On Lusztig-Dupont homology of flag complexes

JOURNAL OF ALGEBRA

卷:531

Article

Meshulam, Roy¹  Zerbib, Shira² 

[1] Technion, Dept Math, IL-32000 Haifa, Israel

[2] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA

关键词: Spherical buildings;    Homology of local systems;   

DOI  :  10.1016/j.jalgebra.2019.04.019

来源: Elsevier

【 摘 要 】

Let V be an n-dimensional vector space over the finite field F-q. The spherical building X-V associated with GL(V) is the order complex of the nontrivial linear subspaces of V. Let g be the local coefficient system on X-V, whose value on the simplex sigma = [V-0 subset of ... subset of V-p] is an element of X-V is given by g(sigma) = V-0. The homology module D-1(V) = (H) over tilde (n-2)(X-V; g) plays a key role in Lusztig's seminal work on the discrete series representations of GL(V). Here, some further properties of g and its exterior powers are established. These include a construction of an explicit basis of D-1(V), a computation of the dimension of D-k(V) = (H) over tilde (n-k-1)(X-V; Lambda(k)g), and the following twisted analogue of a result of Smith and Yoshiara: For any 1 <= k <= n - 1, the minimal support size of a non-zero (n - k - 1)-cycle in the twisted homology (H) over tilde (n-k-1) (X-V; boolean AND(k)g) is (n-k+2)!/2. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free