The strange duality conjecture for generic curves

Let SUX(r) be the moduli space of semi-stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve X of genus g ≥ 1 over C. Recall that a vector bundle E on X is called semi-stable if for any subbundle V , deg(V )/ rk(V ) ≤ deg(E)/ rk(E). Points of SUX(r) correspond to isomorphism classes of semi-stable rank r vector bundles with trivial determinant up to an equivalence relation. For any line bundle L of degree g−1 onX define ΘL = {E ∈ SUX(r), h(E⊗L) ≥ 1}. This turns out be a non-zero Cartier divisor whose associated line bundle L = O(ΘL) does not depend upon L. It is known that L generates the Picard group of SUX(r) (for this and the precise definition of L in terms of determinant of cohomology see [DN]). Let U∗ X(k) be the moduli space of semi-stable rank k and degree k(g−1) bundles on X. Recall that on U∗ X(k) there is a canonical non-zero theta (Cartier) divisor Θk whose underlying set is {F ∈ U∗ X(k), h(X,F ) = 0}. Put M = O(Θk). Consider the natural map τk,r : SUX(r)× U∗ X(k) → U∗ X(kr) given by tensor product. From the theorem of the square, it follows that τ∗ k,rM is isomorphic to L M. The canonical element Θkr ∈ H0(U∗ X(kr),M) and the Kunneth theorem give a map well defined up to scalars: