Résumé
Given a smooth genus two curve CC, the moduli space SUC(3)SU_C(3) of rank three semi-stable vector bundles on CC with trivial determinant is a double cover in P8{\mathbb {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 CC. In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of GL9GL_9, initiated by Gruson, Sam and Weyman. In this spirit, starting from a general trivector vv in ∧3C9\wedge ^3{\mathbb {C}}^9, we construct a Fano manifold DZ10(v)D_{Z_{10}}(v) in G(3,9)G(3,9) as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in SUC(3)SU_C(3). We deduce that DZ10(v)D_{Z_{10}}(v) is isomorphic to the odd moduli space SUC(3,OC(c))SU_C(3, {\mathcal {O}}_C(c)) of rank three stable vector bundles on CC with fixed effective determinant of degree one. We deduce that the intersection of DZ10(v)D_{Z_{10}}(v) with a general translate of G(3,7)G(3,7) in G(3,9)G(3,9) is a K3 surface of genus nineteen.