Thus { \atop d_{∗}(dO_{A_{0}})} is smooth (in fact it is analytic!) and symmetric with respect to the S_{n} action. Thus to verify the formula it suffices by symmetry and continuity to verify the case when ξ = (ξ_{1}, …, ξ_{n}) for ξ_{1} > ξ_{2} > … > ξ_{n}.

We can write out the Fourier transform as

{ \atop d_{∗}(dO_{A_{0}})}(ξ) = ∫_{O_{A_{0}}} e^{−2πid(A)·ξ} dO_{A_{0}}(A).

Introducing the Killing form

⟨X, Y ⟩ = tr(XY),

which makes u(n)^{∗} into a Hilbert space, we have

{ \atop d_{∗}(dO_{A_{0}})}(ξ) = ∫_{O_{A_{0}}} e^{−2πi⟨A,·ξ⟩} dO_{A_{0}}(A),

where ξ is now thought of as the diagonal matrix diag(ξ_{1}, …, ξ_{n}). Writing ξ = λw for some fixed w we thus see

{ \atop d_{∗}(dO_{A_{0}})}(λw) = ∫_{O_{A_{0}}} e^{iλϕ(A)} dO_{A_{0}}(A)

where ϕ(A) = −2π⟨A, w⟩.

1. Stationary points

Our first observation is that the phase ϕ is only stationary when A is diagonal.

To see this, first observe that the tangent space T_{A}O_{A_{0}} of O_{A_{0}} at A is generated by the infinitesimal action of U(n). Since an infinitesimal element of U(n) looks like I + εix + O(ε^{2}) for some Hermitian matrix x, we thus see that

T_{A}O_{A_{0}} = {i[A, x] : x ∈ u(n)^{∗}}.

In order for ϕ to be stationary, we must have

T_{A}O_{A_{0}} ⊥ w.

In other words, we want

tr(i[A, x]w) = 0 for all x ∈ u(n)^{∗}.

But by the trace identity tr([A, B]C) = tr(A[B, C]) this is equivalent to

tr(ix[A, w]) = 0 for all x ∈ u(n)^{∗}.

Since the Killing form is non-degenerate, this implies that

[A, w] = 0,

i.e. that A commutes with w. Since w is diagonal with distinct entries, this implies that A is diagonal. Since A is conjugate to A_{0} and A_{0} is diagonal with distinct entries, this implies that A = σ(A_{0}) for some σ ∈ S_{n}.

By the principle of non-stationary phase we therefore have

{ \atop d_{∗}(dO_{A_{0}})}(λw) = ∑_{σ∈S_{n}} ∫_{O_{A_{0}}} e^{iλϕ(A)} ψ_{σ}dO_{A_{0}}(A) + O(λ^{−N})

for any N > 0, where ψ_{σ} is a cutoff function which equals 1 near σ(A) and vanishes away from σ(A).

2. The Kirrilov-Kostant symplectic form

The co-adjoint orbit O_{A_{0}} comes with a standard symplectic form ω, known as the Kirrilov-Kostant symplectic form. If A is a point on O_{A_{0}} and i[A, x], i[A, y] are tangent vectors at A, the symplectic form is defined by the formula

ω_{A}(i[A, x], i[A, y]) = tr(iA[x, y]).

It is easy to show that this is a well-defined, anti-symmetric non-degenerate 2-form on O_{A_{0}}. To show that it is a symplectic form, we need to make sure it is closed, i.e.

dω(i[A, x], i[A, y], i[A, z]) = 0

at every point A and every x, y, z ∈ u(n)^{∗}.

Fix A, x, y, z, and define the vector fields X, Y, Z on O_{A_{0}} by

X(B) = i[B, x], X(B) = i[B, y], Z(B) = i[B, z]

for all B in O_{A_{0}}. From elementary differential geometry we have

dω(X(A), Y (A), Z(A)) = X(ω(Y, Z))(A) + Y (ω(Z, X))(A) + Z(ω(X, Y))(A)

so we have

dω(i[A, x], i[A, y], i[A, z]) = X(tr(i·[y, z]))(A)+Y (tr(i·[z, x]))(A)+Z(tr(i·[x, y]))(A)

which equals

tr(i[A, x][y, z]) + tr(i[A, y][z, x]) + tr(i[A, z][x, y])

which by the trace identities equals

tr(iA([x, [y, z]] + [y, [z, x]] + [z, [x, y]]))

which equals zero by the Jacobi identity.

