  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.

Group Theory