Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras

Abstract

Abstract. Let W = Λ W• be an Iwahori-Weyl group of a connected reductive group G over a non-archimedean local field. The subgroup W• is a finite Weyl group and the subgroup Λ is a finitely-generated abelian group (possibly containing torsion) which acts on a certain real affine space by translations.I prove that if w ∈ W and w / ∈ Λ then one can apply to w a sequence of conjugations by simple reflections, each of which is length-preserving, resulting in an element w for which there exists a simple reflection s such that

The Bernstein Presentation for general connected reductive groups

“…Now we prove spanning. Elements of W will sometimes be denoted by pairs using the semidirect product It is now routine to extend the scope of [Ros13] to reducible root systems:…”

“…A similar proof is used for linear-independence of the set of all z O . To prove spanning, we apply the main theorem from [Ros13].…”

“…This notation is used only in §6.2 to establish that the main results of [Ros13] apply to reducible root systems, despite the assumption in [Ros13] of irreducibility.…”

“…Compatibility with[Ros13]. In the next subsection §6.3, the main theorem from[Ros13] is used to prove that{z O } O∈ΩM /W• spans Z[H(G; I)].In[Ros13], it was assumed for convenience that the root system defining the affine Weyl group was irreducible, and this subsection shows that this assumption is not really necessary to the conclusion. Denote by Aut(A ss i ) the group of invertible affine transformations of A ssi (see §2.4).…”

Center of $\mathcal{H}_R(0,c_{\tilde{s}})$ associated to a pro-$p$-Iwahori Weyl group

“…The length ℓ is constant on each W 0 -conjugacy class in Λ. By Lemma 2.1 in[12], a conjugacy class of W is finite if and only if it is contained in Λ, and infinite if and only if it is disjoint from Λ.We'll later use the following geometric characterization of length (see Lemma 5.1.1 in[10]):…”