Spherical distributions on Lie groups and 𝐶^{∞} vectors

Penney, R.

doi:10.1090/S0002-9947-1976-0457632-5

Spherical distributions on Lie groups and $C^{\infty }$ vectors
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by R. Penney

Trans. Amer. Math. Soc. 223 (1976), 367-384

DOI: https://doi.org/10.1090/S0002-9947-1976-0457632-5

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Abstract:

Given a Lie group G (not necessarily unimodular) and a subgroup K of G (not necessarily compact), it is shown how to associate with every finite-dimensional unitary irreducible representation $\delta$ of K a class of distributions analogous to the class of spherical functions of height $\delta$ familiar from the unimodular-maximal compact case. The two concepts agree as nearly as possible. A number of familiar theorems are generalized to our situation. As an application we obtain a generalization of the Frobenius reciprocity theorem and of Plancherel’s theorem to arbitrary induced representations of Lie groups.

References

Roger Godement, A theory of spherical functions. I, Trans. Amer. Math. Soc. 73 (1952), 496–556. MR 52444, DOI 10.1090/S0002-9947-1952-0052444-2
Roe Goodman, Complex Fourier analysis on a nilpotent Lie group, Trans. Amer. Math. Soc. 160 (1971), 373–391. MR 417334, DOI 10.1090/S0002-9947-1971-0417334-3

The theory of group representations

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Richard Penney, Entire vectors and holomorphic extension of representations. I, II, Trans. Amer. Math. Soc. 198 (1974), 107–121; ibid. 191 (1974), 195–207. MR 364556, DOI 10.1090/S0002-9947-1974-0364556-9
Richard Penney, Abstract Plancherel theorems and a Frobenius reciprocity theorem, J. Functional Analysis 18 (1975), 177–190. MR 0444844, DOI 10.1016/0022-1236(75)90023-3
Neils Skovhus Poulsen, On $C^{\infty }$-vectors and intertwining bilinear forms for representations of Lie groups, J. Functional Analysis 9 (1972), 87–120. MR 0310137, DOI 10.1016/0022-1236(72)90016-x