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# Algebraic Geometry and Representation Theory Seminar

In analysis, a convolution of two functions usually results in a smoother, better behaved function. Given two morphisms f,g from algebraic varieties X,Y to an algebraic group G, one can define a notion of convolution of these morphisms. Analogously to the analytic situation, this operation yields a morphism (from X x Y to G) with improved smoothness properties.

In this talk, I will define a convolution operation and discuss some of its properties. I will then present a recent result; if G is an algebraic group, X is smooth and absolutely irreducible, and f:X-->G is a dominant map, then after finitely many self convolutions of f, we obtain a morphism with the property of being flat with fibers of rational singularities (a property which we call (FRS)).

Moreover, Aizenbud and Avni showed that the (FRS) property has an equivalent analytic characterization, which leads to various applications such as counting points of schemes over finite rings, representation growth of certain compact p-adic groups and arithmetic groups of higher rank, and random walks on (algebraic families of) finite groups. We will discuss some of these applications, and maybe some of the main ideas of the proof of the above result.

Joint with Yotam Hendel.

The celebrated Gan-Gross-Prasad conjectures aim to describe the branching behavior of representations of classical groups, i.e., the decomposition of irreducible representations when restricted to a lower rank subgroup.

These conjectures, whose global/automorphic version bear significance in number theory, have thus far been formulated and resolved for the generic case.

In this talk, I will present a newly formulated rule in the p-adic setting (again conjectured by G-G-P) for restriction of representations in non-generic Arthur packets of GL_n.

Progress towards the proof of the new rule takes the problem into the rapidly developing subject of quantum affine algebras. These techniques use a version of the Schur-Weyl duality for affine Hecke algebras, combined with new combinatorial information on parabolic induction extracted by Lapid-Minguez.