The space O_{A_{0}} has dimension n^{2} − n = 2N, where N = (n^{2} − n)/2. (This is best seen by observing that u(n)^{∗} has dimension n^{2}, and the set of eigenvalues of a matrix lives in an n-dimensional space).

The symplectic form ω is a 2-form. If we wedge it with itself N times, we get a 2N-form (i.e. a top form) ω^{N}. Top forms are identifiable with measures, so we thus obtain a canonical measure on O_{A_{0}}, called Louville measure. Actually, strictly speaking Louiville measure is normalized to be ω^{N}/N! instead of ω^{N}. Since ω is non-degenerate, Louville measure is non-zero.

The form ω is easily seen to be invariant under the action A → U^{∗}AU of the unitary group U(n). This implies that the Louiville measure is similarly invariant. However, any invariant measure on O_{A_{0}} has to be a constant multiple of the standard measure dO_{A_{0}}. Thus we have

dO_{A_{0}} = cω^{N}

for some constant c (which may depend on A_{0}). Therefore we have

{ \atop d_{∗}(dO_{A_{0}})}(λw) = ∑_{σ∈S_{n}} c ∫_{O_{A_{0}}} e^{iλϕ(A)} ψ_{σ}(A)ω^{N}(A) + O(λ^{−N}).

3. Hamiltonian flows

We have a smooth real-valued function ϕ on the symplectic manifold (M, ω), with M = O_{A_{0}}, and ω equal to the Kirrilov-Kostant form. Any such situation generates a Hamiltonian vector field X on M, defined by the formula

X¬ω = dϕ,

or in other words that

ω(X, Y) = Y (ϕ) (2)

for all vector fields Y.

Any such vector field generates a group e^{tX} of Hamiltonian flows, defined by

e^{0X}(x) = x

\frac{d}{dt}e^{tX}(x) = X[e^{tX}](x).

Example (simple harmonic oscillator, canonical co-ordinates): If the symplectic manifold is M = R^{2} = {(p, q) : p, q ∈ R} with symplectic form ω = dp ∧ dq, and ϕ = \frac{1}{2}(p^{2} + q^{2}), then

X(p, q) = (−q, p)

and

e^{tX}(p, q) = (p cos t + q sin t, q cos t − p sin t).

Easy fact: if two Hamiltonians ϕ, ϕ′ generate the same Hamiltonian flows, then they can only differ by a constant.

The flow e^{tX} always preserves the symplectic form ω. The Lie derivative

L_{X}ω = \frac{d}{dt}(e_{∗}^{tX}ω)

is always given by the fundamental formula

L_{X}ω = X¬dω + d(X¬ω). (3)

But in our case this is just

X¬0 + d(dϕ) = 0.

Now let’s work out the Hamiltonian flow in our particular case when M = O_{A_{0}} and ϕ = −2π⟨A, w⟩. Suppose at a given A ∈ O_{A_{0}}, the Hamiltonian vector field X is given by X(A) = i[A, x] for some Hermitian matrix x. By (2) we have

ω_{A}(i[A, x], Y) = Y (ϕ)(A)

for any Y. If we set Y (A) = i[A, y], we thus obtain

tr(iA[x, y]) = −2π⟨i[A, y], w⟩

for all y ∈ u(n)^{∗}. By the trace identity, this becomes

tr(i[A, x]y) = 2πtr(i[A, w]y)

which implies that

x = 2πw.

In other words, the Hamiltonian vector field is given by

X(A) = 2πi[A, w],

so the flow is

e^{tX}(A) = e^{−2πitw}Ae^{2πitw}.

In other words, the flow is just conjugation by the one-parameter Lie group generated by 2πiw.

4. Flattening the manifold O_{A_{0}} via the exponential map

Fix σ ∈ S_{n}. We’re trying to evaluate the integral

∫_{O_{A_{0}}} e^{iλϕ(A)} ψ_{σ}(A)ω^{N}(A).

