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

This paper studies affine Weyl groups through the geometry of geodesic tubes and their hyperbolic stabilizers. We introduce a geometric framework relating affine Weyl group factorizations to Hecke-algebra-valued Poincar'e series and zeta functions on quotient complexes. Using this framework, we prove a conjecture describing the connection between geodesic tubes and factorizations of Poincar'e series for affine Weyl groups. We also establish partial results toward a Hecke-algebra-valued zeta identity for simply connected groups, including the cases of types ${\tilde{A}}_{n-1}$ and ${\tilde{C}}_n$. These results highlight the interaction between affine Weyl group geometry, Hecke algebras, and zeta functions arising from Bruhat–Tits buildings.