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.

Abstract: Given a family g : X -> S of smooth projective algebraic varieties over a number field K, one often wants to constrain the points s in S where the fibre X_s acquires "extra" algebraic structure. A basic sort of constraint which is important in unlikely intersection theory is that of a Galois-orbit lower bound: an inequality h(s) <= poly([K(s) : K]), where h is some logarithmic Weil height and K(s) is the field of definition of s. Recent work has focused on how to use G-functions constructed from degenerations of g to produce such inequalities.

We describe some new results in the case where g is a one-parameter degeneration of surfaces, and the central role played by rigid and "adelic" geometry.

Abstract: In this talk, we will give a description of the depth-r Bernstein center of a connected reductive p-adic group  for rational depths, as an inverse limit of algebras which we call depth-r standard parahoric Hecke algebras. We will introduce depth-r restricted Langlands parameters attached to smooth irreducible representations and use our description to construct projectors and give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted Langlands parameters. This is based on a joint work with Tsao-Hsien Chen.