(g,K)-module

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.

References

• Doran, Robert S.; Varadarajan, V. S., eds. (2000), The mathematical legacy of Harish-Chandra, Proceedings of Symposia in Pure Mathematics, vol. 68, AMS, ISBN 978-0-8218-1197-9, MR 1767886
• Wallach, Nolan R. (1988), Real reductive groups I, Pure and Applied Mathematics, vol. 132, Academic Press, ISBN 978-0-12-732960-4, MR 0929683