Theta Functions for Sl(n) versus Gl(n)

Over a smooth complex projective curve C of genus g one may consider two types of moduli spaces of vector bundles, M := M(n, d), the moduli space of semistable bundles of rank n and degree d on C, and SM := SM(n, L), the moduli space of those bundles whose determinant is isomorphic to a fixed line bundle L on C. We call the former a full moduli space and the latter a fixed-determinant moduli space. Since the spaces SM(n, L) are all isomorphic as L varies in Pic(C), we also write SM(n, d) to denote any one of them. On both moduli spaces there are well-defined theta bundles, as we recall in Section 1. While the theta bundle θ on SM is uniquely defined, the theta bundles θF on M depend on the choice of complementary vector bundles F of minimal rank over C. For any positive integer k, sections of θ F generalize the classical theta functions of level k on the Jacobian of a curve, and so we call sections of θ over SM and θ F over M theta functions of level k for SL(n) and GL(n) respectively. Our goal is to study the relationship between these two spaces of theta functions. We prove a simple formula relating their dimensions, and then formulate a conjectural duality between these two spaces.