MIT Lie Groups Seminar

Abstract: The algebra of functions on an affine algebraic variety is infinite-dimensional; but it is has (not canonically!) an increasing filtration by finite-dimensional subspaces. The growth of the dimensions of those subspaces is a polynomial of degree the dimension of the variety. Much more information about the variety can be captured by trying to pin down the filtration.

The same words apply to (finite length) representations of real and $p$-adic reductive groups; theu degree of the polynomial is the ``Gelfand-Kirillov dimension'' of the representation. For real groups, Wilfried Schmid and Kari Vilonen have shown how to find a canonical increasing filtration. and so a canonical polynomial. I'll discuss some known and conjectural properties of their filtration and the corresponding polynomials, with the hope of encouraging someone to produce similar results for $p$-adic groups.

Abstract: In the first part of this talk, I will focus on the question of equidistribution of commutators in finite groups of Lie type. I will talk about some known results, as well as a proof of the Shalev conjecture for finite symplectic groups.

In the second part of the talk, I will discuss my approach to Howe duality for finite fields based on endomorphism algebras and interpolation. As an application, I will also discuss a result on the stable part of the Gurevich-Howe approach to equidistribution of commutators.

Abstract:Let G be an unramified group over a p-adic field F, and F_r/F an unramified extension of degree r. Let H(G) (resp. H(G(F_r)) denote the Hecke algebra of G(F) (resp. G(F_r)). Roughly speaking, we say two functions \phi\in H(G(F_r)) and f\in H(G) are associated (or matching functions) if they have the same stable orbital integrals. One main question is: how can we construct matching functions? In 1986, Kottwitz proved the unit elements of some Hecke algebras are associated. In 1990, Clozel defined a base change map between spherical Hecke algebras and proved the two functions corresponded by the base change map are associated. Later in 2009 and 2012, Haines generalized Clozel's result to centers of parahoric Hecke algebras and Bernstein centers of depth zero principal series block. In this talk, we will briefly introduce the history and set up of base change fundamental lemma, and focus on how we can generalize the result to general principal series blocks. This requires the concrete constructions of types for principal series blocks of unramified groups, and some concrete computations of root groups, which might give some inspirations on future study on deeper level structures.

Abstract: In 1996, Serre asked about the existence of motives with exceptional motivic Galois group. Such motives are now known to exist for all exceptional groups, due to work of Gross--Savin, Katz, Dettweiler--Reiter, Yun, Patrikis, Boxer--Calegari--Emerton--Levin--Patrikis--Madapusi Pera, Guralnick--Lubeck--Yu, and Faegerman (and possibly others). For all exceptional groups except \(E_6\), families of such motives were known, but until recently this was open for \(E_6\). I'll survey this story, explain joint work with Krämer--Maculan constructing families of exceptional \(E_6\)-motives---arising from the classical geometry of cubic \(3\)-folds---and speculate about similar constructions for other exceptional groups.

Abstract: The Kazhdan--Lusztig isomorphism, which relates the affine Hecke algebra of a $p$-adic group to the equivariant $K$-theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne--Langlands conjecture on the classification of tamely ramified irreducible representations. For a spherical variety $X$, we can construct two modules over the affine Hecke algebra: the first by considering Iwahori-invariant functions on $X$, and the second using relative Langlands duality and equivariant $K$-theory. It is natural to expect a relationship between these modules. I will discuss this relationship for $X = \mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$ and its application to the study of distinguished representations.