The Trace of an Automorphism H~0(J, O(nΘ))

摘要

Let X be a projective smooth complex curve with group of automorphisms G. Let J be the Jacobian of X and let Θ be the theta divisor of J. Then G acts on J and Θ is invariant under the action of G. Given h ∈ G, our goal is to compute the trace of h on H~0(J, O(nΘ)) in order to decompose this space into a sum of irreducible representations of G. Dolgachev computed the decomposition of H~0 (J, O(2Θ)) when X is the Klein quartic and used it to study some invariant vector bundles on this curve; see the proof of Corollary 6.3 in [4].