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.