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

HTML articles powered by AMS MathViewer

- by Roberto J. Miatello and Jorge A. Vargas PDF
- Trans. Amer. Math. Soc.
**279**(1983), 63-75 Request permission

## 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.## References

- J. Frank Adams,
*Lectures on Lie groups*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0252560** - M. Welleda Baldoni Silva,
*The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one*, Trans. Amer. Math. Soc.**261**(1980), no. 2, 303–368. MR**580893**, DOI 10.1090/S0002-9947-1980-0580893-X - Jochen Brüning and Ernst Heintze,
*Representations of compact Lie groups and elliptic operators*, Invent. Math.**50**(1978/79), no. 2, 169–203. MR**517776**, DOI 10.1007/BF01390288 - J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan,
*Spectra of compact locally symmetric manifolds of negative curvature*, Invent. Math.**52**(1979), no. 1, 27–93. MR**532745**, DOI 10.1007/BF01389856 - David L. DeGeorge and Nolan R. Wallach,
*Limit formulas for multiplicities in $L^{2}(\Gamma \backslash G)$. II. The tempered spectrum*, Ann. of Math. (2)**109**(1979), no. 3, 477–495. MR**534759**, DOI 10.2307/1971222 - I. M. Gel′fand,
*Automorphic functions and the theory of representations*, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 74–85. MR**0175997** - I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro,
*Representation theory and automorphic functions*, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by K. A. Hirsch. MR**0233772** - R. Gangolli,
*Asymptotic behavior of spectra of compact quotients of certain symmetric spaces*, Acta Math.**121**(1968), 151–192. MR**239000**, DOI 10.1007/BF02391912 - Ramesh Gangolli and Garth Warner,
*On Selberg’s trace formula*, J. Math. Soc. Japan**27**(1975), 328–343. MR**399354**, DOI 10.2969/jmsj/02720328 - Sigurdur Helgason,
*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561** - Dale Husemoller,
*Fibre bundles*, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR**0229247** - A. W. Knapp and E. M. Stein,
*Intertwining operators for semisimple groups*, Ann. of Math. (2)**93**(1971), 489–578. MR**460543**, DOI 10.2307/1970887 - A. W. Knapp and N. R. Wallach,
*Szegö kernels associated with discrete series*, Invent. Math.**34**(1976), no. 3, 163–200. MR**419686**, DOI 10.1007/BF01403066 - Roberto J. Miatello,
*The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature*, Trans. Amer. Math. Soc.**260**(1980), no. 1, 1–33. MR**570777**, DOI 10.1090/S0002-9947-1980-0570777-5 - Roberto J. Miatello,
*On the Plancherel measure for linear Lie groups of rank one*, Manuscripta Math.**29**(1979), no. 2-4, 249–276. MR**545044**, DOI 10.1007/BF01303630 - Roberto J. Miatello,
*An alternating sum formula for multiplicities in $L^{2}(\Gamma \backslash G)$*, Trans. Amer. Math. Soc.**269**(1982), no. 2, 567–574. MR**637710**, DOI 10.1090/S0002-9947-1982-0637710-0 - Kiyosato Okamoto,
*On the Plancherel formulas for some types of simple Lie groups*, Osaka Math. J.**2**(1965), 247–282. MR**219668** - Nolan R. Wallach,
*An asymptotic formula of Gelfand and Gangolli for the spectrum of $G\backslash G$*, J. Differential Geometry**11**(1976), no. 1, 91–101. MR**417340**
E. J. Whittaker and G. N. Watson, - D. P. Zhelobenko,
*Kompaktnye gruppy Li i ikh predstavleniya*, Izdat. “Nauka”, Moscow, 1970 (Russian). MR**0473097**

*A course of modern analysis*, Cambridge Univ. Press, London and New York, 1927.

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**279**(1983), 63-75 - MSC: Primary 22E45; Secondary 11F70, 22E40, 58G25
- DOI: https://doi.org/10.1090/S0002-9947-1983-0704602-9
- MathSciNet review: 704602