Representations and Lie Theory SeminarYear 2014  2015
Time: Wednesday, 16:30  17:30


September 17: We study representations of GL(n) appearing as quotients of a tensor of exceptional representations, in the sense of Kazhdan and Patterson. Such representations are called distinguished. We characterize distinguished principal series representations in terms of their inducing data. In particular, we complete the proof of a conjecture of Savin, relating distinguished spherical representations to the image of the tautological lift from a suitable classical group.
September 24: We explain a method due to BernsteinReznikov to obtain estimates for the (generalized) Fourier coefficients of the restriction of an automorphic function on a locally symmetric space to a (compact) totally geodesic subspace. This method depends in a crucial way on the multiplicity one property for the corresponding pair of groups and the knowledge of explicit intertwining operators between representations of these groups.
Applying this method to hyperbolic manifolds, we explain how the relevant intertwining operators can be constructed and how they can be used to obtain new estimates for the restriction of automorphic functions to compact hyperbolic subspaces.
This is joint work with B. Ørsted and Y. Oshima.
October 1: We describe a method to reduce certain questions (like tensor product decomposition, dimension formulas) about finite dimensional representations of the algebraic supergroup GL(mn) resp. the Lie superalgebra gl(mn) to lower rank cases such as gl(mrnr) for some r. This is done in the following way: To every representation and every odd nilpotent element of of gl(mn) we associate an infinite complex whose cohomology groups are representations of gl(mr,nr). We compute the cohomology of this complex if the representation is irreducible and give applications.
October 29: The gamma factors are arithmetic invariants we can attach to generic representations of reductive groups over local fields. We will be interested in the exterior square gamma factor of a supercuspidal representation of GL(n). Stability is the phenomenon that the gamma factor stabilizes upon twists by highly ramified characters. We will define these gamma factors, then explain their relation with Shahidi's local coefficients and how this gives an integral formula for the gamma factors as a Mellin transform of a (partial) Bessel function. Stability then comes about by understanding the asymptotics of these Bessel functions. This we do by thinking of the Bessel function as an orbital integral and adapting the theory of Shalika germs. This result has as an application the preservation of the associated local epsilon factors under the local Langlands correspondence.
November 5: Let F be a nonarchimedean local field of characteristic zero. The local converse problem for GL(n,F) asks to what extent the twisted gamma factors determine a representation of GL(n,F). Jacquet has formulated a conjecture on precisely which family of twisted gamma factors should uniquely determine a representation of GL(n,F). Because of recent work of Dihua Jiang, Chufeng Nien, and Shaun Stevens, this conjecture is now settled in many cases. I will report on recent work to complete the proof of Jacquet's conjecture, joint work with Baiying Liu, Shaun Stevens, and Peng Xu. I will then report on a refined version of Jacquet's conjecture in the case of GL(l,F), where l is prime. This latter work is joint with Baiying Liu and Geo Kam Tam Fai.
November 12: The cohomology of real flag manifolds (real split case) can be seen to be encoded in the Toda Lattice flow and its singularities. It also encodes relations between principal series representations. I will survey, mostly with examples, joint work with Yuji Kodama which built on previous joint work with Bob Stanton.
November 19: Let V be a 2ndimensional vector space over a field k of characteristic not 2 or 3 and let g be a nondegenerate symmetric bilinear form on V of maximal Witt index. We will describe a construction of exceptional Lie algebras and some of their fundamental finite dimensional representations using the spinors of V. This is joint work with Marcus Slupinski.
January 14: Given an arbitrary complex number delta, Deligne defined a symmetric monoidal category Rep(GL_delta) containing an object of dimension delta, which is in some sense universal among all such categories. In this talk I will explain how Rep(GL_delta) is defined in terms of oriented Brauer diagrams. Then I will describe a classification of the indecomposable objects in Rep(GL_delta) by bipartitions. Finally, I will discuss the ideals of Deligne's category that arise as kernels of tensor functors to representations of general linear supergroups and what is known about all ideals in Rep(GL_delta).
