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

Let F_λ^a 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 F_λ^a have rational singularities, are normal and locally complete intersections, and construct a desingularization R_λ of F_λ^a. The varieties R_λ can be viewed as towers of successive ℙ¹-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties R_λ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel-Weil type theorem for F_λ^a. Using the Atiyah-Bott-Lefschetz formula for R_λ, we compute the q-characters of the highest weight sl_n-modules.