Samiksha Jaiswal (Editor)

(g,K) module

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In mathematics, more specifically in the representation theory of reductive Lie groups, a ( 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 ) -modules, where 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 g be its Lie algebra, and K a maximal compact subgroup with Lie algebra k . A ( g , K ) -module is defined as follows: it is a vector space V that is both a Lie algebra representation of g and a group representation of K (without regard to the topology of K) satisfying the following three conditions

1. for any vV, kK, and X g k ( X v ) = ( Ad ( k ) X ) ( k v ) 2. for any vV, Kv spans a finite-dimensional subspace of V on which the action of K is continuous 3. for any vV and Y k ( d d t exp ( t Y ) v ) | t = 0 = Y v .

In the above, the dot, , denotes both the action of g on V and that of K. The notation Ad(k) denotes the adjoint action of G on g , and Kv is the set of vectors k 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 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

k X v = k X k 1 k v = ( k X k 1 ) k v .

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

References

(g,K)-module Wikipedia


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