2024/07/22 by Karol Kozioł, Koziol, Karol, Rachel Ollivier +3
Mathematics · #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.2407.15699
openalex publication_date 2024/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrakF be a nonarchimedean local field of residual characteristic p, and let G denote the group of \mathfrakF-points of a connected reductive group over \mathfrakF. For an open compact subgroup U of G and a unital commutative ring k, we let XU denote the space of compactly supported k-valued functions on G/U. Building on work of Ollivier--Schneider, we investigate the graded \textrmExt-algebra EU^* := \textrmExtG^*(XU,XU)^\textrmop. 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 = \textrmSL2(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^*) \longrightarrow 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 ℓ-modular characteristics.