Discriminants of Theta-Representations

TL;DR

This paper extends Tevelev's explicit discriminant formula from simple Lie algebras to graded Lie algebras, linking dual hypersurfaces to complex reflection groups and explaining geometric properties like the Grassmannian's codegree.

Contribution

It generalizes the discriminant formula to graded Lie algebras and relates dual hypersurfaces to reflections in little Weyl groups, revealing new geometric insights.

Findings

Discriminant formulas for graded Lie algebras derived

Connection established between dual hypersurfaces and complex reflection groups

Explanation of Grassmannian codegree in terms of root systems

Abstract

Tevelev has given a remarkable explicit formula for the discriminant of a complex simple Lie algebra, which can be defined as the equation of the dual hypersurface of the minimal nilpotent orbit, or of the so-called adjoint variety. In this paper we extend this formula to the setting of graded Lie algebras, and express the equation of the corresponding dual hypersurfaces in terms of the reflections in the little Weyl groups, the associated complex reflection groups. This explains for example why the codegree of the Grassmannian $G(4, 8)$ is equal to the number of roots of $\mathfrak{e}_7$ .