Our integral is currently on a 2N-dimensional curved manifold O_{A_{0}}. We’re going to change variables to the 2N-dimensional vector space

V = {x ∈ u(n)^{∗} : d(x) = 0}

of Hermitian matrices with vanishing diagonal, under the map f : V → O_{A} defined by

f(x) = e^{−ix}σ(A_{0})e^{ix}.

Note that we have the Taylor expansion

f(x) = σ(A_{0}) + i[σ(A_{0}), x] + O(x^{2}).

The map x ↦ i[σ(A_{0}), x] is injective, since the only element of V which commutes with σ(A_{0}) (i.e. is diagonal) is the zero matrix by definition of V. Since V has the same dimension as O_{A_{0}}, the inverse function theorem thus tells us that the map f is a local diffeomorphism. Thus we have

∫_{O_{A_{0}}} e^{iλϕ(A)} ψ_{σ}(A)ω^{N}(A) = ∫_{V} e^{iλϕ̃(x)} ψ̃_{σ}(x)ω̃^{N}(x) (4)

where

ϕ̃(x) = ϕ(f(x)) = −2π⟨e^{−ix}σ(A_{0})e^{ix}, w⟩
ψ̃_{σ}(x) = ψ_{σ}(f(x))

and

ω̃ = f_{∗}ω.

Unfortunately, the symplectic form ω̃ and the phase function ϕ̃ now are pretty horrible expressions. However, two things can still be easily computed. Firstly, the form ω̃ can still be calculated at 0 by the formula

ω̃_{0}(x, y) = w_{f(0)}(Df(0)(x), Df(0)(y)) = ω_{σ(A_{0})}(i[σ(A_{0}), x], i[σ(A_{0}), y]) = tr(iσ(A_{0})[x, y]).

Secondly, the Hamiltonian flow e^{tX̃} of ϕ̃ with respect to the form w̃ can be easily computed, because it is just the pullback of the flow e^{tX} of ϕ with respect to w:

f(e^{tX̃}(x)) = e^{tX}(f(x)).

Since we know f and e^{tX}, it is a routine matter to see that

e^{tX̃}(x) = e^{−2πitw}xe^{2πitw}.

The key observation is that exp(UAU^{∗}) = Uexp(A)U^{∗} for any unitary U and any skew-Hermitian matrix A. Differentiating this, we see that

X̃(x) = 2πi[x, w]. (5)

In particular, the one-parameter group action of Hamiltonian flows on V is a linear action.

The symplectic form ω̃ at a point x ∈ V can be written as

ω̃(x) = ω_{0} + O(|x|),

where ω_{0} is the constant-coefficient symplectic form

ω_{0}(x, y) = tr(iσ(A_{0})[x, y]).

In a similar vein, the Hamiltonian ϕ̃ can be Taylor expanded as

ϕ̃(x) = −2π⟨σ(A_{0}), w⟩ − 2π⟨i[σ(A_{0}), x], w⟩ − 2π⟨−\frac{1}{2}[[σ(A_{0}), x], x], w⟩ + O(|x|^{3}).

The linear term vanishes from the trace identity

⟨i[σ(A_{0}), x], w⟩ = tr(i[σ(A_{0}), x]w) = −tr(ix[σ(A_{0}), w]) = 0

since σ(A_{0}) and w commute. (This is no surprise, since we are at a stationary point of ϕ). Thus we have

ϕ̃(x) = ϕ_{0}(x) + O(|x|^{3})

where

ϕ_{0}(x) = −2π⟨σ(A_{0}), w⟩ + πtr([[σ(A_{0}), x], x]w).

It would be very convenient if the variable-coefficient symplectic form ω̃ was replaced by the constant coefficient ω_{0}, and the complicated Hamiltonian ϕ̃ was replaced by the quadratic Hamiltonian ϕ_{0}. Then our integral will be very easy to compute. It turns out that we can do exactly that! There is a change of variables that converts ω̃ to ω_{0} and ϕ̃ to ϕ_{0}. This may seem totally miraculous, but there is some justification for it. One key ingredient is the following observation:

- The Hamiltonian flow generated by ϕ_{0} with respect to ω_{0} is the same as the Hamiltonian flow generated by ϕ̃ with respect to ω̃.

