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Article

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

Mathematische Zeitschrift. 2013. Vol. 275. No. 1-2. P. 55-77.

Let $Fl^a_\lambda$ 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 $Fl^a_\lambda$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda$ of $Fl^a_\lambda$. The varieties $R_\lambda$ can be viewed as towers of successive $P^1$-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 $Fl^a_\lambda$. Using the Atiyah-Bott-Lefschetz formula for $R_\la$, we compute the $q$-characters of the highest weight $\msl_n$-modules.