Smooth Representations of Reductive p-Adic Groups over Arbitrary Fields
Time: 2019-12-05
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.
I will give lectures on the basic theory of smooth C-representations of the reductive p-adic group G := G(F).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.
