(g,K)-module

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

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

  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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.