(g,K)-module

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, more specifically in the representation theory of reductive Lie groups, a -module is an algebraic object, first introduced by Harish-Chandra,[1] used to deal with continuous infinite-dimensional representations using algebraic techniques. Harish-Chandra showed that the study of irreducible unitary representations of a real reductive Lie group, G, could be reduced to the study of irreducible -modules, where is the Lie algebra of G and K is a maximal compact subgroup of G.[2]

Definition[edit]

Let G be a real Lie group. Let be its Lie algebra, and K a maximal compact subgroup with Lie algebra . A -module is defined as follows:[3] it is a vector space V that is both a Lie algebra representation of and a group representation of K (without regard to the topology of K) satisfying the following three conditions

1. for any vV, kK, and X
2. for any vV, Kv spans a finite-dimensional subspace of V on which the action of K is continuous
3. for any vV and Y

In the above, the dot, , denotes both the action of on V and that of K. The notation Ad(k) denotes the adjoint action of G on , and Kv is the set of vectors as k varies over all of K.

The first condition can be understood as follows: if G is the general linear group GL(n, R), then is the algebra of all n by n matrices, and the adjoint action of k on X is kXk−1; condition 1 can then be read as

In other words, it is a compatibility requirement among the actions of K on V, on V, and K on . The third condition is also a compatibility condition, this time between the action of on V viewed as a sub-Lie algebra of and its action viewed as the differential of the action of K on V.

Notes[edit]

  1. ^ Page 73 of Wallach 1988
  2. ^ Page 12 of Doran & Varadarajan 2000
  3. ^ This is James Lepowsky's more general definition, as given in section 3.3.1 of Wallach 1988

References[edit]

  • Doran, Robert S.; Varadarajan, V. S., eds. (2000), The mathematical legacy of Harish-Chandra, Proceedings of Symposia in Pure Mathematics, 68, AMS, ISBN 978-0-8218-1197-9, MR 1767886 
  • Wallach, Nolan R. (1988), Real reductive groups I, Pure and Applied Mathematics, 132, Academic Press, ISBN 978-0-12-732960-4, MR 0929683