Stable Discrete Series Characters at Singular Elements

http://dx.doi.org/10.4153/CJM-2009-065-x
Canad. J. Math. 61(2009), 1375-1382

  Published:2009-12-01
 Printed: Dec 2009

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.