January 21: Let L be a complex semi simple Lie algebra and U(L) the universal enveloping algebra. In the first part of the lecture I will describe the free abelian category Ab [U(L)] over U(L), and its relationship to definability. Then we will localize Ab [U(L)] at the Serre subcategory S associated to the finite dimensional representations of U(L). Using the existence of contravariant positive definite forms, due to Jantzen and KacPeterson, we will explain how the localization Ab[U(L)]/S is itself the free abelian category over a mysterious von Neumann regular algebra that we denote by U'(L) and how this equivalence of categories is induced by an inclusion of U(L) in U'(L), which is an epimorphism of algebras. (This work is joint with Sonia L'Innocente.)
February 18: A semisimple algebraic group G acts on the function field of its Lie algebra k(g) via the adjoint action, and a basic question is this: can k(g) be generated by algebraically independent elements over the invariant field k(g)^G? In 2011, ColliotThelene, Kunyavskii, Popov, and Reichstein found the answer for all groups not containing a factor of type G2. In joint work with Florence and Reichstein, we recently settled this last case. I will give a rough overview of the problem, and sketch the solution in the G2 case concretely.
February 25: Consider a pair of exceptional representations in the sense of Kazhdan and Patterson, of a metaplectic double cover of GL(n). The tensor of these representations is a (very large) representation of GL(n). We characterize its irreducible generic quotients. In the squareintegrable case, these are precisely the representations whose symmetric square Lfunction has a pole at s=0. Our proof of this case involves a new globalization result. In the general case these are the representations induced from distinguished data or pairs of representations and their contragradients. The combinatorial analysis is based on a complete determination of the twisted Jacquet modules of exceptional representations. As a corollary, an exceptional representation is shown to admit a new ``metaplectic Shalika model".
March 4: In the remarkable work of Nakajima, it was shown that the study of infinite dimensional Heisenberg algebra over the homology group of moduli space of torsion free rank 1 sheaves on a surface S and certain operators, known as Nakajima operators, given by the generators of this Heisenberg algebra, provides a tool to study the cohomology of Hilb^n(S) (the absolute Hilbert scheme of n points on S). Later Okounkov and Carlsson generalized the work of Nakajima and constructed a rather different set of operators acting on homology groups of Hilb^n(S). These operators, known as "vertex operators", depend on choice of a fixed line bundle, M, over S, and could be explicitly written with respect to the Nakajima operators. In this talk, I will talk about joint work with Amin Gholampour on obtaining the "relative version" of OkounkovCarlsson generating series for certain top intersection numbers of "relative" Hilbert schemes of points on a surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the nonrelative Hilbert schemes. In particular, we obtain a relative version of the explicit formula found by CarlssonOkounkov for the Euler class of the twisted tangent bundle of Hilbert schemes. We also prove a relative version of (conjectural) generating series of the top Segre class of tautological bundles associated to a line bundle on S. These integrals were studied by Lehn, as well as Donaldson, in connection with the computation of instanton invariants.
March 11: I will develop a local theory of symmetric square Lfunctions for general linear groups and prove a certain characterization of a pole of symmetric square Lfactors of squareintegrable representations, a uniqueness of certain trilinear forms and nonexistence of Whittaker models of higher exceptional representations.
April 1: In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications. This is joint work with Dmitry Gourevitch and Siddhartha Sahi.
April 22: Higher theta functions are the residues of Eisenstein series on covers of reductive groups. On the one hand, they generalize the Jacobi theta function, which comes from the double cover of GL_2. On the other, their Whittaker coefficients are not understood, even for higher covers of GL_2. In this talk I explain why one should expect a series of relations between the coefficients of theta functions on different groups and some new constructions which come close to establishing this, and explain how one may construct an automorphic form on the 4fold cover of GL_2 with algebraic Fourier coefficients. This is based on ongoing joint work with David Ginzburg.