This is just a calculation, actually. We already know the Hamiltonian vector field generated by ϕ̃ with respect to ω̃, it’s the field X̃ defined in (5). To verify that this is also the Hamiltonian vector field generated by ϕ_{0} with respect to ω_{0}, we have to check (2), i.e. that

ω_{0}(X̃, Y)(x) = Y (ϕ_{0})(x)

for all vector fields Y. Writing y = Y (x) we obtain by the product rule

tr(iσ(A_{0})[X̃(x), y]) = πtr([[σ(A_{0}), x], y]w) + πtr([[σ(A_{0}), y], x]w). (6)

Substituting (5), and always using the trace equality to push y outside the Lie brackets, we eventually get

tr([iσ(A_{0}), 2πi[x, w]]y) = πtr([w, [σ(A_{0}), x]]y) + πtr([[x, w], σ(A_{0})]y).

So to finish the proof we just need to show that

−2[σ(A_{0}), [x, w]] = [w, [σ(A_{0}), x]] + [[x, w], σ(A_{0})].

But this is just the Jacobi identity together with the fact that w and σ(A_{0}) commute.

5. The equivariant Darboux theorem

To recap, we have two symplectic forms ω̃ and ω_{0} on a vector space V, and two Hamiltonians ϕ̃ and ϕ_{0}. We also have the linearly action of a one-parameter group G = {e^{tX̃}} which is the Hamiltonian flow for ϕ̃ with respect to ω̃, as well as the flow for ϕ_{0} with respect to ω_{0}. In particular the linear map e^{tX̃} preserves both ω̃ and ω_{0}.

Oh, and we’re trying to estimate the integral

∫_{V} e^{iλϕ̃(x)} ψ̃_{σ}(x)ω̃^{N}(x)

.

We now utilize

Lemma 5.1 (Equivariant Darboux theorem). Let ω̃ and ω_{0} be symplectic forms on a vector space V such that ω̃(0) = ω_{0}(0), and let G be a Lie group acting linearly on V so that the group action preserves both ω̃ and ω_{0}. Then there exists a local diffeomorphism f defined near 0 such that f_{∗}(ω̃) = ω_{0}, and f commutes with the G action.

Proof Since ω̃ − ω_{0} is a closed two-form, we can write

ω̃ − ω̃_{0} = dΦ

where the one-form Φ is given by the formula in Poincaré’s lemma:

Φ(x) = ∫_{0}^{1} x¬ω(tx) dt.

Note that since ω_{0} and ω̃ are G-invariant, and the G action is linear, the one-form Φ is also G-invariant.

We now define, for each t ∈ [0, 1], the form ω_{t} by

ω_{t} = tω̃ + (1 − t)ω_{0}

and the flow X_{t} by

X_{t}¬ω_{t} = Φ.

In other words, X_{t} is the time-dependent Hamiltonian flow with a fixed Hamiltonian Φ and a variable symplectic form ω_{t}. If the support is restricted sufficiently close to 0, then ω_{t} will always be non-degenerate and this will make sense. Also, since ω_{t} and Φ are G-invariant, so is X_{t}.

We now compute

\frac{d}{dε}((e^{εX_{t}})_{∗}ω_{t})|_{ε=0}.

This is the Lie derivative of ω_{t} with respect to X_{t}. By (3) this is

X_{t}¬dω_{t} + d(X_{t}¬ω_{t}) = X_{t}¬0 + dΦ = ω̃ − ω̃_{0} = \frac{d}{dt}ω_{t}.

In other words, the form ω_{t} evolves according to the flow X_{t} at time t. Thus we have a continuous family of G-invariant deformations from ω_{0} to ω̃, as desired.

Applying the diffeomorphism guaranteed by the equivariant Darboux theorem, we can rewrite the integral (if the support of ψ_{σ} was chosen sufficiently small) as

∫_{V} e^{iλϕ′(x)} ψ′_{σ}(x)ω_{0}^{N}(x)

where ϕ′ and ψ′_{σ} are the pullbacks of ϕ̃ and ψ̃_{σ} under the diffeomorphism.

