Theta blocks related to root systems

Abstract

© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.Gritsenko, Skoruppa and Zagier associated to a root system R a theta block ϑR, which is a Jacobi form of lattice index. We classify the theta blocks ϑR of q-order 1 and show that their Gritsenko lift is a strongly-reflective Borcherds product of singular weight, which is related to Conway’s group Co. As a corollary we obtain a proof of the theta block conjecture by Gritsenko, Poor and Yuen for the pure theta blocks obtained as specializations of the functions ϑR.