Friday, Feb. 13

Speaker: Benedict Gross (Harvard)

Title: On the restriction of irreducible representations of the group U_n(k) to the subgroup U_{n-1}(k)

Let k be a local field, and let K be a separable quadratic field extension of k. It is known that an irreducible complex representation p1 of the unitary group G1 = U_n(k) has a multiplicity free restriction to the subgroup G2 = U_{n-1}(k) fixing a non-isotropic line in the corresponding Hermitian space over K. More precisely, if p2 is an irreducible representation of G2, then p = p1 x p2 is an irreducible representation of the product G = G1 x G2 which we can restrict to the subgroup H = G2, diagonally embedded in G. The space of H-invariant linear forms on p has dimension = 1. In this talk, I will use the local Langlands correspondence and some number theoretic invariants of the Langlands parameter of p to predict when the dimension of H-invariant forms is equal to 1, ie. when the dual of p2 occurs in the restriction of p1. I will also illustrate this prediction with several examples, including the classical branching formula for representations of compact unitary groups. This is joint work with Wee Teck Gan and Dipendra Prasad.