Duality for dormant opers of classical types B and C

TL;DR

This paper extends a known duality between dormant opers of type A to classical Lie algebras of types B and C, establishing a canonical isomorphism between their moduli spaces under certain conditions.

Contribution

It generalizes the duality phenomenon for dormant opers from type A to types B and C, providing new structural insights and tools for these cases.

Findings

Constructed a canonical isomorphism between moduli spaces of dormant B and C type opers.

Extended duality to classical Lie algebras of types B and C under specific numerical conditions.

Enhanced understanding of the structure and enumeration of dormant opers in these types.

Abstract

A $\mathfrak{g}$-oper for a simple Lie algebra $\mathfrak{g}$ is a specific type of flat principal bundle on an algebraic curve. When the base field is of prime characteristic $p$, those with vanishing $p$-curvature are called dormant $\mathfrak{g}$-opers, and they form finite and geometrically meaningful moduli spaces. In earlier work, a canonical duality was established between dormant $\mathfrak{sl}_n$-opers and dormant $\mathfrak{sl}_{p-n}$-opers. This duality has provided effective tools for the study of higher-rank cases, as well as for the computation and structural understanding of the associated enumerative invariants. The main result of this paper extends this duality phenomenon to classical Lie algebras of type B and C. More precisely, under the numerical condition $p-1 = 2 (\ell +m)$, we construct a canonical isomorphism between the moduli spaces of dormant $\mathfrak{so}_{2\ell +1}$-opers and dormant and $\mathfrak{sp}_{2m}$-opers with prescribed symmetric radii.