Generalized Verma module

Let ${\displaystyle {\mathfrak {g}}}$ be a semisimple Lie algebra and ${\displaystyle {\mathfrak {p}}}$ a parabolic subalgebra of ${\displaystyle {\mathfrak {g}}}$. For any irreducible finite-dimensional representation ${\displaystyle V}$ of ${\displaystyle {\mathfrak {p}}}$ we define the generalized Verma module to be the relative tensor product

${\displaystyle M_{\mathfrak {p}}(V):={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {p}})}V}$.

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

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

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

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

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

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

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

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 ${\displaystyle M_{\lambda }\to M_{\mathfrak {p}}(\lambda )}$ is

${\displaystyle (1)\quad K_{\lambda }:=\sum _{\alpha \in S}M_{s_{\alpha }\cdot \lambda }\subset M_{\lambda }}$

where ${\displaystyle S\subset \Delta }$ is the set of those simple roots α such that the negative root spaces of root ${\displaystyle -\alpha }$ are in ${\displaystyle {\mathfrak {p}}}$ (the set S determines uniquely the subalgebra ${\displaystyle {\mathfrak {p}}}$), ${\displaystyle s_{\alpha }}$ is the root reflection with respect to the root α and ${\displaystyle s_{\alpha }\cdot \lambda }$ is the affine action of ${\displaystyle s_{\alpha }}$ on λ. It follows from the theory of (true) Verma modules that ${\displaystyle M_{s_{\alpha }\cdot \lambda }}$ is isomorphic to a unique submodule of ${\displaystyle M_{\lambda }}$. In (1), we identified ${\displaystyle M_{s_{\alpha }\cdot \lambda }\subset M_{\lambda }}$. The sum in (1) is not direct.

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

The GVM ${\displaystyle M_{\mathfrak {p}}(\lambda )}$ is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight ${\displaystyle {\tilde {\lambda }}}$. In other word, there exist an element w of the Weyl group W such that

${\displaystyle \lambda =w\cdot {\tilde {\lambda }}}$

where ${\displaystyle \cdot }$ is the affine action of the Weyl group.

The Verma module ${\displaystyle M_{\lambda }}$ is called singular, if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight ${\displaystyle {\tilde {\lambda }}}$ so that ${\displaystyle {\tilde {\lambda }}+\delta }$ is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).

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

For any two weights ${\displaystyle \lambda ,\mu }$ a homomorphism

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

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

${\displaystyle dim(Hom(M_{\mathfrak {p}}(\mu ),M_{\mathfrak {p}}(\lambda )))}$

may be larger than one in some specific cases.

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

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