Lie Groups and Representation Theory Archives for Academic Year 2019


Organizational Meeting

When: Wed, August 29, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Jeffrey Adams () -


Hodge Theory and Unitary Representations

When: Wed, September 5, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Jeffrey Adams (University of Maryland) -
Abstract: I will describe an algorithm to compute the Hodge filtration on a representation of a real reductive group. This is a generalization of an algorithm to compute signatures of Hermitian forms, and is related to a conjecture of Schmid and Vilonen relating the unitary dual and Hodge theory.

Types and unitary representations of reductive p-adic groups

When: Mon, September 17, 2018 - 2:00pm
Where: Kirwan Hall 1313

Speaker: Dan Ciubotaru (University of Oxford) - https://www.maths.ox.ac.uk/people/dan.ciubotaru

Abstract: I show that for every Bushnell–Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. Moreover, I'll explain that every irreducible smooth G-representation contains a rigid type. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors and the main new ingredient is a rigid trace Paley-Wiener theorem proved in joint work with Xuhua He.

Arakawa-Suzuki functors for Whittaker modules

When: Mon, September 24, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Adam Brown (University of Utah) -


Abstract: The category of Whittaker modules is a category of Lie algebra
representations which generalizes other well-studied categories of
representations, such as the Bernstein-Gelfand-Gelfand category O. In this
talk we will construct a family of exact functors from the category of
Whittaker modules to the category of finite-dimensional graded affine
Hecke algebra modules, for type A_n. These algebraically defined functors
provide us with a representation theoretic analogue of certain geometric
relationships, observed independently by Zelevinsky and Lusztig, between
the flag variety and the variety of graded nilpotent classes. Using this
geometric perspective and the corresponding Kazhdan-Lusztig conjectures
for each category, we will prove that these functors map simple modules to
simple modules (or zero). Moreover, we will see that each simple module
for the graded affine Hecke algebra can be realized as the image of a
simple Whittaker module.

Unitary Representations and Hodge Theory II

When: Wed, October 10, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Jeffrey Adams (University of Maryland) -


The Langlands-Kottwitz-Scholze method for Shimura varieties of abelian type

When: Fri, November 2, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Alex Youcis (UC Berkeley) -
Abstract: The local (and global) Langlands conjectures attempt to bridge the major areas of harmonic analysis and number theory by forming a correspondence between representations which naturally appear in both areas. A key insight due to Langlands and Kottwitz is that one could attempt to understand such a conjectural correspondence by comparing the traces of natural operators on both sides of the bridge. Moreover, it was realized that Shimura varieties present a natural means of doing this. For global applications, questions of reduction type (at a particular prime $p$) for these Shimura varieties can often be avoided, and for this reason the methods of Langlands and Kottwitz focused largely on the setting of good reduction. But, for local applications dealing with the case of bad reduction is key. The setting of bad reduction was first dealt with, for some simple Shimura varieties, by Harris and Taylor which they used, together with the work of many other mathematicians, to prove the local Langlands conjecture for GL_n. A decade later Scholze gave an alternative, more geometric, way to understand the case of bad reduction for certain Shimura varieties and was able to reprove the local Langlands conjecture for GL_n. In this talk we will discuss an extension of the ideas of Scholze to a wider class of Shimura varieties, as well as the intended application of these ideas to the local Langlands conjectures for more general groups.

From Schur duality to quantum symmetric pairs

When: Fri, November 16, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Huanchen Bao (University of Maryland)
Abstract:
The classical Schur(-Weyl) duality relates the representation theory of general linear Lie algebras and symmetric groups. Drinfeld and Jimbo independently introduced quantum groups in their study of exactly solvable models, which leads to a quantization of the Schur duality relating quantum groups of general linear Lie algebras and Hecke algebras of symmetric groups.
In this talk, I will explain the generalization of the (quantized) Schur duality to other classical types, algebraically, geometrically and categorically. This new duality leads to a theory of canonical bases arising from quantum symmetric pairs generalizing Lusztig's canonical bases on quantum groups.

