Wikipedia / Generalized Verma module
Wikipedia / Generalized Verma module
In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the 1970s. 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
be a semisimple Lie algebra a parabolic subalgebra. For any irreducible finite-dimensional representation we define the generalized Verma module to be the relative tensor product
.
.
.
-integral weights (see weight.
It is well known that a parabolic subalgebra. It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space-module-module),
.
In further text, we will denote a generalized Verma module simply by GVM.
Properties of GVMs
GVM's are highest weight modules and their highest weight.
GVM's are weight modules, i.e. they are direct sum of its weight spaces and these weight spaces are finite-dimensional.
As all highest weight modules, GVM's are quotients of Verma modules. The kernel of the projection is
is the set of those simple roots is the root reflection is the affine action on λ. It follows from the theory of (true) Verma modules. The sum in (1) is not direct.
is the Borel subalgebra and the GVM is isomorphic to the inducing representation V.
is called regular. In other word, there exist an element w of the Weyl group W such that
is the affine action of the Weyl group.
is called singular is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).
Homomorphisms of GVMs
-homomorphism.
are linked with an affine action of the Weyl group. This follows easily from the Harish-Chandra theorem on infinitesimal central characters.
Unlike in the case of (true) Verma modules, the homomorphisms of GVM's are in general not injective and the dimension
may be larger than one in some specific cases.
. Such a homomorphism (that is a factor of a homomorphism of Verma modules) is called standard. However, the standard homomorphism may be zero in some cases.
Standard
be the set of those simple roots (like in section Properties). The following theorem is proved by Lepowsky:
is the corresponding root reflection is the affine action).
-integral weight can be described explicitly. If W is the Weyl group is a bijection and the set of GVM's with highest weights on the affine orbit in the Bruhat ordering and the standard homomorphism does not make sense, see Homomorphisms of Verma modules).
:
Theorem. for some positive root and the length (see Bruhat ordering.
Theorem.
-integral weights on its affine orbit.
is on the wall of the fundamental Weyl chamber.
Nonstandard
is called nonstandard, if it is not standard. It may happen that the standard homomorphism of GVMs is zero but there still exists a nonstandard homomorphism.
Bernstein–Gelfand–Gelfand resolution
Examples
The fields of conformal field theory belong to generalized Verma modules of the conformal algebra.