Generalized Verma module

Generalized Verma modules are object in 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 subgalgebra 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 sence 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 $1\otimes v_{\lambda }$ 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 withy 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 GVM's

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

For any two weights $\lambda ,\mu$ a homomorphism

$M_{\mathfrak {p}}(\mu )\rightarrow M_{\mathfrak {p}}(\lambda )$

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.

See also

Verma module