(g,K)-module

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}

: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.
• {{Citation | url-access=registration

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