Hecke transformation for orthogonal bundles over curves

TL;DR

This paper introduces a Hecke transformation for orthogonal bundles over curves, extending Tyurin's duality theorem to this setting and analyzing specific low-rank cases.

Contribution

It defines a Hecke transformation for orthogonal bundles and proves a duality theorem analogous to Tyurin's, with detailed study of low-rank cases.

Findings

Hecke transformation for orthogonal bundles is well-defined.

Tyurin's duality theorem extends to orthogonal bundles.

Detailed analysis of ranks 2, 3, 4, and 6 cases.

Abstract

Given an orthogonal bundle $E$ over a smooth projective curve $X$ we define a Hecke transformation in the moduli space of orthogonal bundles by performing an elementary transformation with respect to a Lagrangian submodule $L \subset E_{2x}$ at some point $x \in X$. We show that the analogue of Tyurin's duality theorem holds for orthogonal bundles. Special cases of orthogonal bundles of ranks $2,3,4$ and $6$ are studied in detail.