Lagrangian subvarieties of hyperspherical varieties related to $G_2$

TL;DR

This paper investigates two hyperspherical varieties associated with G_2 and SL(2), verifying a conjecture about the equality of the number of irreducible components of their Lagrangian subvarieties related to duality.

Contribution

It provides a verification of a conjecture linking the Lagrangian subvarieties of two S-dual hyperspherical varieties of G_2, expanding understanding of their geometric and representation-theoretic properties.

Findings

Confirmed the equality of the number of irreducible components of Lagrangian subvarieties.

Validated a conjecture relating dual hyperspherical varieties.

Enhanced understanding of the geometric structure of G_2-related varieties.

Abstract

We consider two $S$-dual hyperspherical varieties of the group $G_2 \times \text{SL}(2)$: an equivariant slice for $G_2$, and the symplectic representation of $G_2 \times \text{SL}_2$ in the odd part of the basic classical Lie superalgebra $\mathfrak{g}(3)$. For these varieties we check the equality of numbers of irreducible components of their Lagrangian subvarieties (zero levels of the moment maps of Borel subgroups' actions) conjectured in arXiv:2310.19770.