# zbMATH — the first resource for mathematics

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