The arithmetic and combinatorics of buildings for

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate both arithmetic and combinatorial aspects of buildings and associated Hecke operators for with 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 .

We then give a natural representation of the local Hecke algebra over acting on the special vertices of the Bruhat-Tits building for . Finally, we give an application of the Hecke operators defined on the building by characterizing minimal walks on the building for .

**1.**A. N. Andrianov and V. G. Zhuravlëv,*Modular forms and Hecke operators*, Translations of Mathematical Monographs, vol. 145, American Mathematical Society, Providence, RI, 1995. Translated from the 1990 Russian original by Neal Koblitz. MR**1349824****2.**Cristina M. Ballantine, John A. Rhodes, and Thomas R. Shemanske,*Hecke operators for 𝐺𝐿_{𝑛} and buildings*, Acta Arith.**112**(2004), no. 2, 131–140. MR**2051373**, https://doi.org/10.4064/aa112-2-3**3.**Kenneth S. Brown,*Buildings*, Springer Monographs in Mathematics, Springer-Verlag, New York, 1998. Reprint of the 1989 original. MR**1644630****4.**P. Cartier,*Representations of -adic groups: a survey*, Automorphic forms, representations and -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 111-155. MR**0546593 (81e:22029)****5.**Paul Garrett,*Buildings and classical groups*, Chapman & Hall, London, 1997. MR**1449872****6.**Mark Ronan,*Lectures on buildings*, Perspectives in Mathematics, vol. 7, Academic Press, Inc., Boston, MA, 1989. MR**1005533****7.**J. P. Serre,*Trees*, Spring-Verlag, 1980. MR**0607504 (82c:20083)****8.**Goro Shimura,*Introduction to the arithmetic theory of automorphic functions*, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. MR**0314766****9.**J. Tits,*Reductive groups over local fields*, Automorphic forms, representations and -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 29-69. MR**0546588 (80h:20064)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20E42,
11F46,
11F60,
11F70

Retrieve articles in all journals with MSC (2000): 20E42, 11F46, 11F60, 11F70

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:
https://doi.org/10.1090/S0002-9947-07-04293-6

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.