(g,K)-module

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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 Template:Harvnb
  2. Page 12 of Template:Harvnb
  3. This is James Lepowsky's more general definition, as given in section 3.3.1 of Template:Harvnb

References

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