Since the flow of ϕ̃ with respect to ω̃ is given by the action of G, and the diffeomorphism commutes with the action of G, we see that the flow of ϕ′ with respect to ω_{0} is also given by the action of G. But the flow of ϕ_{0} with respect to ω_{0} is again given by the action of G. Hence ϕ′ and ϕ_{0} differ by a constant. Since these two functions are both equal to −2π⟨σ(A_{0}), w⟩ at the origin, we therefore have ϕ′ = ϕ_{0}.

Furthermore, since ω_{0} is a constant coefficient symplectic form, ω_{0}^{N} is just a multiple of Lebesgue measure dx. The integral is now

c ∫_{V} e^{iλϕ_{0}(x)} ψ′_{σ}(x) dx.

In other words, our phase function is now quadratic, our measure is Lebesgue measure, and our cutoff function is identically equal to 1 near the origin.

6. Quadratic phase integrals

To conclude we will need

Lemma 6.1. Let ϕ(x) = ∑_{j=1}^{m} a_{j}x_{j}^{2} be a quadratic function on R^{m} for some nonzero constants a_{j}, and let ψ be a bump function which equals 1 near the origin. Then we have

∫ e^{iλϕ(x)}ψ(x) dx = λ^{−m/2} ∏_{j=1}^{m}(2πi/a_{j})^{1/2} + O(λ^{−N})

for any constant N.

Proof We may replace the cutoff ψ by a tensor product

ψ̃(x) = ∏_{j=1}^{m} ψ_{0}(x_{j})

where ψ_{0} is a one-dimensional bump function which equals 1 near the origin. This is because the error incurred in the integral by estimating ψ by ψ̃ is O(λ^{−N}) by the principle of non-stationary phase. Thus it suffices to prove the one-dimensional estimate

∫ e^{iλax^{2}}ψ_{0}(x) dx = λ^{−1/2}\sqrt{\frac{2πi}{a}} + O(λ^{−N}).

We have already proved the identity

∫ e^{iλax^{2}}e^{−λεx^{2}} dx = λ^{−1/2}\sqrt{\frac{2πi}{a + iε}}

for any ε ≠ 0 by shifting the contour. This implies that

∫ e^{iλax^{2}}e^{−λεx^{2}}ψ_{0}(x) dx = λ^{−1/2}\sqrt{\frac{2πi}{a + iε}} + O(λ^{−N})

since the error can be controlled by the principle of non-stationary phase. Furthermore, by inspecting the non-stationary phase argument (i.e. explicitly doing the integration by parts) we see that the error term can be bounded independently of ε (idea: integrate by parts N times, and then appeal to Van der Corput). The estimate now follows by sending ε → 0.

The space V can be parameterized by n^{2} − n real co-ordinates x_{jk}, y_{jk} for 1 ≤ j < k ≤ n by requiring that the jk co-ordinate of x be equal to x_{jk} + iy_{jk}. A computation then shows that

ϕ_{0}(x) = −2π⟨σ(A_{0}), w⟩ + 2π ∑_{j<k}(w_{k} − w_{j})(a_{σ(k)} − a_{σ(j)})(x_{jk}^{2} + y_{jk}^{2}).

Thus our integral is

ce^{−2πλ⟨σ(A_{0}),w⟩}λ^{−(n^{2}−n)/2} ∏_{j<k} \frac{1}{(w_{k} − w_{j})(a_{σ(k)} − a_{σ(j)})} + O(λ^{−N}).

By definition of signum, we have

∏_{j<k}(a_{σ(k)} − a_{σ(j)}) = sgn(σ) ∏_{j<k}(a_{k} − a_{j}) = csgn(σ)

and we can rewrite the integral as

c\frac{sgn(σ)e^{−2πi⟨σ(A_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})} + O(|ξ|^{−N}).

Adding up all the contributions from all the σ we finally get

{ \atop d_{∗}(dO_{A_{0}})}(ξ) = c\frac{∑_{σ∈S_{n}} sgn(σ)e^{−2πi⟨σ(A_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})} + O(|ξ|^{−N}). (7)

7. Piecewise polynomiality

Let μ be the measure whose Fourier transform is equal to the main term on the right-hand side of (7), i.e.

μ̂(ξ) = c\frac{∑_{σ∈S_{n}} sgn(σ)e^{−2πi⟨σ(A_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})} .

We claim that μ is piecewise polynomial and compactly supported.

