(g,K)-module

In mathematics, more specifically in the representation theory of reductive Lie groups, a ${\displaystyle ({\mathfrak {g}},K)}$-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 ${\displaystyle ({\mathfrak {g}},K)}$-modules, where ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ be its Lie algebra, and K a maximal compact subgroup with Lie algebra ${\displaystyle {\mathfrak {k}}}$. A ${\displaystyle ({\mathfrak {g}},K)}$-module is defined as follows:[3] it is a vector space V that is both a Lie algebra representation of ${\displaystyle {\mathfrak {g}}}$ 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 ∈ ${\displaystyle {\mathfrak {g}}}$

${\displaystyle k\cdot (X\cdot v)=(\operatorname {Ad} (k)X)\cdot (k\cdot v)}$

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 ∈ ${\displaystyle {\mathfrak {k}}}$

${\displaystyle \left.\left({\frac {d}{dt}}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v.}$

In the above, the dot, ${\displaystyle \cdot }$, denotes both the action of ${\displaystyle {\mathfrak {g}}}$ on V and that of K. The notation Ad(k) denotes the adjoint action of G on ${\displaystyle {\mathfrak {g}}}$, and Kv is the set of vectors ${\displaystyle k\cdot v}$ 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 ${\displaystyle {\mathfrak {g}}}$ 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

${\displaystyle kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv.}$

In other words, it is a compatibility requirement among the actions of K on V, ${\displaystyle {\mathfrak {g}}}$ on V, and K on ${\displaystyle {\mathfrak {g}}}$. The third condition is also a compatibility condition, this time between the action of ${\displaystyle {\mathfrak {k}}}$ on V viewed as a sub-Lie algebra of ${\displaystyle {\mathfrak {g}}}$ and its action viewed as the differential of the action of K on V.

Notes

1. Page 73 of Wallach 1988
2. Page 12 of Doran & Varadarajan 2000
3. 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, MR 1767886
• Wallach, Nolan R. (1988), Real reductive groups I, Pure and Applied Mathematics, 132, Academic Press, ISBN 978-0-12-732960-4, MR 0929683