The pro--Iwahori Hecke algebra of a reductive -adic group I

Abstract

Let $R$ be a commutative ring, let $F$ be a locally compact non-archimedean field of finite residual field $k$ of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. We show that the pro-$p$-Iwahori Hecke $R$-algebra of $G=\mathbf{G}(F)$ admits a presentation similar to the Iwahori–Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was previously known only for a $F$-split group $\mathbf{G}$.

A comparison between pro-p Iwahori–Hecke modules and mod p representations

“…The algebra H is called pro-p Iwahori-Hecke algebra. The structure of this algebra is studied by Vignéras [Vig16].…”

“…Define T * w as in [Vig16,4.3] for w ∈ W (1). This is also a basis of H and it satisfies the following:…”

Cocenters and representations of pro-𝑝 Hecke algebras

“…After the paper was finished, we learned from Vignéras that Theorem 4.2 is also proved in her paper [24] and her preprint [23]. We thank her for sending us [23] and for many useful comments.…”

“…We thank her for sending us [23] and for many useful comments. We also thank the referee for a very careful reading and many helpful comments.…”

“…If one considers the right module instead of left module, then our construction becomes − ⊗H− JH , while the construction in [24] uses − ⊗H+ JH . We do not know the explicit relationship between the two constructions.…”