Generalized Verma module

Generalized Verma module

Generalized Verma modules are object in the representation theory of Lie algebras, a field in mathematics. They were studied originally by James Lepowsky in 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{g} be a semisimple Lie algebra and mathfrak{p} a parabolic subalgebra of mathfrak{g}. For any irreducible finite dimensional representation V of mathfrak{p} we define the generalized Verma module to be the relative tensor product

:M_{mathfrak{p(V):=mathcal{U}(mathfrak{g})otimes_{mathcal{U}(mathfrak{p})} V.

The action of mathfrak{g} is left multiplication in mathcal{U}(mathfrak{g}).

If λ is the highest weight of V, we sometimes denote the Verma module by M_{mathfrak{p(lambda).

Note that M_{mathfrak{p(lambda) makes sense only for mathfrak{p}-dominant and mathfrak{p}-integral weights (see weight) lambda.

It is well known that a parabolic subalgebra mathfrak{p} of mathfrak{g} determines a unique grading mathfrak{g}=oplus_{j=-k}^k mathfrak{g}_j so that mathfrak{p}=oplus_{j>0} mathfrak{g}_j. Let mathfrak{g}_-:=oplus_{j<0} mathfrak{g}_j. It follows from the Poincaré-Birkhoff-Witt theorem that, as a vector space (and even as a mathfrak{g}_--module and as a mathfrak{g}_0-module),

:M_{mathfrak{p(V)simeq mathcal{U}(mathfrak{g}_-)otimes V.

In further text, we will denote a generalized Verma module simply by GVM.

Properties of GVM's

GVM's are highest weight modules and their highest weight λ is the highest weight of the representation V. If v_lambda is the highest weight vector in V, then 1otimes v_lambda is the highest weight vector in M_{mathfrak{p(lambda).

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 M_lambda o M_{mathfrak{p(lambda) is :(1)quad K_lambda:=sum_{alphain S} M_{s_alphacdot lambda}subset M_lambdawhere SsubsetDelta is the set of those simple roots α such that the negative root spaces of root -alpha are in mathfrak{p} (the set S determines uniquely the subalgebra mathfrak{p}), s_alpha is the root reflection with respect to the root α ands_alphacdot lambda is the affine action of s_alpha on λ. It follows from the theory of (true) Verma modules that M_{s_alphacdotlambda} is isomorphic to a unique submodule of M_lambda. In (1), we identified M_{s_alphacdotlambda}subset M_lambda. The sum in (1) is not direct.

In the special case when S=emptyset, the parabolic subalgebra mathfrak{p} is the Borel subalgebra and the GVM coincides with (true) Verma module. In the other extremal case when S=Delta, mathfrak{p}=mathfrak{g} and the GVM is isomorphic to the inducing representation V.

The GVM M_{mathfrak{p(lambda) is called "regular", if its highest weight λ is on the affine Weyl orbit of a dominant weight ildelambda. In other word, there exist an element w of the Weyl group W such that:lambda=wcdot ildelambda where cdot is the affine action of the Weyl group.

The Verma module M_lambda is called "singular", if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight ildelambda so that ildelambda+delta is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).

Homomorphisms of GVM's

By a homomorphism of GVM's we mean mathfrak{g}-homomorphism.

For any two weights lambda, mu a homomorphism

:M_{mathfrak{p(mu) ightarrow M_{mathfrak{p(lambda)

may exist only if mu and lambda are linked with an affine action of the Weyl group W of the Lie algebra mathfrak{g}. 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

:dim(Hom(M_{mathfrak{p(mu), M_{mathfrak{p(lambda)))

may be larger than one in some specific cases.

If f: M_mu o M_lambda is a homomorphism of (true) Verma modules, K_mu resp. K_lambda is the kernels of the projection M_mu o M_{mathfrak{p(mu), resp. M_lambda o M_{mathfrak{p(lambda), then there exists a homomorphism K_mu o K_lambda and f factors to a homomorphism of generalized Verma modules M_{mathfrak{p(mu) o M_{mathfrak{p(lambda). 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.

tandard

Let us suppose that there exists a nontrivial homomorphism of true Verma moduls M_mu o M_lambda.Let SsubsetDelta be the set of those simple roots α such that the negative root spaces of root -alpha are in mathfrak{p} (like in section Properties).The following theorem is proved by Lepowsky [Lepowski J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra, 49 (1977), 496-511.] :

The standard homomorphism M_{mathfrak{p(mu) o M_{mathfrak{p(lambda) is zero if and only if there exists alphain S such that M_mu is isomorphic to a submodule of M_{s_alphacdot lambda} (s_alpha is the corresponding root reflection and cdot is the affine action).

The structure of GVM's on the affine orbit of a mathfrak{g}-dominant and mathfrak{g}-integral weight ildelambda can be described explicitely. If W is the Weyl group of mathfrak{g}, there exists a subset W^{mathfrak{psubset W of such elements, so that win W^{mathfrak{pLeftrightarrow w( ildelambda) is mathfrak{p}-dominant. It can be shown that W^{mathfrak{psimeq W_{mathfrak{packslash W where W_{mathfrak{p is the Weyl group of mathfrak{p} (in particular, W^{mathfrak{p does not depend on the choice of ildelambda). The map win W^{mathfrak{p mapsto M_{mathfrak{p(wcdot ildelambda) is a bijection between W^{mathfrak{p and the set of GVM's with highest weights on the affine orbit of ildelambda. Let as suppose that mu=w'cdot ildelambda, lambda=wcdot ildelambda and wleq w' in the Bruhat ordering (otherwise, there is no homomorphism of (true) Verma modules M_mu o M_lambda and the standard homomorphism does not make sense, see Homomorphisms of Verma modules).

The following statements follow from the above theorem and the structure of W^{mathfrak{p:

"Theorem." If w'=s_gamma w for some positive root gamma and the length (see Bruhat ordering) l(w')=l(w)+1, then there exists a nonzero standard homomorphism M_{mathfrak{p(mu) o M_{mathfrak{p(lambda).

"Theorem". The standard homomorphism M_{mathfrak{p(mu) o M_{mathfrak{p(lambda) is zero if and only if there exists w"in W such that wleq w"leq w' and w" otin W^{mathfrak{p.

However, if ildelambda is only dominant but not integral, there may stil exist mathfrak{p}-dominant and mathfrak{p}-integral weights on its affine orbit.

The situation is even more complicated if the GVM's have singular character, i.e. there mu and lambda are on the affine orbit of some ildelambda such that ildelambda+delta is on the wall of the fundamental Weyl chamber.

Nonstandard

A homomorphism M_{mathfrak{p(mu) o M_{mathfrak{p(lambda) is called nonstandard, if it is not standard. It may happen that the standard homomorphism of GVM's is zero but there stil exists a nonstandard homomorphism.

Bernstein-Gelfand-Gelfand resolution

Examples

References

ee also

*Verma module


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