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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: [email protected]
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Abstract

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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

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