Degenerate flag varieties of type A: Frobenius splitting and BW theorem

Abstract

Let $\mathcal F ^a_\lambda$ be the PBW degeneration of the flag varieties of type $A_{n-1}$. These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$. We prove that $\mathcal F ^a_\lambda$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda$ of $\mathcal F ^a_\lambda$. The varieties $R_\lambda$ can be viewed as towers of successive $\mathbb{P }^1$-fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties $R_\lambda$ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for $\mathcal F ^a_\lambda$. Using the Atiyah–Bott–Lefschetz formula for $R_\lambda$, we compute the $q$-characters of the highest weight $\mathfrak sl _n$-modules.