Representation Theory Seminar

Representation Theory Seminar
2010-2011
Wednesdays at 2:00pm in JWB 333

2009-2010: 2009-2010
2008-2009: 2008-2009
2007-2008: 2007-2008

Date	Speaker	Title
September 15	Moshe Adrian	Local Langlands conjecture for p-adic groups I
September 22	Moshe Adrian	Local Langlands conjecture for p-adic groups II
September 29	Dan Ciubotaru	Formal degrees of discrete series of classical p-adic groups
October 6	Mike Woodbury (Wisconsin)	Subconvexity, the triple product L-function and trilinear forms
October 29	Wilfried Schmid (Harvard)	Invariant hermitian forms on Harish Chandra modules
November 3	Ben Trahan	Functors for the metaplectic group and graded affine Hecke algebras
December 1	Matt Housley	Associated cycles of Harish-Chandra modules for SU(p,q)
January 19	Jeffrey Adams (Maryland)	The Contragredient
January 26	Moshe Adrian	An introduction to rational forms of linear algebraic groups
February 2	Moshe Adrian	A possible first step towards strong rational forms of p-adic groups
February 9	Moshe Adrian	Strong rational forms II
March 2	Marty Weissman (UC Santa Cruz)	Structure and representations of metaplectic groups
March 23	Tommaso Centeleghe (Heidelberg)	On certain automorphic forms on the quaternion with prime discriminant
April 1, Friday 2pm	Gopal Prasad (Michigan)	Colloquium (JWB 335): Number theoretic techniques in the study of Lie groups and locally symmetric spaces
April 6	Dragan Milicic	Geometry and unitarity
April 13	Wee Liang Gan (UC Riverside)	Infinitesimal Hecke algebras

Maintained by Dan Ciubotaru.

October 29, 2010
Wilfried Schmid
Title: Invariant hermitian forms on Harish Chandra modules
Abstract: Understanding the unitary dual of a reductive Lie group G is equivalent to knowing which irreducible Harish Chandra modules carry a positive definite hermitian form invariant under the action of the Lie algebra of G. Such hermitian forms do not fit well into the D-module approach to Harish Chandra modules. Vogan, Trapa, et al. have pointed out that - subject to a (technically innocuous) additional hypothesis - an irreducible Harish Chandra module carries a nontrivial hermitian form invariant under the action of a compact real form of the complexification of G; knowing the latter helps in approaching the unitarity problem. I shall recall the relevant facts about invariant hermitian forms, and describe a D-module construction of the second type of hermitian form. This is joint work with Kari Vilonen.

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March 2, 2011
Marty Weissman (UC Santa Cruz)
Title: Structure and representations of metaplectic groups
Abstract: When G is the group of K-points of a connected reductive group over a local field K, G is naturally a locally compact topological group whose representations are widely studied. I will describe a class of topological central extensions of G (by finite abelian groups), which have "algebraic origin" though they do not arise as K-points of algebraic groups over K. These "metaplectic groups" (or "nonlinear covers") have attracted much attention, and we struggle to incorporate them into the Langlands program, where they play a mysterious role in functoriality. I will discuss some foundational problems when attempting to parameterize representations of metaplectic groups, some results for tori, and some results for depth zero representations of tame covers of p-adic groups.

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March 23, 2011
Tommaso Centeleghe
Title: On certain automorphic forms on the quaternion with prime discriminant
Abstract: Let p be a prime number, and D the Q-quaternion algebra ramified at p and infinity. In this seminar we consider unitary, automorphic representations of the multiplicative group D^* that are trivial at infinity and whose local component at p contains a nonzero vector invariant by the maximal pro-p group of D_p^*. This condition at p arises in a natural way when considering the reduction mod p of systems of eigenvalues associated to unitary automorphic representations on D^* that are trivial at infinity. For an integer N>0 not divisible by p, let A(p,N) the number of automorphic representations of the type considered and whose conductor away from p (defined locally as in the GL_2 case) is equal to N. The main theorem that we present is a closed formula for A(p,N). We conclude the seminar indicating how a theorem of Serre motivates the study of the reduction mod p of systems of eigenvalues arising from automorphic forms on D^* that are trivial at infinity.

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April 13, 2011
Wee Liang Gan
Title: Infinitesimal Hecke algebras
Abstract: Infinitesimal Hecke algebras were introduced by Etingof, Gan and Ginzburg around 2005. They are analogs of degenerate Hecke algebras in which the role of the finite group is replaced by a Lie algebra. I will give an introduction and discuss their representation theory in a simplest case.

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