(g,K)-module

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title: "(g,K)-module" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["representation-theory-of-lie-groups"] topic_path: "general/representation-theory-of-lie-groups" source: "https://en.wikipedia.org/wiki/(g,K)-module" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, more specifically in the representation theory of reductive Lie groups, a (\mathfrak{g},K)-module is an algebraic object, first introduced by Harish-Chandra, 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 (\mathfrak{g},K)-modules, where \mathfrak{g} is the Lie algebra of G and K is a maximal compact subgroup of G.

Definition

Let G be a real Lie group. Let \mathfrak{g} be its Lie algebra, and K a maximal compact subgroup with Lie algebra \mathfrak{k}. A (\mathfrak{g},K)-module is defined as follows: it is a vector space V that is both a Lie algebra representation of \mathfrak{g} and a group representation of K (without regard to the topology of K) satisfying the following three conditions :1. for any v ∈ V, k ∈ K, and X ∈ \mathfrak{g} ::k\cdot (X\cdot v)=(\operatorname{Ad}(k)X)\cdot (k\cdot v) :2. for any v ∈ V, Kv spans a finite-dimensional subspace of V on which the action of K is continuous :3. for any v ∈ V and Y ∈ \mathfrak{k} ::\left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v. In the above, the dot, \cdot, denotes both the action of \mathfrak{g} on V and that of K. The notation Ad(k) denotes the adjoint action of G on \mathfrak{g}, and Kv is the set of vectors k\cdot v 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 \mathfrak{g} 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 :kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv. In other words, it is a compatibility requirement among the actions of K on V, \mathfrak{g} on V, and K on \mathfrak{g}. The third condition is also a compatibility condition, this time between the action of \mathfrak{k} on V viewed as a sub-Lie algebra of \mathfrak{g} and its action viewed as the differential of the action of K on V.

Notes

References

  • {{Citation | editor1-last=Doran | editor1-first=Robert S. | editor2-last=Varadarajan | editor2-first=V. S. | title=The mathematical legacy of Harish-Chandra | publisher=AMS | series=Proceedings of Symposia in Pure Mathematics | volume=68 | year=2000 | mr=1767886 | isbn=978-0-8218-1197-9
  • {{Citation | last=Wallach | first=Nolan R. | title=Real reductive groups I | year=1988 | publisher=Academic Press | series=Pure and Applied Mathematics | volume=132 | mr=0929683 | isbn=978-0-12-732960-4 | url-access=registration | url=https://archive.org/details/realreductivegro0000wall

References

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

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