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Generalized Verma module : ウィキペディア英語版
Generalized Verma module

In mathematics, generalized Verma modules are a generalization of a (true) Verma module,〔Named after Daya-Nand Verma.〕 and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the nineteen seventies. The motivation for their study is that their homomorphisms correspond to invariant differential operators over generalized flag manifolds. The study of these operators is an important part of the theory of parabolic geometries.
==Definition==

Let \mathfrak be a semisimple Lie algebra and \mathfrak a parabolic subalgebra of \mathfrak. For any irreducible finite-dimensional representation V of \mathfrak we define the generalized Verma module to be the relative tensor product
:M_(\mathfrak)\otimes_)} V.
The action of \mathfrak is left multiplication in \mathcal(\mathfrak).
If λ is the highest weight of V, we sometimes denote the Verma module by M_}(\lambda) makes sense only for \mathfrak-dominant and \mathfrak-integral weights (see weight) \lambda.
It is well known that a parabolic subalgebra \mathfrak of \mathfrak determines a unique grading \mathfrak=\oplus_^k \mathfrak_j so that \mathfrak=\oplus_ \mathfrak_j.
Let \mathfrak_-:=\oplus_ \mathfrak_j.
It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space (and even as a \mathfrak_--module and as a \mathfrak_0-module),
:M_(\mathfrak_-)\otimes V.
In further text, we will denote a generalized Verma module simply by GVM.

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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