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

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 ${N_\sigma }(r) = \Sigma _{\omega \in P(\sigma ),|{\lambda _{{\omega }}| < r}}{n_\Gamma }(\omega )$ 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.