Generalized Verma module

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In

. The study of these operators is an important part of the theory of parabolic geometries.

Definition

Let ${\displaystyle {\mathfrak {g}}}$ be a semisimple Lie algebra and ${\displaystyle {\mathfrak {p}}}$ a

of ${\displaystyle {\mathfrak {g}}}$. For any ${\displaystyle V}$ of ${\displaystyle {\mathfrak {p}}}$ we define the generalized Verma module to be the

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

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

${\displaystyle {\mathfrak {p}}}$ 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.

Properties of GVMs

GVM's are

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

and these weight spaces are finite-dimensional.

As all

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

).

Homomorphisms of GVMs

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

.

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.

Standard

Let us suppose that there exists a nontrivial homomorphism of true Verma modules ${\displaystyle M_{\mu }\to M_{\lambda }}$. Let ${\displaystyle S\subset \Delta }$ be the set of those

α such that the negative root spaces of root ${\displaystyle -\alpha }$ are in ${\displaystyle {\mathfrak {p}}}$ (like in section Properties). The following theorem is proved by Lepowsky:^[2]

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

root reflection

and ${\displaystyle \cdot }$ is the affine action).

The structure of GVMs on the affine orbit of a ${\displaystyle {\mathfrak {g}}}$-dominant and ${\displaystyle {\mathfrak {g}}}$-integral weight ${\displaystyle {\tilde {\lambda }}}$ can be described explicitly. If W is the Weyl group of ${\displaystyle {\mathfrak {g}}}$, there exists a subset ${\displaystyle W^{\mathfrak {p}}\subset W}$ of such elements, so that ${\displaystyle w\in W^{\mathfrak {p}}\Leftrightarrow w({\tilde {\lambda }})}$ is ${\displaystyle {\mathfrak {p}}}$-dominant. It can be shown that ${\displaystyle W^{\mathfrak {p}}\simeq W_{\mathfrak {p}}\backslash W}$ where ${\displaystyle W_{\mathfrak {p}}}$ is the Weyl group of ${\displaystyle {\mathfrak {p}}}$ (in particular, ${\displaystyle W^{\mathfrak {p}}}$ does not depend on the choice of ${\displaystyle {\tilde {\lambda }}}$). The map ${\displaystyle w\in W^{\mathfrak {p}}\mapsto M_{\mathfrak {p}}(w\cdot {\tilde {\lambda }})}$ is a bijection between ${\displaystyle W^{\mathfrak {p}}}$ and the set of GVM's with highest weights on the affine orbit of ${\displaystyle {\tilde {\lambda }}}$. Let as suppose that ${\displaystyle \mu =w'\cdot {\tilde {\lambda }}}$, ${\displaystyle \lambda =w\cdot {\tilde {\lambda }}}$ and ${\displaystyle w\leq w'}$ in the

(otherwise, there is no homomorphism of (true) Verma modules ${\displaystyle M_{\mu }\to M_{\lambda }}$ 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 ${\displaystyle W^{\mathfrak {p}}}$:

Theorem. If ${\displaystyle w'=s_{\gamma }w}$ for some

positive root

${\displaystyle \gamma }$ and the length (see

Bruhat ordering

) l(w')=l(w)+1, then there exists a nonzero standard homomorphism ${\displaystyle M_{\mathfrak {p}}(\mu )\to M_{\mathfrak {p}}(\lambda )}$.

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

However, if ${\displaystyle {\tilde {\lambda }}}$ is only dominant but not integral, there may still exist ${\displaystyle {\mathfrak {p}}}$-dominant and ${\displaystyle {\mathfrak {p}}}$-integral weights on its affine orbit.

The situation is even more complicated if the GVM's have singular character, i.e. there ${\displaystyle \mu }$ and ${\displaystyle \lambda }$ are on the affine orbit of some ${\displaystyle {\tilde {\lambda }}}$ such that ${\displaystyle {\tilde {\lambda }}+\delta }$ is on the wall of the

.

Nonstandard

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

Bernstein–Gelfand–Gelfand resolution

This section is empty. You can help by adding to it. (July 2010)

Examples

• The fields of
  conformal algebra.^[3]

See also

• Verma module
• Parabolic geometry

External links

• Code for constructing the BGG resolution of Lie algebra modules and computing its cohomology

References