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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the distribution of the principal series in $L^{2}(\Gamma \backslash G)$
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by Roberto J. Miatello and Jorge A. Vargas PDF
Trans. Amer. Math. Soc. 279 (1983), 63-75 Request permission


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 ${N_\sigma }(r) = \Sigma _{\omega \in P(\sigma ),|{\lambda _{{\omega }}| < r}}{n_\Gamma }(\omega )$ and $G$ is without connected compact normal subgroups, then \[ {N_\sigma }(r)\;\sim {C_G}\; \cdot \; |Z(G) \cap \Gamma |\; \cdot \;{\text {vol}}(\Gamma \backslash G)\; \cdot \; \dim (\sigma )\; \cdot \; {r^c} \qquad (c = \frac {1} {2} \dim G/K),\] 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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 279 (1983), 63-75
  • MSC: Primary 22E45; Secondary 11F70, 22E40, 58G25
  • DOI:
  • MathSciNet review: 704602