## 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 $Sp_n$.## References

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## Additional Information

**Thomas R. Shemanske**- Affiliation: Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, New Hampshire 03755
- Email: thomas.r.shemanske@dartmouth.edu
- Received by editor(s): January 5, 2004
- Received by editor(s) in revised form: July 12, 2005
- Published electronically: January 30, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 3409-3423 - MSC (2000): Primary 20E42; Secondary 11F46, 11F60, 11F70
- DOI: https://doi.org/10.1090/S0002-9947-07-04293-6
- MathSciNet review: 2299461