The arithmetic and combinatorics of buildings for 𝑆𝑝_{𝑛}

Shemanske, Thomas

doi:10.1090/S0002-9947-07-04293-6

The arithmetic and combinatorics of buildings for $Sp_n$
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by Thomas R. Shemanske PDF

Trans. Amer. Math. Soc. 359 (2007), 3409-3423 Request permission

Abstract:

In this paper, we investigate both arithmetic and combinatorial aspects of buildings and associated Hecke operators for $Sp_n(K)$ with $K$ 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 $\mathbb {Z}^{n+1}/\mathbb {Z}(2,1,\dots ,1)$. We then give a natural representation of the local Hecke algebra over $K$ acting on the special vertices of the Bruhat-Tits building for $Sp_n(K)$. Finally, we give an application of the Hecke operators defined on the building by characterizing minimal walks on the building for $Sp_n$.

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