Asymptotic -support and restrictions of representations

Authors:
Sönke Hansen, Joachim Hilgert and Sameh Keliny

Journal:
Represent. Theory **13** (2009), 460-469

MSC (2000):
Primary 22E46; Secondary 46F10

Published electronically:
September 25, 2009

MathSciNet review:
2550473

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The restriction, from a compact Lie group to a closed subgroup, of a polynomially bounded representation remains polynomially bounded provided a geometric assumption on the asymptotic -support of the representation is satisfied. This is a theorem of T. Kobayashi. We give a proof of this theorem using microlocal analysis in the setting of distribution rather than hyperfunction theory. The proof is based on a characterization, up to the natural action, of the wavefront set of a distribution on in terms of the asymptotic behavior of its Fourier coefficients.

**1.**J. J. Duistermaat,*Fourier integral operators*, Courant Institute of Mathematical Sciences, New York University, New York, 1973. Translated from Dutch notes of a course given at Nijmegen University, February 1970 to December 1971. MR**0451313****2.**Lars Hörmander,*Fourier integral operators. I*, Acta Math.**127**(1971), no. 1-2, 79–183. MR**0388463****3.**-,*The analysis of linear partial differential operators*, vol. III, Springer-Verlag, Berlin and New York, 1985.**4.**Roger Howe,*Wave front sets of representations of Lie groups*, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 117–140. MR**633659****5.**M. Kashiwara and M. Vergne,*𝐾-types and singular spectrum*, Noncommutative harmonic analysis (Proc. Third Colloq., Marseille-Luminy, 1978) Lecture Notes in Math., vol. 728, Springer, Berlin, 1979, pp. 177–200. MR**548330****6.**Toshiyuki Kobayashi,*Discrete decomposability of the restriction of 𝐴_{𝔮}(𝜆) with respect to reductive subgroups. II. Micro-local analysis and asymptotic 𝔎-support*, Ann. of Math. (2)**147**(1998), no. 3, 709–729. MR**1637667**, 10.2307/120963**7.**R. T. Seeley,*Integro-differential operators on vector bundles*, Trans. Amer. Math. Soc.**117**(1965), 167–204. MR**0173174**, 10.1090/S0002-9947-1965-0173174-1

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
22E46,
46F10

Retrieve articles in all journals with MSC (2000): 22E46, 46F10

Additional Information

**Sönke Hansen**

Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Email:
soenke@math.upb.de

**Joachim Hilgert**

Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Email:
hilgert@math.upb.de

**Sameh Keliny**

Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Email:
sameh@math.upb.de

DOI:
https://doi.org/10.1090/S1088-4165-09-00362-8

Received by editor(s):
May 4, 2009

Received by editor(s) in revised form:
August 25, 2009

Published electronically:
September 25, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.