Canonical extensions of Harish-Chandra modules to representations of \(G\).

*(English)*Zbl 0702.22016Let \(G\) 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)].

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:

22E46 | Semisimple Lie groups and their representations |