Comparison of generalized theta functions
We extend the Verlinde formula to reductive groups by relating the dimension of the space of sections for a reductive group to the corresponding dimension for its semisimple part. In addition, we reduce the comparison question for two arbitrary reductive groups to the question of comparing the spaces of theta functions for isogenous semisimple groups. The latter turns out to be quite complicated. After proving a Kodaira-type vanishing theorem we outline an alternative approach that uses the holomorphic Lefschetz-Riemann-Roch formula. We show in an example that the branch locus of the map between moduli spaces gives non-trivial contributions to the spaces of theta-functions.
Pantev, Tony, "Comparison of generalized theta functions" (1994). Dissertations available from ProQuest. AAI9427595.