Canonical Extensions of Harish-Chandra Modules to Representations of G

Let G be the group of R-rational points on a reductive, Zariskiconnected, algebraic group defined over R, let K be a maximal compact subgroup, and let g be the corresponding complexified Lie algebra of G. It is a curious fault of the current representation theory of G that for technical reasons one very rarely works with representations of G itself, but rather with a certain category of simultaneous representations of g and K. The reasons for this are, roughly speaking, that for a given (g,K)-module of finite length there are clearly any number of overlying rather distinct continuous G-representations, whose ‘essence’ is captured by the (g, K)-module alone. At any rate, this paper will propose a remedy for this inconvenience, and define a category of smooth representations of G of finite length which will, I hope, turn out to be as easy to work with as representations of (g, K) and occasionally much more convenient. It is to be considered a report on what has been to a great extent joint work with Nolan Wallach, and is essentially a sequel to 38.