2018/08/28 by Ollivier, Rachel, Schneider, Peter
#16E30 #20C08 #20J06 #22D35 #22E50 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1808.09503
Let \mathfrak F be a locally compact nonarchimedean field of positive residue characteristic p and k a field of characteristic p. Let G be the group of \mathfrakF-rational points of a connected reductive group over \mathfrakF which we suppose \mathfrak F-split. Given a pro-p Iwahori subgroup I of G, we consider the space \mathbf X of k-valued functions with compact support on G/I. It is naturally an object in the category Mod(G) of all smooth k-representations of G. We study the graded Ext-algebra E^*=ExtMod(G)^*(\mathbf X, \mathbf X). Its degree zero piece E0 is the usual pro-p Iwahori-Hecke algebra H. We describe the product in E^* and provide an involutive anti-automorphism of E^*. When I is a Poincaré group of dimension d, the Ext-algebra E^* is supported in degrees i∈\0… d\ and we establish a duality theorem between Ei and Ed-i. Under the same hypothesis (and assuming that \mathbf G is almost simple and simply connected), we compute Ed as an H-module on the left and on the right. We prove that it is a direct sum of the trivial character, and of supersingular modules.