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# Stable Discrete Series Characters at Singular Elements

Published:2009-12-01
Printed: Dec 2009
• Steven Spallone
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## Abstract

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 $\Phi_M(\gm,\Theta^E)$ Arthur's extension of $|D_M^G(\gm)|^{\lfrac 12} \Theta^E(\gm)$ to $T(\R)$. In this paper we give a simple explicit expression for $\Phi_M(\gm,\Theta^E)$ when $\gm$ is elliptic in $G$. We do not assume $\gm$ is regular.
 MSC Classifications: 22E47 - Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also 17B10]

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