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In mathematics, more specifically in the representation theory of reductive Lie groups, a -module is an algebraic object, first introduced by Harish-Chandra,〔Page 73 of 〕 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 -modules, where is the Lie algebra of ''G'' and ''K'' is a maximal compact subgroup of ''G''.〔Page 12 of 〕 ==Definition== Let ''G'' be a real Lie group. Let be its Lie algebra, and ''K'' a maximal compact subgroup with Lie algebra . A -module is defined as follows:〔This is James Lepowsky's more general definition, as given in section 3.3.1 of 〕 it is a vector space ''V'' that is both a Lie algebra representation of 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'' ∈ :: :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'' ∈ :: In the above, the dot, , denotes both the action of on ''V'' and that of ''K''. The notation Ad(''k'') denotes the adjoint action of ''G'' on , and ''Kv'' is the set of vectors 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 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'', on ''V'', and ''K'' on . The third condition is also a compatibility condition, this time between the action of on ''V'' viewed as a sub-Lie algebra of and its action viewed as the differential of the action of ''K'' on ''V''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「(g,K)-module」の詳細全文を読む スポンサード リンク
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