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(g,K)-module
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(g,K)-module : ウィキペディア英語版
(g,K)-module
In mathematics, more specifically in the representation theory of reductive Lie groups, a (\mathfrak,K)-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 (\mathfrak,K)-modules, where \mathfrak 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 \mathfrak be its Lie algebra, and ''K'' a maximal compact subgroup with Lie algebra \mathfrak. A (\mathfrak,K)-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 \mathfrak 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
::k\cdot (X\cdot v)=(\operatorname(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
::\left.\left(\frac\exp(tY)\cdot v\right)\right|_=Y\cdot v.
In the above, the dot, \cdot, denotes both the action of \mathfrak on ''V'' and that of ''K''. The notation Ad(''k'') denotes the adjoint action of ''G'' on \mathfrak, 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 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^kv=\left(kXk^\right)kv.
In other words, it is a compatibility requirement among the actions of ''K'' on ''V'', \mathfrak on ''V'', and ''K'' on \mathfrak. The third condition is also a compatibility condition, this time between the action of \mathfrak on ''V'' viewed as a sub-Lie algebra of \mathfrak 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)
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