2017/03/15 by Rachel Ollivier, Ollivier, Rachel, Marie‐France Vignéras +1
Mathematics · #11E95 #20C08 #20G25 #22E50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1703.04921
openalex publication_date 2017/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be the group of rational points of a reductive connected group over a finite field (resp. nonarchimedean local field of characteristic p) and R a commutative ring. The unipotent (resp. pro-p Iwahori) invariant functor takes a smooth representation of G to a module over the unipotent (resp. pro-p Iwahori) Hecke R-algebra H of G. We prove that these functors for G and for a Levi subgroup of G commute with the parabolic induction functors, as well as with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show in the local case that an irreducible admissible R-representation V of G is supercuspidal (or equivalently supersingular) if and only if the H-module VI of its invariants by the pro-p Iwahori I admits a supersingular subquotient, if and only if VI is supersingular.