# Verma modules and Borel–Weil

Let $$\mathfrak{g}$$ be a semisimple Lie algebra and fix a root system. Let $$\mathfrak{b}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in R^+}\mathfrak{g}_\alpha$$. The complex irreducible representation of $$\mathfrak{g}$$ with highest weight $$\lambda \in P^+$$ can be obtained via the following procedure:

1. Form the Verma module $$V_\lambda := U(\mathfrak{g})\otimes_{U(\mathfrak{b})} k_\lambda$$, where $$k_\lambda$$ is a 1-dimensional representation of $$\mathfrak{b}$$. Here $$h\in\mathfrak{h}$$ acts by $$\lambda(h)$$ on $$k_\lambda$$ and the positive roots act trivially.
2. Take the quotient of $$V_\lambda$$ by the maximal submodule $$M_\lambda \subset V_\lambda$$ to get the representation.

Now let $$G$$ be a connected semisimple algebraic group over $$\mathbb{C}$$. if I understood correctly, the Borel–Weil theorem states that irreducible representations of $$G$$ with highest weight $$\lambda$$ can be obtained as follows:

1. Take the negative Borel subgroup $$B^- \subset G$$ and form the equivariant line bundle $$L_\lambda := G\times_{B^-}\mathbb{C_\lambda}$$, for $$\mathbb{C}_\lambda$$ the $$B^-$$-representation pulled back from $$T = B^-/U^-$$.
2. Take $$\Gamma_{\text{hol}}(L_\lambda) \subset \Gamma(L_\lambda)$$ to get the representation.

It's hard to not notice the similarities between these constructions: both are "induced" from the corresponding representation of $$\mathfrak{h}$$, or $$T$$. However, the first construction is applied to the positive Borel subalgebra, while the second is applied to the negative Borel subgroup.

Is there a more precise relation between the two constructions, and is there an explanation for this discrepancy of choice of Borel subalgebra/subgroup?

• It may be better to think of $G/B^-$ as the variety of all Borel subgroups. Any $T$ fixed point $B$ on this variety will do. If one has such a fixed point and an equivariant line bundle, it becomes a puzzle to compare representations of $G$ and $B$, where $B$ acts on the fiber. Sep 19, 2022 at 7:25

I don't think the $$\pm$$ issue is too deep, and I'm punting on it in favor of answering the other question.
You can get a hold of dual Verma modules by considering distributions on $$G/B$$ supported on a $$B$$-orbit. Note that $$G$$ will, naturally, not act on this space, but $$B$$ and $$\mathfrak g$$ will. If the $$B$$-orbit is the open dense one, then this space of distributions is simply functions, or, sections of a line bundle (trivialized) over the big cell. This $$(\mathfrak g,B)$$-module contains a minimal submodule (dual to the Verma quotient you asked about), which is the sections that extend over the whole $$G/B$$.