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Asymptotic expansions of generalized matrix entries of representations of real reductive groups. (English) Zbl 0553.22005
Lie group representations I, Proc. Spec. Year, Univ. Md., College Park 1982-83, Lect. Notes Math. 1024, 287-369 (1983).
This paper contains several important results on the global structure of representations of reductive Lie groups. A big part of it is joint work with W. Casselman. Let G be a reductive Lie group in Harish-Chandra’s class, and \(KA_ 0N_ 0\) its Iwasawa decomposition. To an admissible Banach space representation (\(\pi\),H) of G one associates the (\({\mathfrak g},K)\)-module of K-finite vectors, the G-module \(H_{\infty}\) of \(C^{\infty}\) vectors in H. Then \(V\subset H_{\infty}\). The following theorem is an extension of a result of Casselman: there exists an injective, continuous G-homomorphism of \(H_{\infty}\) into the space of \(C^{\infty}\)-vectors of some principal series (Theorem 5.10). Let \(\lambda\) be a \({\mathfrak n}_ 0\)-finite functional on V. Then \(\lambda\) extends to a continuous functional on \(H_{\infty}\) (here G is linear; cf. Corollary (5.11)). Consider the matrix coefficient \(g\to \lambda (\pi (g)v).\) Fix \(H\in {\mathfrak n}_ 0\), so that \(\exp (tH)\in {\mathcal C}\ell (A^+_ 0),\) \(t\in {\mathbb{R}}\). Theorem (5.8) asserts that the function \(t\to \lambda (\pi (\exp (tH)v)\) is asymptotically equal to a formal exponential-polynomial series, if \(v\in H_{\infty}\), and \(\lambda\) is in K-finite dual of H. This result generalizes a theorem of Harish- Chandra [W. Casselman, D. Miličić, Duke Math. J. 49, 869-930 (1982; Zbl 0524.22014)], in which both v and \(\lambda\) are assumed to be K-finite. Theorem (7.2) is a variant of (5.8): \(\lambda\) has ”moderate growth” along a parabolic P. This answers affirmatively a question of Piatetski-Shapiro. These, and many other results (discussion of completions of (\({\mathfrak g},K)\)-modules, for example) are often interdependent. The paper, which is expository in nature, contains also discussions of essentially known results (spaces of analytic vectors, Jacquet modules, etc.).
Reviewer: H.Hecht

22E46 Semisimple Lie groups and their representations