We first observe that the right-hand side is analytic. We can expand the numerator as

∑_{m} ∑_{σ∈S_{n}} sgn(σ)\frac{1}{m!}(−2πi⟨σ(A_{0}), ξ⟩)^{m}.

The summand for a given m is a polynomial of degree m in ξ. Clearly it vanishes whenever two of the ξ_{j} co-incide, and so the polynomial must be a multiple of the denominator ∏_{j<k}(ξ_{k} −ξ_{j}). Thus one can divide the series term by term (much like one does with sin(x)/x); the 1/k! factor ensures that the series is very convergent.

Since μ̂ is analytic, μ is therefore rapidly decreasing. (One can also see this from (7) and the fact that d_{∗}(dO_{A_{0}}) is rapidly decreasing).

Next, we observe that

∏_{j<k}(ξ_{k} − ξ_{j})μ̂(ξ)

is a finite linear combination of plane waves. This implies that

∏_{j<k}(∂_{k} − ∂_{j})μ

is a finite linear combination of Dirac delta functions, which we will call ν. Since μ is rapidly decreasing at infinity, we therefore have from the Fundamental theorem of Calculus that

μ = ∏_{j<k}(∂_{k} − ∂_{j})^{−1}ν

where (∂_{k} − ∂_{j})^{−1} is the integration operator

(∂_{k} − ∂_{j})^{−1}f(x) = ∫_{−∞}^{0} f(x + t(e_{k} − e_{j})) dt.

It is easy to see by induction that μ vanishes when ξ · (1, 2, …, n) is sufficiently positive, and is piecewise polynomial. Since it is piecewise polynomial and rapidly decreasing, it must be compactly supported.

8. Elimination of the error term

There is some cohomology machinery which shows that d_{∗}(dO_{A_{0}}) is also piecewise polynomial. Combining this with the previous section this shows that d_{∗}(dO_{A_{0}})−μ is piecewise polynomial and compactly supported. But from (7) this expression is also smooth. The only way all three of these properties can hold is if the error is zero.

A full explanation of why this fact holds would take us rather far afield: see [3] though for a nice exposition. A model example is provided by Archimedes’ theorem, in which case dO_{A_{0}} is the sphere S^{2} in R^{3} and d is the projection onto the z-axis. Archimedes’ theorem states that the projection of the sphere is uniform measure on the unit interval. (Of course, he didn’t phrase it that way, and talked about how horizontal slices of a vertical cylinder and an inscribed sphere had equal surface area).

Here is a very insufficient sketch of how the argument works. The measure d_{∗}(dO_{A_{0}}) can be written as a function f(x) times Lebesgue measure dx, where f(x) is the induced measure of the fiber d^{−1}(x). For most points x, d^{−1}(x) is a smooth manifold that varies smoothly with x, although there is a codimension 1 set on which things are bad. (That’s why we’re only going to get piecewise polynomiality of f).

The symplectic form ω is a closed non-degenerate two-form on O_{A_{0}}, and it therefore restricts to a closed two-form on d^{−1}(x). However, it isn’t non-degenerate; there are a bunch of directions in which the two-form vanishes completely. However, one can quotient out these directions and end up with something called the ”symplectic fiber” M_{x} at x, which is a symplectic manifold with some form ω_{x}. f(x) then just becomes the Louville volume of this manifold, i.e. f(x) = c ∫_{M_{x}}(ω_{x})^{N} for some N (N = (n − 1)(n − 2)/2, if you must know).

Now one can choose co-ordinates so that M_{x} is independent of x, at least locally, i.e. M_{x} = M. But ω_{x} will then be variable. However, a bunch of differential geometry (which is related to the stuff used in the equivariant Darboux theorem) shows that the derivative of ω_{x} in any direction v is equal to the curvature Ω_{v} of a certain connection on the bundle of M’s over the base space R^{n}. However, it is a basic fact of cohomology theory that the cohomology class of the curvature of a connection is independent of the base point (this class is called the Chern class of the connection). The upshot of this is that the cohomology class [Ω_{v}] is locally independent of x, which means that [ω_{x}] is locally linear in x. On the other hand, the other basic fact of cohomology theory is that global integrals depend only on the cohomology class of the integrand, i.e.

f(x) = c ∫_{M_{x}}(ω_{x})^{N} = c ∫_{M_{x}}[ω_{x}]^{N}.

