Lie Groups and Representation Theory Archives for Fall 2022 to Spring 2023
Self-dual cuspidal representations
When: Wed, September 8, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Manish Mishra (IISER Pune) - https://sites.google.com/site/manishmishra/
Abstract: Let G be a connected reductive group defined over a finite or a non-archimedean local field F. We show that G(F) admits cuspidal representations when F is finite and supercuspidal representations when F is non-archimedean local. We also determine precisely when G(F) admits self-dual representations. For the results on self-duality, we assume some hypothesis on G. These hypotheses disallow G to have certain small rank factors when the field (in case F is finite) or the residue field (in case F is non-archimedean local) is of cardinality ≤ 5. When F is non-archimedean local and G is ramified, these hypotheses impose some additional restrictions on G. This is a joint work with Jeff Adler.
Arthur and ABV packets
When: Wed, September 22, 2021 - 2:00pmWhere: Kirwan Hall 3206 and https://umd.zoom.us/j/96890967721
Speaker: Jeffrey Adams (University of Maryland) -
Abstract: In 1983 Jim Arthur conjectured the existence of certain sets of representations
of a reductive group G over a local field F, associated to homomorphisms from the Weil-Deligne group x SL(2) into the L-group of G. In the case of split classical groups he defined these "Arthur" packets in his 2010 book. For F=R Adams, Barbasch and Vogan gave a definition in their 1992 book, commonly referred to as ABV. Nicolas Arancibia and Paul Mezo, with some help from the speaker, have proved these two definitions agree when they are both defined.
The talk will be available online at https://umd.zoom.us/j/96890967721
Arthur and ABV packets II
When: Wed, September 29, 2021 - 2:00pmWhere: Math 3206 and on zoom: https://umd.zoom.us/j/96890967721 (see seminar page for the password)
Speaker: Jeffrey Adams (University of Maryland) -
The smooth locus of twisted affine Schubert varieties
When: Fri, October 22, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Jiuzu Hong (UNC) - https://hong.web.unc.edu/
Abstract: Let G be a special parahoric group scheme of twisted type, excluding the absolutely special case for ramified odd unitary group. Using the methods and results of Zhu, we prove a duality theorem for G: there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in the affine Schubert varieties of G. Along the way, we also establish the duality theorem for untwisted E_6. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of G. In particular, this confirms a conjecture of Haines and Richarz. This talk is based on the joint work with Marc Besson.
A geometric approach to unipotent representations
When: Mon, November 22, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Dmytro Matvieievskyi (Yale University) -
Abstract: For a real reductive group G, the set of unipotent representations is a finite set of irreducible representations that are supposed to be “building blocks” for all unitary representations of G. Moreover, unipotent representations are expected to be indexed by nilpotent G-orbits, possibly with some additional data. There were several attempts to give a definition of a unipotent representation, most notably the set of special unipotent representations, due to Barbasch-Vogan and Arthur. However, this notion does not include some interesting unitary representations, such as a metaplectic representation. In joint work with Ivan Losev and Lucas Mason-Brown, we propose a new definition of unipotent representations of a complex semisimple Lie group that extends the set of special unipotent ones. Using these definitions we are able to parameterize the unipotent representations and prove their key properties, including some that were not known for special unipotent representations before. In this talk, I will explain the definition and the parametrization of unipotent representations. Time permitting I will sketch what we expect to be a definition of a unipotent representation of a real semisimple Lie group, and our approach to studying them. The talk is based on arXiv:2108.03453.
Derived Satake equivalence for Godement-Jacquet monoids, I
When: Mon, December 13, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Jonathan Wang (Perimeter Institute) -
Abstract: Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. I will explain how the local unramified part of this theory can be geometrized to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. This confirms a particular case of a general conjecture of Ben-Zvi-Sakellaridis-Venkatesh. I will discuss how this equivalence fits into the general framework of (relative) Langlands duality and to conjectures of Braverman-Kazhdan and Ngo on constructions of general automorphic L-functions. In the second talk I will give an overview of the proof of the equivalence and some of its finer properties, which have connections to invariant theory on a "dual" variety related to the standard L-function. This is joint work with Tsao-Hsien Chen (in preparation).
