Hecke Algebra-valued Poincar\'e Series and Geometric Factorization of Affine Weyl Groups

TL;DR

This paper studies affine Weyl groups and their Hecke algebra-valued Poincaré series, revealing their connections to zeta functions and geometric structures, and confirming key conjectures in the field.

Contribution

It establishes a link between the Poincaré series and geodesic tubes, confirming a conjecture, and provides partial evidence for another related conjecture.

Findings

Confirmed the conjecture relating Poincaré series and geodesic tubes.

Extended the understanding of zeta functions in the context of affine Weyl groups.

Provided partial evidence for a conjecture on zeta identities for simply connected groups.

Abstract

This paper explores affine Weyl groups and their associated Hecke algebras, concentrating on the Poincar\'e series with coefficients in Hecke algebra. We investigate its relationship with zeta functions on complexes and extend existing research on geodesic tubes to encompass higher dimensions. Our main findings confirm a conjecture that elucidates the connection between the Poincar\'e series and geodesic tubes. Additionally, we provide partial evidence for another conjecture related to the zeta identity for simply connected groups. These contributions deepen our understanding of the interactions among algebraic groups, Hecke algebras, and the geometry of related complexes.