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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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
  • 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, DOI 10.1090/mmono/145
  • Cristina M. Ballantine, John A. Rhodes, and Thomas R. Shemanske, Hecke operators for $\textrm {GL}_n$ and buildings, Acta Arith. 112 (2004), no. 2, 131–140. MR 2051373, DOI 10.4064/aa112-2-3
  • Kenneth S. Brown, Buildings, Springer Monographs in Mathematics, Springer-Verlag, New York, 1998. Reprint of the 1989 original. MR 1644630
  • P. Cartier, Representations of $p$-adic groups: a survey, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
  • Paul Garrett, Buildings and classical groups, Chapman & Hall, London, 1997. MR 1449872, DOI 10.1007/978-94-011-5340-9
  • Mark Ronan, Lectures on buildings, Perspectives in Mathematics, vol. 7, Academic Press, Inc., Boston, MA, 1989. MR 1005533
  • Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
  • J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588
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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