Fourier inversion of invariant integrals on semisimple real Lie groups

Herb, Rebecca A.

doi:10.1090/S0002-9947-1979-0525674-X

Fourier inversion of invariant integrals on semisimple real Lie groups
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by Rebecca A. Herb PDF

Trans. Amer. Math. Soc. 249 (1979), 281-302 Request permission

Abstract:

Let G be a connected, semisimple real Lie group with finite center. Associated with every regular semisimple element g of G is a tempered invariant distribution ${ \Lambda _g}$ given by an orbital integral. This paper gives an inductive formula for computing the Fourier transform of ${ \Lambda _g}$ in terms of the space of tempered invariant eigendistributions of G.

References

Harish-Chandra, A formula for semisimple Lie groups, Amer. J. Math. 79 (1957), 733–760. MR 96138, DOI 10.2307/2372432
Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math. 86 (1964), 534–564. MR 180628, DOI 10.2307/2373023
Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math. 86 (1964), 534–564. MR 180628, DOI 10.2307/2373023
Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241–318. MR 219665, DOI 10.1007/BF02391779
Harish-Chandra, Two theorems on semi-simple Lie groups, Ann. of Math. (2) 83 (1966), 74–128. MR 194556, DOI 10.2307/1970472
Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 219666, DOI 10.1007/BF02392813

Fourier inversion on semisimple Lie groups of real rank two

R. A. Herb and P. J. Sally Jr., Singular invariant eigendistributions as characters, Bull. Amer. Math. Soc. 83 (1977), no. 2, 252–254. MR 480875, DOI 10.1090/S0002-9904-1977-14287-0
Ronald L. Lipsman, On the characters and equivalence of continuous series representations, J. Math. Soc. Japan 23 (1971), 452–480. MR 289718, DOI 10.2969/jmsj/02330452

On a method of computing multiplicities in

Paul J. Sally Jr. and Garth Warner, The Fourier transform on semisimple Lie groups of real rank one, Acta Math. 131 (1973), 1–26. MR 450461, DOI 10.1007/BF02392034
Wilfried Schmid, On the characters of the discrete series. The Hermitian symmetric case, Invent. Math. 30 (1975), no. 1, 47–144. MR 396854, DOI 10.1007/BF01389847

Harmonic analysis on semisimple Lie groups

Joseph A. Wolf, The action of a real semisimple Lie group on a complex flag manifold. II. Unitary representations on partially holomorphic cohomology spaces, Memoirs of the American Mathematical Society, No. 138, American Mathematical Society, Providence, R.I., 1974. MR 0393350