Algebraic representations and vector bundles

Question

This might seem like a silly question considering my relatively elementary knowledge of representation theory.

The question is regarding Eugen Hellman 's paper titled "On the derived category of the Iwahori-Hecke algebra". Specifically on the paragraph above Lemma 2.18. First, let $G$ be a reductive group over a field $C$ of characteristic zero, $B$ a Borel subgroup and $U$ the unipotent radical of $B$. The paper states:

... recall that an algebraic representation of $B$ defines a $G$-equivariant vector bundles on $G/B$. We write $\mathcal{U}^\vee$ for the $G$-equivaraiant vector bundles on $G/B$ defined by the canonical $B$-representation on $\mathfrak{u}^\vee$. Here $\mathfrak{u}$ denotes the Lie algebra of $U$ (considered as a $C$-vector space), and $\mathfrak{u}^\vee$ denotes it's dual. In particular $\mathcal{U}^\vee$ admits a filtration who graded pieces are line bundles $\mathcal{L}_\alpha$ on $G/B$ associated to negative roots (with respect to $B$) roots $\alpha$ of $G$.

My questions are the following:

1. Firstly, I believe by an algebraic representation, it just means a homomorphism of group schemes $B\rightarrow \operatorname{GL}_n$. Is that correct?

2. How does such a representation define a vector bundle on $G/B$ ? I am guessing it has something to do with constructing a $\operatorname{GL}_n$-torsor using the representation, but I am not sure how.

3. Why does $\mathcal{U}^\vee$ admit a filtration into line bundles? (this question might seem silly considering I don't understand how $\mathcal{U}^\vee$ is constructed in the first place)?

Thanks in advance.

Answer 1

The definition of algebraic representation should be exactly what you suggested in 1).

For what concerns 2), you can construct a $\operatorname{Gl}_n$-torsor over $G/B$ given a representation $B \to \operatorname{Gl}_n$ as you suggested. The reasoning should be the following: from the homomorphism $B \to \operatorname{Gl}_n$ you get an action of $B$ over $V$ a complex vector space of dimension $n$. You then consider the product variety $G \times V$ with the $B$ (right) action $$b \cdot(g,x)=(gb,b^{-1}x) .$$

You can check that this action is free and there is a quotient variety $G \times V/B$ usually denoted $G \times_B V$. The projection onto the first factor $p:G \times V \to G$ pasisng to the quotient induces a well defined map $p: G\times_B V \to G/B$ which can be checked to be a vector bundle. You can see that the fiber of $p$ is actually isomorphic to $V$ and so the vector bundle has dimension $n$.

Notice that given a $B$-invariant subspace $W$ you get an embedding of vector bundles $G \times_B W \subseteq G \times_B V$ such that the following isomorphism of vector bundles holds $$G \times_B V/W \cong (G \times_B V)/ (G \times_B W) .$$

You then need to find a filtration of $B$ representations of $\mathfrak{u}^{\vee}$ such that the associated graded piece is isomorphic to the $1$-dimensional space $\mathfrak{u}_{\alpha}$ for $\alpha$ negative roots. This should come from reprentation theory of reductive Lie algebras and reductive groups.