The Verlinde traces for $$SU_{X}(2,\xi )$$ and blow-ups

Abstract

Given a compact Riemann surface X of genus at least 2 with automorphism group G, we provide formulae that enable us to compute traces of automorphisms of X on the space of global sections of G-linearized line bundles defined on certain blow-ups of projective spaces along the curve X. The method is an adaptation of one used by Thaddeus to compute the dimensions of those spaces. In particular, we can compute the traces of automorphisms of X on the Verlinde spaces corresponding to the moduli space $$SU_{X}(2,\xi )$$ S U X ( 2 , ξ ) when $$\xi $$ ξ is a line bundle G-linearized of suitable degree.