Representation Theory

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…