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Canonical extensions of Harish-Chandra modules to representations of $$G$$. (English) Zbl 0702.22016
Let $$G$$ be the group of $$\mathbb R$$-rational points on a connected reductive algebraic group defined over $$\mathbb R$$. Let $$K$$ be a maximal compact subgroup and $${\mathfrak g}$$ the complexified Lie algebra of $$G$$. Assume $$G$$ to be algebraically embedded as a closed subset of a finite-dimensional matrix algebra, and let $$g\mapsto \| g\|$$ be the associated norm. A continuous representation $$\pi$$ of $$G$$ on a topological vector space $$V$$ is called of moderate growth if $$V$$ is a Fréchet space and for every continuous semi-norm $$\rho$$ on $$V$$ there exists a positive integer $$N$$ and a continuous semi-norm $$\nu$$ such that $$\rho (\pi (g)v)\leq \| g\|^ N\nu (v)$$ for all $$g\in G$$ and $$v\in V$$.
The author proves that to each finitely generated Harish-Chandra module $$V$$ one can attach (up to canonical topological isomorphism) a unique smooth representation of $$G$$ of moderate growth whose underlying ($${\mathfrak g},K)$$-module is isomorphic to $$V$$. This assignment is an exact functor from the category of finitely generated Harish-Chandra modules into the category of smooth representations of moderate growth. As an application, he obtains results on asymptotic behavior of (not necessarily $$K$$-finite) matrix coefficients of smooth representations of moderate growth.
These results are in part joint work with Nolan Wallach, and this paper is essentially a sequel to N. Wallach [in Lie Group Representations I, Lect. Notes Math. 1024, 287–369 (1983; Zbl 0553.22005)].
Reviewer: D. Miličić

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