Abstract
Given a smooth genus two curve C, the moduli space SU_C (3) of rank three semistable vector bundles on C with trivial determinant is a double cover in P^8 branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of C. In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of GL_9, initiated in [GSW13] and pursued in [GS15, RS18, RS19, BMT21]. Starting from a general trivector v in ∧^3 C^9 , we construct a Fano manifold D_Z_10 (v) in G(3, 9) as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in SU_C (3). We deduce that D_Z_10 (v) is isomorphic to the odd moduli space SU_C (3, O_C (c)) of rank three stable vector bundles on C with fixed effective determinant of degree one. We deduce that the intersection of D_Z_10 (v) with a general translate of G(3, 7) in G(3, 9) is a K3 surface of genus 19.