Derived Satake equivalence for Godement-Jacquet monoids, II
When: Tue, December 14, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Jonathan Wang (Perimeter Institute) -
Abstract: Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. I will explain how the local unramified part of this theory can be geometrized to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. This confirms a particular case of a general conjecture of Ben-Zvi-Sakellaridis-Venkatesh. I will discuss how this equivalence fits into the general framework of (relative) Langlands duality and to conjectures of Braverman-Kazhdan and Ngo on constructions of general automorphic L-functions. In the second talk I will give an overview of the proof of the equivalence and some of its finer properties, which have connections to invariant theory on a "dual" variety related to the standard L-function. This is joint work with Tsao-Hsien Chen (in preparation).
Product structure and regularity theorem for totally nonnegative flag varieties
When: Mon, April 4, 2022 - 8:30pmWhere: Online
Speaker: Xuhua He (CUHK) -
Abstract: The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this talk, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.
We show that the $J$-totally nonnegative flag variety has a cellular decomposition into totally positive $J$-Richardson varieties. Moreover, each totally positive $J$-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive $J$-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the $J$-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.
This talk is based on a joint work with Huanchen Bao.
On Hecke algebras for p-adic groups and local Langlands correspondances I
When: Mon, May 2, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Yujie Xu (Harvard) -
Abstract: I will talk about my recent joint work with A.-M. Aubert on Hecke algebras attached to Bernstein components of reductive p-adic groups, and applications to local Langlands correspondances, illustrated by some explicit examples.
On Hecke algebras for p-adic groups and local Langlands correspondances II
When: Wed, May 4, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Yujie Xu (Harvard) -
Abstract: I will talk about my recent joint work with A.-M. Aubert on Hecke algebras attached to Bernstein components of reductive p-adic groups, and applications to local Langlands correspondances, illustrated by some explicit examples.
How to restrict representations from a complex reductive group to a real form
When: Mon, May 9, 2022 - 2:00pmWhere: Online
Speaker: Lucas Mason-Brown (Oxford University) -
Abstract: Let G(R) be the real points of a complex connected algebraic group G. There are many difficult questions about admissible representations of real reductive groups which have (relatively) easy answers in the case of complex groups. Thus, it is natural to look for a relationship between representations of G and representations of G(R). In this talk, I will introduce a functor from admissible representations of G to admissible representations of G(R). This functor interacts nicely with many natural invariants, including infinitesimal character, associated variety, and restriction to a maximal compact subgroup, and it takes unipotent representations of G to unipotent representations of G(R).
Arithmetic modularity and geometry of unitary Shimura varieties at parahoric levels
When: Thu, May 19, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Zhiyu Zhang (MIT) - https://math.mit.edu/~zzyzhang/
Abstract: In this talk, I will present a general method to establish the modularity of arithmetic theta series for many “parahoric” hermitian lattices, which live on integral unitary Shimura varieties at parahoric levels. Via complex and p-adic uniformizations, the arithmetic modularity is proved in good cases. We study irreducible components and nice stratifications of the mod p fiber and its basic locus to reduce general cases to good cases by modifications. I will explain new geometric phenomena when the group is non-quasi-split.
Arithmetic transfer identities and singularities at parahoric levels
When: Fri, May 20, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Zhiyu Zhang (MIT) - https://math.mit.edu/~zzyzhang/
Abstract: For some p-adic parahoric hermitian lattices, I will formulate and prove some arithmetic transfer identities relating derived intersection numbers on local integral unitary Shimura varieties to central derivatives of orbital integrals, in the context of the arithmetic GGP conjecture (and its p-adic analogs). A key feature at parahoric levels is the singularity of related moduli spaces, which we resolve via nice stratifications to define well-behaved intersection numbers. The proof is local-global, using our new arithmetic modularity results as a key input.