Parahoric Hecke Ext-algebras in characteristic p

Let F be a nonarchimedean local field of residual characteristic p, and let G denote the group of F-points of a connected reductive group over F. For an open compact subgroup U of G and a unital commutative ring k, we let Xₔ denote the space of compactly supported k-valued functions on G/U. Building on work of Ollivier--Schneider, we investigate the graded Ext-algebra Eₔ^*: = ExtG^* (Xₔ, Xₔ) ^op. In particular, we describe the Yoneda product, an involutive anti-automorphism, and (when k is a field of characteristic p and U has no p-torsion) a duality operation. We allow for the reductive group to be non-split, and for the open compact subgroup U to be non-pro-p. Specializing further to the case G = SL₂ (Qₚ) with p 5 and a coefficient field of characteristic p, we obtain more precise results when U is equal to an Iwahori subgroup J or a hyperspecial maximal compact subgroup K. In particular, we compute the structure of EJ^* as an EJ⁰-bimodule, obtain an explicit description of the center Z (EJ^*) of EJ^*, and construct a surjective morphism of algebras Z (EJ^*) EK^* (analogous to the compatibility between Bernstein and Satake isomorphisms in characteristic 0). From this we deduce the (somewhat surprising) fact that EK^* is not graded-commutative, contrary to what happens for almost all -modular characteristics.