68^th Algebra Day

Let l be a rational prime. In a work with Khare and Larsen we proved that the groups of Lie type B_n, C_n and G₂ over the finite field of order l^k appear as Galois groups over rational numbers for infinitely many k. The main tools are l-adic representations attached to automorphic representations. Our approach consists of carefully constructing automorphic representations so that the corresponding l-adic representations give desired Galois groups.

We study Chow rings and motives of the varieties of totally isotropic subspaces of a hermitian form (over a field). These are projective homogeneous varieties under an action of the corresponding projective unitary group. Applications to quadratic forms, incompressibility, and the question of isotropy of involutions on central simple algebras are provided. This work is joint (in part) with Maksim Zhykhovich.

Until a few years ago it was not known that there are infinitely many cusp forms on a group such as SL_n beyond very small values of n. Weyl's law refers to an asymptotic formula for the number of cusp forms on a given connected reductive group, in particular establishing their infinitude. I will discuss some work-in-progress, joint with Werner Mueller of Bonn, establishing Weyl's law with remainder terms for classical groups and the results we have so far. Without a remainder, Weyl's law was recently established by Lindenstrauss and Venkatesh in a rather general setting.

We will discuss a new interpretation of symplectic branching which, unlike the branching for the other towers of classical groups, is not multiplicity-free. Our first result shows how to reduce questions about symplectic branching to analogous ones concerning the general linear groups. This is achieved via a canonical isomorphism of "branching algebras". Our second result asserts that each multiplicity space that arises in the restriction of an irreducible representation of Sp_2n(C) to Sp_2n-2(C) has a canonical irreducible action of the n-fold product of SL₂(C). As an application we obtain a Gelfand-Zetlin type basis for all irreducible finite-dimensional representations of Sp_2n(C)

Affine coordinate rings of conjugacy classes of nilpotent matrices play a useful role in other branches of mathematics and so do their reflexive modules of rank one. We shall give descriptions and properties of them using line bundles of explicit desingularisations.