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

Abstract

Let $\Fl^a_\la$ 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 $S{L}_n^a$. We prove that $\Fl^a_\la$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\la$ of $\Fl^a_\la$. The varieties $R_\la$ can be viewed as towers of successive $\bP^1$-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties $R_\la$ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel-Weyl type theorem for $\Fl^a_\la$. Using the Atiyah-Bott-Lefschetz formula for $R_\la$, we compute the $q$-characters of the highest weight $\msl_n$-modules.

arXiv comment27 pages

References in corpus (3)

• ${\mathbb{G}}_a^M$ degeneration of flag varieties
• PBW filtration and bases for irreducible modules in type $A_n$
• Flag varieties as equivariant compactifications of G_a^n

Cited by in corpus (7)

• Degenerate flag varieties: moment graphs and Schröder numbers
• Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations
• PBW--filtration over $\bz$ and compatible bases for $V_\bz(\la)$ in type ${\mathtt{A}}_n$ and ${\mathtt{C}}_n$
• The median Genocchi numbers, Q-analogues and continued fractions
• Desingularizations of quiver Grassmannians via graded quiver varieties
• Degenerate $S{L}_n$: representations and flag varieties
• Degenerate coordinate rings of flag varieties and Frobenius splitting