## Smooth Representations of Reductive p-Adic Groups over Arbitrary Fields

Time: 2019-12-05
Views: 673
Published By: He Liu

**Speaker(s): ** Marie-France Vignéras (Université Paris Diderot)

**Time: **January 14 - January 17, 2020

**Venue: ** Room 77201, Jingchunyuan 78, BICMR

Date: 10:00-12:00, January 14

10:00-12:00, January 16

10:00-11:00, January 17

Let (F, G, C) be a triple consisting of a local non-archimedean field F of characteristic a and residue field of characteristic p, a connected reductive linear group G defined over F and a field C of characteristic c. Either c = 0 or c is a prime number p or l≠p. The field of complex numbers is denoted by **C**.

According to the time, the audience and my energy, the lectures may include:

- a revised version of certain parts of my book [V96] where c ≠p, specially II.4 on rationality and integral structures.

- the irreducible cuspidal C-representations of G which are compactly induced, when c ≠p.

- the Bernstein center and the second adjoint theorem (when C =

**C**the book [R10]) when c ≠p is banal ([V96] II.3.9).

- when c = p, cuspidal = supersingular [AHHV16] [OV18].

- when c = p and a = 0, the existence a subgroup Γ of G, discrete, cocompact modulo Z(G) and such that the smooth part of C[ΓZ(G)\G] admits an admissible irreducible cuspidal constituent [HKV19].

References.

Two books in french:[V96] M.-F. Vignéras. Représentations l-modulaires d’un groupe réductif p-adique avec l≠p. Progress in Math. 137. Birkhauser. 1996.

[R10] D. Renard. Représentations des groupes réductifs p-adiques. Cours spécialisés 17. S.M.F. 2010.

Three articles in english:

[AHHV16] N. Abe, G. Henniart, Fl. Herzig, M.-F. Vignéras. Journal of the A.M.S. 30 (2016) no.2, 495-559.

[OV18] R. Ollivier, M.-F. Vignéras. Parabolic Induction in characteristic p. Selecta Mathematica, Volume 24, Issue 5, (2018), 3973-4039.

[HKV19] Fl. Herzig, K. Koziol, M.-F. Vignéras. On the existence of admissible supersingular representations of p-adic reductive groups. Accepted in Forum of Mathematics, Sigma. 2019.