Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-07-08T00:57:52.881Z Has data issue: false hasContentIssue false

Stable Discrete Series Characters at Singular Elements

Published online by Cambridge University Press:  20 November 2018

Steven Spallone*
Affiliation:
Purdue University, West Lafayette, IN email: sspallon@math.purdue.edu
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.

Write ${{\Theta }^{E}}$ for the stable discrete series character associated with an irreducible finite-dimensional representation $E$ of a connected real reductive group $G$. Let $M$ be the centralizer of the split component of a maximal torus $T$, and denote by ${{\Phi }_{M}}\left( \gamma ,\,{{\Theta }^{E}} \right)$ Arthur’s extension of $|D_{M}^{G}\,\left( \gamma \right)|{{\,}^{1/2}}\,{{\Theta }^{E}}\,\left( \gamma \right)$ to $T\left( \mathbb{R} \right)$. In this paper we give a simple explicit expression for ${{\Phi }_{M}}\left( \gamma ,\,{{\Theta }^{E}} \right)$ when $\gamma $ is elliptic in $G$. We do not assume $\gamma $ is regular.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Arthur, J., The L 2 -Lefschetz numbers of Hecke operators. Invent. Math. 97(1989), no. 2, 257–290.Google Scholar
[2] Bourbaki, N., Lie Groups and Lie Algebras. Chapters 4-6, Springer-Verlag, Berlin, 2002.Google Scholar
[3] Goresky, M., Kottwitz, R., and MacPherson, R., Discrete series characters and the Lefschetz formula for Hecke operators. Duke Math. J. 89(1997), no. 3, 477–554.Google Scholar
[4] Herb, R., Characters of averaged discrete series on semisimple real Lie groups. Pacific J. Math. 80(1979), no. 1, 169–177.Google Scholar