Asymptotic 𝐾-support and restrictions of representations

Hansen, Sönke; Hilgert, Joachim; Keliny, Sameh

doi:10.1090/S1088-4165-09-00362-8

Asymptotic $K$-support and restrictions of representations
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by Sönke Hansen, Joachim Hilgert and Sameh Keliny

Represent. Theory 13 (2009), 460-469

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

Published electronically: September 25, 2009

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Abstract:

The restriction, from a compact Lie group $K$ to a closed subgroup, of a polynomially bounded representation remains polynomially bounded provided a geometric assumption on the asymptotic $K$-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 $K\times K$ action, of the wavefront set of a distribution on $K$ in terms of the asymptotic behavior of its Fourier coefficients.

References

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
Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
—, The analysis of linear partial differential operators, vol. III, Springer-Verlag, Berlin and New York, 1985.
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, Springer-Verlag, Berlin-New York, 1981, pp. 117–140. MR 633659
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
Toshiyuki Kobayashi, Discrete decomposability of the restriction of $A_{\mathfrak {q}}(\lambda )$ 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, DOI 10.2307/120963
R. T. Seeley, Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117 (1965), 167–204. MR 173174, DOI 10.1090/S0002-9947-1965-0173174-1