Equivariant Maximal Cohen-Macaulay sheaves on the minimal orbit closures

Abstract

In this paper, we study maximal Cohen-Macaulay sheaves on closures of minimal nilpotent orbits in simple Lie algebras. For singularities of type $A_n$, we first classify vector bundles on their symplectic resolutions whose pushforwards are maximal Cohen-Macaulay. We then construct equivariant maximal Cohen-Macaulay sheaves via irreducible representations of the stabilizer group. We compare these two approaches in the case of maximal Cohen-Macaulay Weil divisors, and extend the equivariant construction to the classical types $B_n$, $C_n$, and $D_n$. Finally, we formulate the construction for an arbitrary simple Lie algebra and carry it out explicitly in the exceptional cases.