The arithmetic and combinatorics of buildings for $Sp_n$

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 $Sp_n$.