Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the distribution of the principal series in $ L\sp{2}(\Gamma \backslash G)$

Authors: Roberto J. Miatello and Jorge A. Vargas
Journal: Trans. Amer. Math. Soc. 279 (1983), 63-75
MSC: Primary 22E45; Secondary 11F70, 22E40, 58G25
MathSciNet review: 704602
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Abstract: Let $ G$ be a semisimple Lie group of split rank one with finite center. If $ \Gamma \subset G$ is a discrete cocompact subgroup, then $ {L^2}(\Gamma \backslash G) = {\Sigma_{\omega \in \mathcal{E}(G)}}{n_\Gamma }(\omega ) \cdot \omega $. For fixed $ \sigma \in \mathcal{E}(M)$, let $ P(\sigma )$ denote the classes of irreducible unitary principal series $ {\pi_{\sigma ,iv}}(v \in {\mathcal{U}^{\ast}})$. Let, for $ s > 0,{\psi_\sigma }(s) = {\Sigma_{\omega \in P(\sigma )}}{n_\Gamma }(\omega ) \cdot {e^{s{\lambda_\omega }}}$, where $ {\lambda_\omega }$ is the eigenvalue of $ \Omega $ (the Casimir element of $ G$) on the class $ \omega $. In this paper, we determine the singular part of the asymptotic expansion of $ {\psi_\sigma }(s)$ as $ s \to {0^ + }$ if $ \Gamma $ is torsion free, and the first term of the expansion for arbitrary $ \Gamma $. As a consequence, if $ {N_\sigma }(r) = \Sigma_{\omega \in P(\sigma ),\vert{\lambda_{{\omega }}\vert < r}}{n_\Gamma }(\omega )$ and $ G$ is without connected compact normal subgroups, then

$\displaystyle {N_\sigma }(r)\;\sim {C_G}\; \cdot\; \vert Z(G) \cap \Gamma \vert... ...\; \cdot\; \dim(\sigma )\; \cdot\; {r^c} \qquad (c = \frac{1} {2}\,\dim \,G/K),$

as $ r \to + \infty $. In the course of the proof, we determine the image and kernel of the restriction homomorphism $ {i^{\ast}}:R(K) \to R(M)$ between representation rings.

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Article copyright: © Copyright 1983 American Mathematical Society