(g,K)-module
In mathematics, more specifically in the representation theory of reductive Lie groups, a [math]\displaystyle{ (\mathfrak{g},K) }[/math]-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 [math]\displaystyle{ (\mathfrak{g},K) }[/math]-modules, where [math]\displaystyle{ \mathfrak{g} }[/math] 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 [math]\displaystyle{ \mathfrak{g} }[/math] be its Lie algebra, and K a maximal compact subgroup with Lie algebra [math]\displaystyle{ \mathfrak{k} }[/math]. A [math]\displaystyle{ (\mathfrak{g},K) }[/math]-module is defined as follows:[3] it is a vector space V that is both a Lie algebra representation of [math]\displaystyle{ \mathfrak{g} }[/math] 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 ∈ [math]\displaystyle{ \mathfrak{g} }[/math]
- [math]\displaystyle{ k\cdot (X\cdot v)=(\operatorname{Ad}(k)X)\cdot (k\cdot v) }[/math]
- 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 ∈ [math]\displaystyle{ \mathfrak{k} }[/math]
- [math]\displaystyle{ \left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v. }[/math]
In the above, the dot, [math]\displaystyle{ \cdot }[/math], denotes both the action of [math]\displaystyle{ \mathfrak{g} }[/math] on V and that of K. The notation Ad(k) denotes the adjoint action of G on [math]\displaystyle{ \mathfrak{g} }[/math], and Kv is the set of vectors [math]\displaystyle{ k\cdot v }[/math] 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 [math]\displaystyle{ \mathfrak{g} }[/math] 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
- [math]\displaystyle{ kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv. }[/math]
In other words, it is a compatibility requirement among the actions of K on V, [math]\displaystyle{ \mathfrak{g} }[/math] on V, and K on [math]\displaystyle{ \mathfrak{g} }[/math]. The third condition is also a compatibility condition, this time between the action of [math]\displaystyle{ \mathfrak{k} }[/math] on V viewed as a sub-Lie algebra of [math]\displaystyle{ \mathfrak{g} }[/math] and its action viewed as the differential of the action of K on V.
Notes
- ↑ Page 73 of Wallach 1988
- ↑ Page 12 of Doran & Varadarajan 2000
- ↑ This is James Lepowsky's more general definition, as given in section 3.3.1 of Wallach 1988
References
- 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
- Wallach, Nolan R. (1988), Real reductive groups I, Pure and Applied Mathematics, 132, Academic Press, ISBN 978-0-12-732960-4, https://archive.org/details/realreductivegro0000wall
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