# $G$-equivariant coherent sheaves on Bott$-$Samelson resolutions

Let $G$ be a Lie group and $B$ a Borel subgroup. $G/B$ is the corresponding flag variety.

Let $w$ be an element of the Weyl group $W$ with a reduced expression $w = s_1 \cdots s_n$. Let $X_w$ be the corresponding Schubert variety in the flag variety, and $BS(s_1,...s_n)$ the corresponding Bott–Samelson resolution.

There is a birational map $$f: BS(s_1,s_2,...,s_n) \rightarrow X_{s_1s_2...s_n} \subset G/B\,.$$.

What can we say about the (derived) pushforward of the structure sheaf $\mathcal{O}$ on $BS(s_1,s_2,...,s_n)$?

Is that the convolution product of some $G$-equivariant coherent sheaves on $X_w \subset G/B$?

• Is the singularity of $X_w$ rational? If it is, then $Rf_*O = O$. – Sasha Oct 5 '14 at 4:43
• @Sasha yes, Schubert varieties have rational singularities. – Piotr Achinger Oct 5 '14 at 5:29
• Are you sure about the equivariance? I would have expected $B$-equivariant objects. The group $G$ acts transitively on the $B$-fixed points of $G/B$, so it does not seem to act on $X_w$. From the question, it seems that you might be interested in looking at the literature on Soergel bimodules. – Matthias Wendt Oct 5 '14 at 9:16
• Yes, this is only $B$-equivariant, and yes, $Rf_*\mathcal O = \mathcal O$, even when the word is not reduced. One place to read about such things is Brion and Kumar's excellent book (on Frobenius splitting). – Allen Knutson Oct 5 '14 at 11:29
• @SeanLawton Maybe it is worth mentioning that some users do not agree to adding MathJax/LaTeX to the titles which are perfectly readable without it. See this discussion on meta or this message in chat. (As far as I can say, there is no clear consensus about this. But since at the moment you are one of the most active editors, I thought that it would be good to make sure you are aware of this.) – Martin Sleziak May 31 '16 at 4:50