Some consequences of Harish-Chandra's submersion principle

Authors:
Cary Rader and Allan Silberger

Journal:
Proc. Amer. Math. Soc. **118** (1993), 1271-1279

MSC:
Primary 22E50

DOI:
https://doi.org/10.1090/S0002-9939-1993-1169888-X

MathSciNet review:
1169888

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a reductive -adic group, a good maximal compact subgroup, any open subgroup, and an admissible representation of of finite type. In *A submersion principle and its applications*, Harish-Chandra proves the theorem that is a finite-rank operator for in the regular set in order to show that the character is a locally constant class function on . From this, the authors derive the formula for any -finite matrix coefficient of a discrete series representation with formal degree . They use another technical result of the paper to prove that invariant integrals of Schwartz space functions converge absolutely. None of these results depends upon a characteristic zero assumption.

**[1]**Laurent Clozel,*Invariant harmonic analysis on the Schwartz space of a reductive*-*adic group*, Harmonic Analysis on Reductive Groups (Bowdoin Conference), Birkhauser, Boston, MA, 1991, pp. 101-121. MR**1168480 (93h:22020)****[2]**Harish-Chandra,*Harmonic analysis on reductive*-*adic groups*, Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, RI, 1974, pp. 167-192.**[3]**-,*A submersion principle and its applications*, Papers dedicated to the memory of V. K. Patodi, Indian Academy of Sciences, Bangalore, and the Tata Institute of Fundamental Research, Bombay, 1980, pp. 95-102. MR**592255 (82e:22032)****[4]**-,*Harmonic analysis on reductive*-*adic groups*, Notes by G. van Dijk, Lecture Notes in Math., vol. 162, Springer-Verlag, Berlin, Heidelberg, and New York, 1970. MR**0414797 (54:2889)****[5]**George Kempf,*Instability in invariant theory*, Ann. of Math. (2)**108**(1978), 299-316. MR**506989 (80c:20057)****[6]**Philip Kutzko,*Character formulas for supercuspidal representations of**a prime*, Amer. J. Math.**109**(1987), 201-222. MR**882420 (88k:22003)****[7]**Paul J. Sally, Jr.,*Some remarks on discrete series characters for reductive*-*adic groups*, Representations of Lie Groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, North-Holland, Amsterdam and New York, 1988, pp. 337-348.**[8]**Allan J. Silberger,*Introduction to harmonic analysis on reductive*-*adic groups*, Math. Notes, vol. 23, Princeton Univ. Press, Princeton, NJ, 1979. MR**544991 (81m:22025)****[9]**Nolan Wallach,*Real reductive groups*. I, Academic Press, Boston, MA, 1988. MR**929683 (89i:22029)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1169888-X

Keywords:
Character,
discrete series,
reductive -adic groups,
Schwartz space,
invariant integral

Article copyright:
© Copyright 1993
American Mathematical Society