Arithmetic and Representation Theory (ART) Seminar

Abstract: In this talk, we will use type theory to construct a family of depth 1/N supercuspidal representations of p-adic GL(2N, F) which we call middle supercuspidal representations. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, we will show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of F* and simple supercuspidal representations of GL(N, F). Furthermore, we will pose a conjecture which refines the Local Converse Theorem for GL(n, F).

Abstract: I will introduce the study of arithmetic subgroups of isometries of hyperbolic space (of the first type) and their quotients which are finite volume hyperbolic orbifolds. These groups are constructed from quadratic forms defined over totally real number fields and the geometry of their quotients reflects the arithmetic properties of the associated forms. I will also discuss joint work with P. Murillo concerning commensurability and the length spectrum of arithmetic hyperbolic manifolds.

Abstract: I will first give a review of many people works on Loop groups, vector bundles on complex P^1 and Langlands duality for reductive groups. Then I will discuss recent progress on extending the story to a more general setting.