The orbit philosophy for Spin groups

When: Mon, December 3, 2018 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Wan-Yu Tsai (University of Ottawa) -

Abstract: Let G be a semisimple Lie group with Lie algebra \g and maximal compact subgroup K. The philosophy of coadjoint orbits suggests a way to study unitary representations of G by their close relations to the coadjoint G-orbits on \g*. In this talk, we study a special part of the orbit philosophy. We provide a comparison between the K-structure of unipotent representations and regular functions of bundles on nilpotent orbits for complex and real groups of type D. More precisely, we provide a list of genuine unipotent representations for a Spin group; separately we compute the K-spectra of the regular functions on certain small nilpotent orbits, and then match them with the K-types of the genuine unipotent representations.This is joint work with Dan Barbasch.

Smoothness of Schubert varieties in twisted affine Grassmannians

When: Wed, February 13, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Tom Haines (UMCP) -
Abstract: This talk will describe results about Schubert varieties in affine Grassmannians. In joint work, Timo Richarz and I gave a complete list of smooth (resp. rationally smooth) Schubert varieties in the twisted affine Grassmannian associated with a tamely ramified group and a special vertex of its Bruhat-Tits building. When the underlying group is not split, there are surprising cases of smoothness not occurring in the split group setting, called cases of *exotic smoothness*. This is closely related to the recent He-Pappas-Rapoport classification of Shimura varieties with good reduction, and to the cases of *exotic good reduction* they describe. Our classification of smooth Schubert varieties verifies a conjectural classification postulated by Rapoport in 2010. The proof uses our theorem on normality of such Schubert varieties, a result which generalizes earlier work of Faltings and Pappas-Rapoport.

Deformation quantization of coadjoint orbits

When: Wed, February 27, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Shilin Yu (Texas A&M) -
Abstract: The coadjoint orbit method/philosophy suggests that irreducible unitary representations of a Lie group can be constructed as quantization of coadjoint orbits of the group. I will propose a geometric way to understand orbit method using deformation quantization, in the case of noncompact real Lie groups. This is joint work with Conan Leung.

Steinberg theory and Robinson-Schensted correspondence for partial permutations

When: Wed, March 6, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Kyo Nishiyama (Aoyama Gakuin University and MIT)
Abstract: RS correspondence is a bijection between permutation and pair of
standard tableaux with the same shape. This is a combinatorial
bijection, which can be described in many different ways. In early
70's, Steinberg found a geometric method to explain this bijection,
and at the same time he generalize it in vast way.

In this talk, we reconsider his geometric consideration to get a
seemingly new bijection between partial permutations and triplets of a
pair of tableaux and a partition called "core". We can also interpret
it by using double flag varieties for symmetric pairs, and get a
general frame work for considering those new combinatorial bijections,
which reveals an interesting interchange between geometry and
combinatorics.

The talk is based on the on-going joint work with Lucas Fresse at IECL
(France).

Special nilpotents and higher Teichmuller components

When: Wed, April 10, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Brian Collier (University of Maryland) -
Abstract: In this talk, we will consider a special class of nilpotent elements of a complex simple Lie algebra and show how they determine two real Lie algebras, one of which is a real form of the initial complex Lie algebra. After classifying these nilpotents, we will discuss the relation with Guichard and Wienhard's work on positivity. Time permitting, we will present a construction of higher Teichmuller components in Higgs bundle moduli spaces associated to every special nilpotent.

Normality and Cohen-Macaulayness of local models

When: Mon, April 29, 2019 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Tom Haines (UMCP) -
Abstract: It is an old problem (from around 1990) to determine the singularities which arise in special fibers of Shimura varieties with bad reduction. In the case of parahoric level, this problem may be studied with the help of local models. In this talk, I will describe a complete solution to the problem for all Kisin-Pappas Shimura varieties (and all corresponding moduli stacks of shutkas): we show these spaces are normal and Cohen-Macaulay, and thereby prove a conjecture of Pappas and Zhu. This is joint work with Timo Richarz.