Generic pro-$p$ Hecke algebras

TL;DR

This paper introduces and studies generic pro-$p$ Hecke algebras, extending the class of Hecke algebras by considering certain extensions of Coxeter groups, and provides structural results including bases and finiteness properties.

Contribution

It defines generic pro-$p$ Hecke algebras, introduces orientations and Bernstein maps, and establishes foundational structural results for affine cases.

Findings

Explicit canonical basis of the center for affine pro-$p$ Hecke algebras

Finiteness of the center and module-structure results

Connections to Bernstein-Zelevinsky-Lusztig and Vignéras results

Abstract

This is an extended and corrected version of the author's Diplomarbeit. A class of algebras called generic pro-$p$ Hecke algebras is introduced, enlarging the class of generic Hecke algebras by considering certain extensions of (extended) Coxeter groups. Examples of generic pro-$p$ Hecke algebras are given by pro-$p$-Iwahori Hecke algebras and Yokonuma-Hecke algebras. The notion of an orientation of a Coxeter group is introduced and used to define `Bernstein maps' intimately related to Bernstein's presentation and to Cherednik's cocycle. It is shown that certain relations in the Hecke algebra hold true, equivalent to Bernstein's relations in the case of Iwahori-Hecke algebras. For a certain subclass called affine pro-$p$ Hecke algebras, containing Iwahori-Hecke and pro-$p$-Iwahori Hecke algebras, an explicit canonical and integral basis of the center is constructed and finiteness results are proved about the center and the module-structure of the algebra over its center, recovering results of Bernstein-Zelevinsky-Lusztig and Vign\'eras.