Twisted Poincaré series and zeta functions on finite quotients of buildings

Abstract

In the case where G =SL 2 (F ) for a non-archimedean local field F and Γ is a discrete torsion-free cocompact subgroup of G, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of G by the action of Γ, and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of G. We show how this can be generalised to other split simple algebraic groups of rank two over F , and formulate a conjecture about how this might be generalised to groups of higher rank. Twisted Poincaré series and Ihara zeta functions2.1. Poincaré series. Let (W, S) be a Coxeter system with a generating set S consisting of elements of order two. For w ∈ W , let ℓ(w) be the shortest length of a word consisting of elements

{Twisted Poincaré Series and Geometric Factorization of Affine Weyl Groups

“…, ω ∨ i can be easily computed from Table 2 in [OV90] (page 295-297). Together with the table of degrees in Theorem 5.0.2 in [KM18], we prove the part (a) of Conjecture 1.0.1 as the following theorem.…”

“…Finally, let us show that X n × T n−1 × • • • × T 1 × X 1 is a length-preserving decomposition of W aff . By Theorem 3.5.1 in [KM18], the decomposition is length-preserving if and only if…”

“…where w 0 is the unique element of maximal length in W . In [KM18], the authors study the alternating product of the Poincaré series of parabolic subgroups for the case when W is an affine Weyl group. They show that there are some positive integers…”

Zeta and L-functions of finite quotients of apartments and buildings

“…In this case, the group G(Q 1 ) is isomorphic to the extended affine Weyl group and its building is a single apartment. Other approaches to study zeta functions for algebraic groups of higher rank are explored in [KM16] and [DKM15], which only concern geodesics in the highest dimension.…”