Let F be a non-archimedean local field (such as ℚ_p). The Langlands philosophy says that the arithmetic of F is intimately related to the category R(G) of smooth complex representations of G(F) where G denotes a reductive F-group (for example the general linear group). The building blocks of R(G) are the "supercuspidal" representations of G(F). I will define this term in the talk. The category R(G) comes equipped with an involution - the "contragradient" or the "dual". The supercuspidal representations of G(F) which are self-dual are of considerable interest in the subject. In this talk, I will talk about a joint work with Jeff Adler about the existence of supercuspidals and self-dual supercuspidals. Specifically, we show that G(F) always admits supercuspidal representations. Under some mild hypotheses on G, we determine precisely when G(F) admits self-dual supercuspidal representations. These results are obtained from analogous results for finite reductive groups which I will also talk about.
Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran.
Yeansu Kim : CLASSIFICATION OF DISCRETE SERIES REPRESENTATIONS AND ITS APPLICATIONS ON THE GENERIC LOCAL LANGLANDS CORRESPONDENCE FOR ODD GSPIN GROUPS- Uploaded by schrett ( 34 Views )
The classification of discrete series is one main subject in Langlands program with numerous applications. We first explain the result on the classification of discrete series of odd GSpin groups, generalizing the Mœglin-Tadi ́c classification for classical groups. Note that our approach will give alternate proof for classical groups. This is a joint work with Ivan Mati ́c. We also explain its application on the generic local Langlands correspondence via Langlands-Shahidi method. If time permits, we will explain possible generalization of those to other groups, which is work in progress