Theta blocks related to root systems

TL;DR

This paper classifies certain theta blocks associated with root systems, proves their lifts are special Borcherds products, and confirms a conjecture linking these functions to the Conway group.

Contribution

It provides a classification of theta blocks of q-order 1 and proves their Gritsenko lifts are strongly-reflective Borcherds products, confirming a conjecture for pure theta blocks.

Findings

Classification of theta blocks of q-order 1.

Gritsenko lifts are strongly-reflective Borcherds products.

Proof of the theta block conjecture for pure theta blocks.

Abstract

Gritsenko, Skoruppa and Zagier associated to a root system $R$ a theta block $\vartheta_R$, which is a Jacobi form of lattice index. We classify the theta blocks $\vartheta_R$ 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 $\operatorname{Co}_0$. 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 $\vartheta_R$.