Generalized Verma module

In mathematics, generalized Verma modules are a generalization of a (true) Verma module,[1] 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

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 \geq 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 GVMs

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 1\otimes 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\to M_{\mathfrak{p}}(\lambda) is

(1)\quad K_\lambda:=\sum_{\alpha\in S} M_{s_\alpha\cdot \lambda}\subset M_\lambda

where S\subset\Delta 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 and s_\alpha\cdot \lambda is the affine action of s_\alpha on λ. It follows from the theory of (true) Verma modules that M_{s_\alpha\cdot\lambda} is isomorphic to a unique submodule of M_\lambda. In (1), we identified M_{s_\alpha\cdot\lambda}\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 \tilde\lambda. In other word, there exist an element w of the Weyl group W such that

\lambda=w\cdot\tilde\lambda

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 \tilde\lambda so that \tilde\lambda+\delta is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).

Homomorphisms of GVMs

By a homomorphism of GVMs we mean \mathfrak{g}-homomorphism.

For any two weights \lambda, \mu a homomorphism

M_{\mathfrak{p}}(\mu)\rightarrow 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\to M_\lambda is a homomorphism of (true) Verma modules, K_\mu resp. K_\lambda is the kernels of the projection M_\mu\to M_{\mathfrak{p}}(\mu), resp. M_\lambda\to M_{\mathfrak{p}}(\lambda), then there exists a homomorphism K_\mu\to K_\lambda and f factors to a homomorphism of generalized Verma modules M_{\mathfrak{p}}(\mu)\to 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.

Standard

Let us suppose that there exists a nontrivial homomorphism of true Verma moduls M_\mu \to M_\lambda. Let S\subset\Delta 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:[2]

The standard homomorphism M_{\mathfrak{p}}(\mu)\to M_{\mathfrak{p}}(\lambda) is zero if and only if there exists \alpha\in S such that M_\mu is isomorphic to a submodule of M_{s_\alpha\cdot \lambda} (s_\alpha is the corresponding root reflection and \cdot is the affine action).

The structure of GVMs on the affine orbit of a \mathfrak{g}-dominant and \mathfrak{g}-integral weight \tilde\lambda can be described explicitly. If W is the Weyl group of \mathfrak{g}, there exists a subset W^{\mathfrak{p}}\subset W of such elements, so that w\in W^{\mathfrak{p}}\Leftrightarrow w(\tilde\lambda) is \mathfrak{p}-dominant. It can be shown that W^{\mathfrak{p}}\simeq W_{\mathfrak{p}}\backslash W where W_{\mathfrak{p}} is the Weyl group of \mathfrak{p} (in particular, W^{\mathfrak{p}} does not depend on the choice of \tilde\lambda). The map w\in W^{\mathfrak{p}} \mapsto M_{\mathfrak{p}}(w\cdot\tilde\lambda) is a bijection between W^{\mathfrak{p}} and the set of GVM's with highest weights on the affine orbit of \tilde\lambda. Let as suppose that \mu=w'\cdot\tilde\lambda, \lambda=w\cdot\tilde\lambda and w\leq w' in the Bruhat ordering (otherwise, there is no homomorphism of (true) Verma modules M_\mu\to 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)\to M_{\mathfrak{p}}(\lambda).

Theorem. The standard homomorphism M_{\mathfrak{p}}(\mu)\to M_{\mathfrak{p}}(\lambda) is zero if and only if there exists w''\in W such that w\leq w''\leq w' and w''\notin W^{\mathfrak{p}}.

However, if \tilde\lambda is only dominant but not integral, there may still 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 \tilde\lambda such that \tilde\lambda+\delta is on the wall of the fundamental Weyl chamber.

Nonstandard

A homomorphism M_{\mathfrak{p}}(\mu)\to M_{\mathfrak{p}}(\lambda) 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.

See also

References

  1. Named after Daya-Nand Verma.
  2. Lepowsky J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra, 49 (1977), 496-511.
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