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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

MSC:
 2.2e+47 Semisimple Lie groups and their representations