The arithmetic and combinatorics of buildings for $Sp_n$

Author:
Thomas R. Shemanske

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

Published electronically:
January 30, 2007

MathSciNet review:
2299461

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Thomas R. Shemanske**

Affiliation:
Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, New Hampshire 03755

Email:
thomas.r.shemanske@dartmouth.edu

Keywords:
Bruhat–Tits building,
symplectic group,
Hecke operators,
representation

Received by editor(s):
January 5, 2004

Received by editor(s) in revised form:
July 12, 2005

Published electronically:
January 30, 2007

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.