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"](https://arxiv.org/abs/2006.03013). 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 $G$.
> 
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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.



Algebraic representations and vector bundles
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.


