Infinite-dimensional modules of semisimple

Lecture 1. (g, k)-modules and weight modules
A famous problem in infinite-dimensional representation theory is the classification
of unitary representation of a real semisimple Lie group G. A closely related
celebrated result is the classification of certain simple modules over the complexified
Lie algebra g, called Harish-Chandra modules. The latter are simple g-modules
on which the Lie algebra k of a complexified maximal compact subgroup of G acts
locally finitely, for short (g, k)-modules.
From the point of view of algebraic representation theory, it is interesting to
consider (g, k)-modules for essentially arbitrary subalgebra k of g. For instance,
one may take k = h to be a Cartan subalgebra of g. The study of (g, h)-modules,
i.e. of weight modules, was initiated by G. Benkart, D. Britten, F. Lemire, A. Joseph
and Yu. Drozd, and was developed further by S. Fernando and V. Futorny. The
breakthrough was achieved in 1998 by O. Mathieu, who obtained a classification of
irreducible weight modules.
In this first lecture, I will recall the necessary preliminaries on (g, k)-modules
and will explain Mathieu’s classification.