Representation Theory
Organizer: Xinwen Zhu and Zhiyu Zhang
Upcoming Events
Lusztig's theory of canonical bases reveals a remarkably rigid and positive algebraic structure on quantum groups and their modules. In symmetric types, it is known that the structure constants for multiplication in the negative part $U^-$, as well as for the action of Chevalley generators $E_i…
Any construction of a quantum computer would require finding good sets of quantum logic gates: finite sets of 2^n-by-2^n unitary matrices that efficiently and computably approximate arbitrary unitary matrices through short products. We explain a connection between constructing these gate…
Past Events
Fargues and Scholze constructed a local Langlands correspondence, parametrizing representations of p-adic reductive groups in terms of Galois theory. On the other hand, such representations can be constructed explicitly in terms of so-called Yu data. I will discuss joint work with Sean Cotner…
I will discuss joint with Robin Bartlett and Bao V. Le Hung on modularity lifting for three-dimensional Galois representations that are potentially crystalline of minimal regular weight at all places above p. I will begin by reviewing the Taylor-Wiles method and the analogous results in…
The Langlands correspondence for real groups is a classification of irreducible admissible representations of a real reductive group in terms of Langlands parameters associated with the dual group. It was conjectured by Soergel (and proved in a restricted setting by Bezrukavnikov and Vilonen)…
We explain a conjectural generalisation of Uglov’s level-rank duality that arises from the theory of d-Harish-Chandra series introduced by Broué-Malle-Michel (which has applications to modular representation theory of finite groups of Lie type). We discuss connections with character sheaves…
Abstract: The universal monodromic affine Hecke category is a family of categories over the dual torus. It is obtained by allowing sheaves on the enhanced affine flag variety with arbitrary monodromy along the torus orbits. I will discuss a Langlands dual coherent realization, which is joint…
(joint work with Eunsu Hur) Fargues and Scholze give a geometric construction of L-parameters attached to smooth irreducible representations of p-adic groups. They furthermore predict an enhancement to a category equivalence, following the philosophy of the geometric Langlands program. I will…
The determination of the unitary dual of a Lie group is a longstanding problem. In this talk I will explain how the unitarity of a representation of a real reductive group can be read off from its Hodge filtration establishing a conjecture made by Wilfried Schmid and myself a while back. This is…
Moy-Prasad filtration subgroups are generalization of congruence subgroups for $GL_n(Q_p)$ to a general $p$-adic reductive group $G(F)$. Moy-Prasad proved that any irreducible smooth representation of $G(F)$ has its restriction to a Moy-Prasad subgroup given by an irreducible representation (…
In 1988, Kazhdan and Laumon constructed an abelian category associated to a reductive group G over a finite field, with the aim of using it to construct discrete series representations of the finite Chevalley group G(F_q). The well-definedness of their construction depended on their conjecture…