Cohomology on Cotangent Bundles of Partial Flag Varieties in Type A

Abstract

Let $G=\mathrm{SL}_n(\mathbf C)$ and let $P\subset G$ be a standard parabolic subgroup with Levi factor $L$. For a $G$-dominant weight $\lambda$, consider the vector bundle on $T^*(G/P)$ obtained by pulling back the vector bundle on $G/P$ associated to the irreducible $L$-module $V_L(\lambda)^*$. We express its cohomology as a direct limit of the cohomology of certain line bundles on a Bott--Samelson variety associated to an affine Kac-Moody group. This comparison yields vanishing of higher cohomology and shows that the global sections are generated over $\mathbf C[\mathfrak g^*]$ by their degree-zero part $V_G(\lambda)^*$. Using these results together with the Braverman--Kazhdan intertwiners constructed in earlier joint work with A. Slipper, we give an explicit generating set for $\mathbf C[T^*(\mathrm{SL}_n/[P,P])]$. Finally, in an appendix joint with Tom Gannon, we combine these results to show the affinization $\mathrm{Spec}(\mathbf{C}[T^*(SL_n/[P,P])])$ has terminal singularities.