(g,K)-module

In mathematics, more specifically in the representation theory of reductive Lie groups, a ( g , K ) {displaystyle ({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 ( g , K ) {displaystyle ({mathfrak {g}},K)} -modules, where g {displaystyle {mathfrak {g}}} is the Lie algebra of G and K is a maximal compact subgroup of G. In mathematics, more specifically in the representation theory of reductive Lie groups, a ( g , K ) {displaystyle ({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 ( g , K ) {displaystyle ({mathfrak {g}},K)} -modules, where g {displaystyle {mathfrak {g}}} is the Lie algebra of G and K is a maximal compact subgroup of G. Let G be a real Lie group. Let g {displaystyle {mathfrak {g}}} be its Lie algebra, and K a maximal compact subgroup with Lie algebra k {displaystyle {mathfrak {k}}} . A ( g , K ) {displaystyle ({mathfrak {g}},K)} -module is defined as follows: it is a vector space V that is both a Lie algebra representation of g {displaystyle {mathfrak {g}}} and a group representation of K (without regard to the topology of K) satisfying the following three conditions In the above, the dot, ⋅ {displaystyle cdot } , denotes both the action of g {displaystyle {mathfrak {g}}} on V and that of K. The notation Ad(k) denotes the adjoint action of G on g {displaystyle {mathfrak {g}}} , and Kv is the set of vectors k ⋅ v {displaystyle kcdot 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 g {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 In other words, it is a compatibility requirement among the actions of K on V, g {displaystyle {mathfrak {g}}} on V, and K on g {displaystyle {mathfrak {g}}} . The third condition is also a compatibility condition, this time between the action of k {displaystyle {mathfrak {k}}} on V viewed as a sub-Lie algebra of g {displaystyle {mathfrak {g}}} and its action viewed as the differential of the action of K on V.