Thus we get that f(x) is locally a polynomial.

9. Determination of the constant

The only thing left to do is to work out exactly what this constant c is. But that’s easily done because we’ve normalized surface measure dO_{A_{0}} to have total mass 1. This implies that the pushforward d_{∗}(dO_{A_{0}}) also has total mass 1, so that

{ \atop d_{∗}(dO_{A_{0}})}(0) = 1.

Combining this with (7) (which is now exact) we get

c^{−1} = lim_{ξ→0} \frac{∑_{σ∈S_{n}} sgn(σ)e^{−2πi⟨σ(A_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})} .

The denominator is a homogeneous polynomial of order m = (n^{2} − n)/2, so what we need to do is work out the m order term in the numerator. As we have already observed, this is

\frac{1}{m!} ∑_{σ∈S_{n}} sgn(σ)(−2πi)^{m}⟨σ(A_{0}), ξ⟩^{m}.

This produces many monomials of the form ξ_{1}^{j_{1}} … ξ_{n}^{j_{n}} with j_{1}+…+j_{n} = (n^{2}−n)/2. If any two of the j_{i} are equal then by symmetry the contribution of this monomial will vanish, hence the only monomials that appear are those for which (j_{1}, …, j_{n}) a1re a permutation of (0, …, n − 1), i.e. j_{k} = α(k) − 1 for some permutation α. Thus we can rewrite the above as

\frac{(−2πi)^{m}}{m!} ∑_{σ∈S_{n}} sgn(σ) ∑_{α∈S_{n}} ∏_{k=1}^{n}(a_{σ(k)}ξ_{k})^{α(k)−1}.

We can rearrange this as

\frac{(−2πi)^{m}}{m!} ∑_{α∈S_{n}} ξ^{α−1} ∑_{σ∈S_{n}} sgn(σ) ∏_{k=1}^{n} a_{σ(k)}^{α(k)−1}.

By the substitution σ = σα, this becomes

\frac{(−2πi)^{m}}{m!} ∑_{α∈S_{n}} ξ^{α−1}sgn(α) ∑_{σ∈S_{n}} sgn(σ) ∏_{k=1}^{n} a_{σ(k)}^{k−1}.

We will shortly prove the identity

∑_{σ∈S_{n}} sgn(σ) ∏_{k=1}^{n} a_{σ(k)}^{k−1} = ∏_{j<k}(a_{k} − a_{j}). (8)

Assuming this for the moment, we rewrite the above as

\frac{(−2πi)^{m}}{m!} ∑_{α∈S_{n}} ∏_{k=1}^{n} ξ_{k}^{α(k)−1}sgn(α) ∏_{j<k}(a_{k} − a_{j})

which in turn becomes (after replacing α by α^{−1})

\frac{(−2πi)^{m}}{m!} ∏_{j<k}(a_{k} − a_{j}) ∑_{α∈S_{n}} sgn(α) ∏_{k=1}^{n} ξ_{k}^{α(k)−1}.

Using (8) again this becomes

\frac{(−2πi)^{m}}{m!} ∏_{j<k}(a_{k} − a_{j})(ξ_{k} − ξ_{j}).

Dividing out the denominator we get

c^{−1} = \frac{(−2πi)^{m}}{m!} ∏_{j<k}(a_{k} − a_{j})

and the theorem follows.

We now prove (8). The right-hand side is clearly anti-symmetric with respect to S_{n}. Thus the only monomials which appear are those of the form in the left-hand side of (8), since everything else cancels. One now easily checks that the a_{2}a_{3}^{2} … a_{n}^{n−1} terms of both sides agree, and the rest follows by symmetry.

10. Sum of two Hermitian matrices

Let A_{0}, B_{0} be two diagonal matrices with real decreasing entries. We consider the map e : O_{A_{0}} × O_{B_{0}} → R^{n} which sends (A, B) to the eigenvalues of A + B, arranged in real decreasing order. The problem of determining, for a given A_{0} and B_{0}, the range of e is a very old problem in linear algebra. The correct answer was conjectured by Horn [4] in the 60’s, but was only proven last year [6], [5]. The proof is extremely involved and requires several different areas of mathematics. However, it is relatively easy to compute the Fourier transform of the (symmetrized) pushforward measure e_{∗}(dO_{A_{0}} × dO_{B_{0}}).

