Let $G$ be a reductive algebraic group defined over an algebraically closed field $k$ of characteristic p, let assume p is good prime for simplicity. Fix $B$ a Borel subgroup of $G$. Then for every $B$-variety $X$, we can define an associated bundle $G\times^B X$. Suppose $X$ has the dualizing sheaf $\omega\_X$. My question is whether there exists a formula for computing dualizing sheaf of $G\times^B X$ in terms of $\omega\_X$.


Any reference is appreciated. Thanks in advance!



Dualizing sheaf of an associated bundle
Yes, there is a formula for this due to Brion (it's Lemma 2 in his paper Multiplicity-Free Subvarieties of Flag Varieties). First, for any $B$-equivariant coherent sheaf $\mathcal F$ on $X$, there is a natural $G$-equivariant coherent sheaf $ G \times^B \mathcal F $ on $G \times^B X$. This assignment in fact is an equivalence of categories. (See Brion's paper for details on the construction.) Now assume that $X$ is Cohen-Macaulay. Then we have
$$ \omega\_{G \times^B X} = G \times^B \big( \omega\_X( 2\rho )\big), $$
where by $ \omega\_X(2\rho) $ we mean the bundle $\omega\_X$ on $X$ twisted by the trivial line bundle on $X$ with $B$-weight $2\rho$. (The twist by $2\rho$ comes from the fact that $\omega\_{G/B}$ is the $G$-equivariant bundle on $G/B$ whose fiber is the 1-dimensional $B$-module with weight $2\rho$. Here I'm using the convention that $B$ corresponds to the positive roots.)


