Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T05:02:10.742Z Has data issue: false hasContentIssue false
  • English
  • Français

A Truncated Integral of the Poisson Summation Formula

Published online by Cambridge University Press:  20 November 2018

Jason Levy
Jason Levy*
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON, K1N 6N5. email: jlevy@science.uottawa.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$, with anisotropic centre. Given a rational action of $G$ on a finite-dimensional vector space $V$, we analyze the truncated integral of the theta series corresponding to a Schwartz-Bruhat function on $V\left( \mathbb{A} \right)$. The Poisson summation formula then yields an identity of distributions on $V\left( \mathbb{A} \right)$. The truncation used is due to Arthur.

Keywords

11F9911F72
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 1 , 01 February 2001 , pp. 122 - 160
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Arthur, J., A trace formula for reductive groups I: terms associated to classes in G(Q). Duke Math J. 45(1978), 911–952.Google Scholar
[2] Arthur, J., The trace formula in invariant form. Ann. of Math. 114(1981), 1–74.Google Scholar
[3] Arthur, J., A measure on the unipotent variety. Canad. J. Math 37(1985), 1237–1274.Google Scholar
[4] Brion, M. and Vergne, M., Residue formulae, vector partition functions, and lattice points in rational polytopes. J. Amer. Math. Soc. 10(1997), 797–833.Google Scholar
[5] Bröndsted, A., An introduction to convex polytopes. Graduate Texts in Math. 90, Springer-Verlag, New York-Berlin, 1983.Google Scholar
[6] Kempf, G., Instability in invariant theory. Ann. of Math. 108(1978), 299–316.Google Scholar
[7] Kudla, S. and Rallis, S., A regularized Siegel-Weil formula: The first term identity. Ann. of Math. 140(1994), 1–80.Google Scholar
[8] Levy, J., A truncated Poisson formula for groups of rank at most two. Amer. J. Math. 117(1995), 1371–1408.Google Scholar
[9] Levy, J., Rationality of orbit closures. Preprint, 2000.Google Scholar
[10] Luna, D., Sur certaines opérations différentiables des groupes de Lie. Amer. J. Math. 97(1975), 172–181.Google Scholar
[11] Rader, C. and Rallis, S., Spherical characters on p-adic symmetric spaces. Amer. J. Math. 118(1996), 91–178.Google Scholar
[12] Richardson, R. W., Conjugacy classes of n-tuples in Lie algebras and algebraic groups. Duke Math. J. 57(1988), 1–35.Google Scholar
[13] Weil, A., Sur la Formule de Siegel dans la Théorie des Groupes Classiques. Acta Math. 113(1965), 1–87.Google Scholar
[14] Yukie, A., Shintani zeta functions. LondonMath. Soc. Lecture Note Ser. 183, CambridgeUniversity Press, 1993.Google Scholar