Automorphisms of the Extended Affine Root System and Modular Property for the Flat Theta Invariants

Abstract

We define the central extension ${\widetilde{\mathrm{Aut}}}^{+}(R)$ of the automorphism group ${\mathrm{Aut}}^{+}(R)$ of the extended affine root system. We give the action of ${\widetilde{\mathrm{Aut}}}^{+}(R)$ on the flat theta invariants (theta functions). This describes the modular property for the flat theta invariants.