2021/04/27 by Rachel Ollivier, Peter Schneider, Ollivier, Rachel +1
Mathematics · #11F85 #16E30 #20C08 #20J06 #22E50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2104.13422
openalex publication_date 2021/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G=\rm SL2(\mathfrak F) where \mathfrak F is a finite extension of \mathbb Qp. We suppose that the pro-p Iwahori subgroup I of G is a Poincaré group of dimension d. Let k be a field containing the residue field of \mathfrak F. In this article, we study the graded Ext-algebra E^*=ExtMod(G)^*(k[G/I], k[G/I]). Its degree zero piece E0 is the usual pro-p Iwahori-Hecke algebra H. We study Ed as an H-bimodule and deduce that for an irreducible admissible smooth representation of G, we have Hd(I,V)=0 unless V is the trivial representation. When \mathfrak F=\mathbb Qp with p≥ 5, we have d=3. In that case we describe E^* as an H-bimodule and give the structure as an algebra of the centralizer in E^* of the center of H. We deduce results on the values of the functor H^*(I, -) which attaches to a (finite length) smooth k-representation V of G its cohomology with respect to I. We prove that H^*(I,V) is always finite dimensional. Furthermore, if V is irreducible, then V is supersingular if and only if H^*(I,V) is a supersingular H-module.