Abstract
Let G be a p-adic group (or lace group over a finite field) and H a compact open subgroup of G. The Hecke ring Z[H\G/H] is well understood for some H (parahoric or their pro-unipotent radicals), but remains mysterious in general. We'll show that, after inversion of p, this ring is finite on its center, which is a Z[1/p]-algebra of finite type. For this, we use the work of Fargues and Scholze, who construct a morphism from the ring of functions on the Langlands parameter moduli space to the center of this Hecke ring. We are then brought back to a natural and interesting problem of finiteness " relative " of character varieties, initiated by Vinberg in characteristic 0 and completely solved by Cotner very recently.
