Monadicity of localization for Lie super-algebras $\mathfrak{gl}(m, n)$

TL;DR

This paper investigates the monadicity of the localization functor for Lie super-algebras of type gl(m, n), establishing an equivalence between certain module categories and monodromic D-modules on flag varieties.

Contribution

It demonstrates that the right localization functor for gl(m, n) super-algebras is monadic, linking the coderived category of modules to modules over an algebra W in monodromic D-modules.

Findings

Right localization is monadic in a suitable sense.

Coderived category of modules is equivalent to ind-completion of W-modules.

Establishes a new connection between super-algebra representations and D-modules.

Abstract

We study the localization functor from the category of representation of Lie super-algebra $\mathfrak{g} = \mathfrak{gl}(m, n)$ into monodromic D-modules on the flag manifold $X = G/B$. We show that the right localization is monadic in a suitable sense, which identifies the coderived category of $\mathfrak{g}$-modules with the ind-completion of compactly generated $W$-modules for some algebra $W$ in monodromic $D_X$-bimodules.