Symplectic degenerate flag varieties

TL;DR

This paper introduces and analyzes degenerate symplectic flag varieties, providing explicit constructions, desingularizations, and proving their geometric properties, leading to new representation-theoretic formulas.

Contribution

It constructs explicit models and desingularizations of degenerate symplectic flag varieties, establishing their geometric properties and deriving a new $q$-character formula for symplectic group modules.

Findings

Varieties are normal, locally complete intersections with terminal and rational singularities.

Varieties are Frobenius split, enabling geometric and representation-theoretic applications.

Derived a $q$-character formula for irreducible $Sp_{2n}$-modules.

Abstract

Let $\SF^a_\lambda$ be the degenerate symplectic flag variety. These are projective singular irreducible $\bG_a^M$ degenerations of the classical flag varieties for symplectic group $Sp_{2n}$. We give an explicit construction for the varieties $\SF^a_\lambda$ and construct their desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $\SF^a_\la$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil-Bott theorem and obtain a $q$-character formula for the characters of irreducible $Sp_{2n}$-modules via the Atiyah-Bott-Lefschetz fixed points formula.