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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: January 30, 2007
MathSciNet review: 2299461
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04293-6
PII: S 0002-9947(07)04293-6
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.