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In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernstein functor is closely related. ==Notation and terminology== *''G'' is a connected reductive real affine algebraic group (for simplicity; the theory works for more general groups), and ''g'' is the Lie algebra of ''G''. ''K'' is a maximal compact subgroup of ''G''. *''L'' is a Levi subgroup of ''G'', the centralizer of a compact connected abelian subgroup, and *''l'' is the Lie algebra of ''L''. *A representation of ''K'' is called K-finite if every vector is contained in a finite-dimensional representation of ''K''. Denote by ''W''''K'' the subspace of ''K''-finite vectors of a representation ''W'' of ''K''. *A (g,K)-module is a vector space with compatible actions of ''g'' and ''K'', on which the action of ''K'' is ''K''-finite. *R(''g'',''K'') is the Hecke algebra of ''G'' of all distributions on ''G'' with support in ''K'' that are left and right ''K'' finite. This is a ring which does not have an identity but has an approximate identity, and the approximately unital R(''g'',''K'')- modules are the same as (''g'',''K'') modules. 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「Zuckerman functor」の詳細全文を読む スポンサード リンク
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