(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,Page 73 of {{harvnb|Wallach|1988}} 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.Page 12 of {{harvnb|Doran|Varadarajan|2000}}

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:This is James Lepowsky's more general definition, as given in section 3.3.1 of {{harvnb|Wallach|1988}} 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

}}

Category:Representation theory of Lie groups