The arithmetic and combinatorics of buildings for 𝑆𝑝_{𝑛}

Abstract

In this paper, we investigate both arithmetic and combinatorial aspects of buildings and associated Hecke operators for S p n ( K ) Sp_n(K) with K K a local field. We characterize the action of the affine Weyl group in terms of a symplectic basis for an apartment, characterize the special vertices as those which are self-dual with respect to the induced inner product, and establish a one-to-one correspondence between the special vertices in an apartment and the elements of the quotient Z n + 1 / Z ( 2 , 1 , … , 1 ) \mathbb {Z}^{n+1}/\mathbb {Z}(2,1,\dots ,1) . We then give a natural representation of the local Hecke algebra over K K acting on the special vertices of the Bruhat-Tits building for S p n ( K ) Sp_n(K) . Finally, we give an application of the Hecke operators defined on the building by characterizing minimal walks on the building for S p n Sp_n .