All our constants are allowed to depend on A_{0} and B_{0}. We begin by considering, for any diagonal matrix ξ, the expression

∫_{O_{A_{0}}} ∫_{O_{B_{0}}} e^{−2πi⟨A+B,ξ⟩} dO_{A_{0}}dO_{B_{0}}.

We can split this into two integrals, and by Harish-Chandra’s formula this is equal to

c\frac{∑_{α∈S_{n}} sgn(α)e^{−2πi⟨α(A_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})} \frac{∑_{β∈S_{n}} sgn(β)e^{−2πi⟨β(B_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})}

which we rewrite as

c\frac{∑_{α,β∈S_{n}} sgn(αβ)e^{−2πi⟨α(A_{0})+β(B_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})^{2}} .

On the other hand, every A + B must belong to a co-adjoint orbit O_{C_{0}} for some C_{0} in the range of e. We can therefore rewrite the integral as

∫ ∫_{O_{C_{0}}} e^{−2πi⟨C,ξ⟩} dO_{C_{0}}(C)e_{∗}(dO_{A_{0}} × dO_{B_{0}})(C_{0}).

By (1) this is equal to

c ∫ ∏_{j<k}(x_{k} − x_{j})^{−1} \frac{∑_{σ∈S_{n}} sgn(σ)e^{−2πi⟨σ(x),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})} e_{∗}(dO_{A_{0}} × dO_{B_{0}})(x).

Equating the two expressions, we obtain

∫ ∑_{σ∈S_{n}} sgn(σ)e^{−2πi⟨σ(x),ξ⟩} \frac{e_{∗}(dO_{A_{0}} × dO_{B_{0}})(x)}{∏_{j<k}(x_{k} − x_{j})} = c\frac{∑_{α,β∈S_{n}} sgn(αβ)e^{−2πi⟨α(A_{0})+β(B_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})} . (9)

The function

f(x)dx = \frac{e_{∗}(dO_{A_{0}} × dO_{B_{0}})(x)}{∏_{j<k}(x_{k} − x_{j})}

is supported on the cone {x : x_{1} ≥ x_{2} ≥ … ≥ x_{n}}. We symmetrize it by defining

\overline{f}(x) = ∑_{σ∈S_{n}} sgn(σ)f(σ(x)),

so that it is now defined on all of R^{n}. We can rewrite the left-hand side of (9) as

∑_{σ∈S_{n}} sgn(σ) ∫ e^{−2πi⟨x,ξ⟩}f(σ(x)) dx = ˆ \overline{f}(ξ).

We therefore have a formula

ˆ\overline{f}(ξ) = c\frac{∑_{α,β∈S_{n}} sgn(αβ)e^{−2πi⟨α(A_{0})+β(B_{0}),ξ⟩}}{∏_{j<k}(ξ_{k} − ξ_{j})}

which can be used to determine \overline{f} (it is a piecewise polynomial function), which in turn determines f (by restricting it to the cone where x is decreasing), which in turn determines the measure e_{∗}(dO_{A_{0}} × dO_{B_{0}})(x). In principle, one could then determine the range of e by working out the support of this measure. But this is combinatorially very complicated; it was only accomplished very recently in [1].

References

1. A. Berenstein, A. Zelevinsky, Triple multiplicites for sl(r +1) and the spectrum of the exterior algebra of the adjoint representation, J. Alg. Comb. 1 (1992), 7–22.

2. J.J. Dusitermaat, G.J. Heckman, On the variation in the coohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 256–269.

3. V. Guillemin, S. Sternberg, Symplectic techniques in physics, Cambridge University Press, 1984.

4. A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225–241.

5. A. Knutson, T. Tao, The honeycomb model of the Berenstein-Zelevinsky cone I: Proof of the Saturation Conjecture, to appear, JAMS.

6. A. A. Klyachko, Stable vector bundles and Hermitian operators, IGM, University of Marnela-Vallee preprint (1994).

7. Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math 79 (1957), 87–120.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

E-mail address: tao@math.ucla.edu