Equivalent definitions of Verma modules

Question

This is a rather basic question. I was reading some notes on geometric representation theory by Gaitsgory and his defition of Verma module is the following: Let $ \lambda $ be a weight of $ \mathfrak{g}$. The Verma module $M_\lambda \in \mathfrak{g} - mod$ is defined such that for any object $ M \in \mathfrak{g} -mod$ we have : $Hom_\mathfrak{g} (M_\lambda , M) = Hom_\mathfrak{b} (\mathbb{C}^\lambda , M)$. Where $\mathfrak{b}$ is the borel subalgebra and $\mathbb{C}^\lambda $ is the one dimensional $\mathfrak{b}$ module.

The definition of Verma module that I am used to is that we define $M_\lambda := U(\mathfrak{g}) \otimes_{U ( \mathfrak{b} )} \mathbb{C}^\lambda$.

Gaitsgory says that his defintion implies this one but gives no proof and I cant see how it should be. My first thought was that maybe I should play around with some adjoint functors but I don't really see how to proceed. An answer would be nice but maybe its a lot to write so I would love even a hint to get started. Thanks.

Answer 1

Let me add on to pre-kidney's answer, which to me is just explaining the tensor-hom adjunction by words; algebraically, it can be expressed in the following lines. (Note that I identify $\mathfrak{g}-mod$ with $U(\mathfrak{g})-mod$, and consider Hom-spaces in the later category)

Let $M(\lambda):= U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_\lambda$, we have $$ \begin{array}{rcl} Hom_{U(\mathfrak{g})}(M(\lambda),M) &=& Hom_{U(\mathfrak{g})}(U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_\lambda,M)\\ &=& Hom_{U(\mathfrak{b})}(\mathbb{C}_\lambda, Hom_{U(\mathfrak{g})}(U(\mathfrak{g}),M)) \\ &=& Hom_{U(\mathfrak{b})}(\mathbb{C}_\lambda, M) \end{array} $$ Conversely, in Gaitsgory's definition, the equality has a hidden "restrict $M$ to a $U(\mathfrak{b})$-module" on the right hand side, which is algebraically given by $Hom_{U(\mathfrak{g})}(U(\mathfrak{g}),M)$ (since restricting is the adjoint of inducing), so reading the above three line backwards shows that $M_\lambda$ is indeed $M(\lambda)$.

Answer 2

The two definitions are seen to be equivalent after considering the universal construction of the tensor product.

Starting from the tensor product definition, we see the Verma module is the most general $U(\mathfrak{b})$-bilinear map that projects down to $U(\mathfrak{g})$ and $\mathbb{C}^{\lambda}$. This is equivalent to forcing the $U(\mathfrak{g})$-linear morphisms of $M_{\lambda}$ to coincide with the $U(\mathfrak{b})$-linear morphisms of $\mathbb{C}^{\lambda}$, which is Gaitsgory's definition.

By the way, I feel it is important to state the action of $\mathfrak{b}$ on $\mathbb{C}^{\lambda}$ because this is the concrete part of the definition.

Answer 3

Here's a version of Aaron's answer using the (closely related) extension/restriction of scalars adjunction.

As noted before and similar to most of the Gaitsgory notes, I write $\mathfrak g$ and $\mathfrak b$ but really mean $U(\mathfrak g)$ and $U(\mathfrak b)$.

Consider the ring map $\varphi: U(\mathfrak b) \to U(\mathfrak g)$. I use the notation $(-)^\mathfrak{g} $ for the extension of scalars/$(-)_\mathfrak{b}$ for the restriction of scalars.

The before mentioned adjunction gives you

$Hom_\mathfrak{b}( \mathbb C^\lambda, M_\mathfrak{b} ) = Hom_\mathfrak{g}\left( (\mathbb C^\lambda)^\mathfrak{g}, M \right)$

which is what is really meant by the equality in the notes.

Recall that $(\mathbb C ^ \lambda)^\mathfrak {g} := U(\mathfrak g) \otimes_{U(\mathfrak{b})} \mathbb C ^ \lambda =: M_\lambda$.

In other words, the Verma module is the induced module by $ \varphi$ and $\mathbb C ^ \lambda$. So one could write $ M_\lambda = \varphi_! (\mathbb C^\lambda )$ (in the notation of the linked the wikipedia article) or $M_\lambda = Ind_\mathfrak b^\mathfrak g \mathbb (C ^\lambda)$, which is the notation Humphreys uses in his book "Representation Theory of Semisimple Lie Algebras in the BGG Category O" (compare this with §1.3.). This is a great book in general that might be worth reading along the Gaitsgory notes.

So morally, Vermas in $\mathfrak g$-mod are determined by their bahavior on $\mathfrak b$. The character $\lambda $ prescribes the action on $\mathfrak h$ and the quotient map $\mathfrak b \to \mathfrak h$ prescribes a trivial action on $\mathfrak n$. So, $M_\lambda$ is defined in such a way that it is a heighest weight module of weight $\lambda$ (which is what you want it to be; c.f. §1.2. of the book).