On the distribution of the principal series in

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a semisimple Lie group of split rank one with finite center. If is a discrete cocompact subgroup, then . For fixed , let denote the classes of irreducible unitary principal series . Let, for , where is the eigenvalue of (the Casimir element of ) on the class . In this paper, we determine the singular part of the asymptotic expansion of as if is torsion free, and the first term of the expansion for arbitrary . As a consequence, if and is without connected compact normal subgroups, then

**[**J. Frank Adams,**A**]*Lectures on Lie groups*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0252560****[**M. Welleda Baldoni Silva,**B**]*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**, 10.1090/S0002-9947-1980-0580893-X**[**Jochen Brüning and Ernst Heintze,**BH**]*Representations of compact Lie groups and elliptic operators*, Invent. Math.**50**(1978/79), no. 2, 169–203. MR**517776**, 10.1007/BF01390288**[**J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan,**DKV**]*Spectra of compact locally symmetric manifolds of negative curvature*, Invent. Math.**52**(1979), no. 1, 27–93. MR**532745**, 10.1007/BF01389856**[**David L. DeGeorge and Nolan R. Wallach,**DW**]*Limit formulas for multiplicities in 𝐿²(Γ\𝐺). II. The tempered spectrum*, Ann. of Math. (2)**109**(1979), no. 3, 477–495. MR**534759**, 10.2307/1971222**[**I. M. Gel′fand,**G**]*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,**GGP**]*Representation theory and automorphic functions*, Translated from the Russian by K. A. Hirsch, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR**0233772****[**R. Gangolli,**Ga**]*Asymptotic behavior of spectra of compact quotients of certain symmetric spaces*, Acta Math.**121**(1968), 151–192. MR**0239000****[**Ramesh Gangolli and Garth Warner,**GW**]*On Selberg’s trace formula*, J. Math. Soc. Japan**27**(1975), 328–343. MR**0399354****[**Sigurdur Helgason,**H**]*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,**Hu**]*Fibre bundles*, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR**0229247****[**A. W. Knapp and E. M. Stein,**KS**]*Intertwining operators for semisimple groups*, Ann. of Math. (2)**93**(1971), 489–578. MR**0460543****[**A. W. Knapp and N. R. Wallach,**KW**]*Szegö kernels associated with discrete series*, Invent. Math.**34**(1976), no. 3, 163–200. MR**0419686****[**Roberto J. Miatello,**M1**]*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**, 10.1090/S0002-9947-1980-0570777-5**[**Roberto J. Miatello,**M2**]*On the Plancherel measure for linear Lie groups of rank one*, Manuscripta Math.**29**(1979), no. 2-4, 249–276. MR**545044**, 10.1007/BF01303630**[**Roberto J. Miatello,**M3**]*An alternating sum formula for multiplicities in 𝐿²(Γ\𝐺)*, Trans. Amer. Math. Soc.**269**(1982), no. 2, 567–574. MR**637710**, 10.1090/S0002-9947-1982-0637710-0**[**Kiyosato Okamoto,**O**]*On the Plancherel formulas for some types of simple Lie groups*, Osaka J. Math.**2**(1965), 247–282. MR**0219668****[**Nolan R. Wallach,**W**]*An asymptotic formula of Gelfand and Gangolli for the spectrum of 𝐺\𝐺*, J. Differential Geometry**11**(1976), no. 1, 91–101. MR**0417340****[**E. J. Whittaker and G. N. Watson,**WW**]*A course of modern analysis*, Cambridge Univ. Press, London and New York, 1927.**[**D. P. Želobenko,**Z**]*Compact Lie groups and their representations*, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 40. MR**0473098**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
22E45,
11F70,
22E40,
58G25

Retrieve articles in all journals with MSC: 22E45, 11F70, 22E40, 58G25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0704602-9

Article copyright:
© Copyright 1983
American Mathematical Society