Jacobi and modular forms on symmetric domains (Q812497)

Suppose \(G\) is a semi-simple Lie group of Hermitian type, with associated symmetric domain \({\mathcal D}=G/K\) (\(K\) a maximal compact subgroup), and \(\rho\colon G\to{\text{ Sp}}(n,{\mathbb R})\) is a Lie group homomorphism. Suppose also that \(\tau\colon {\mathcal D}\to {\mathcal H}_n\), where \({\mathcal H}_n\) is the Siegel upper half-plane, is an equivariant map. Then one can define a Jacobi group \(G^J\), a semidirect product \(G\ltimes H\) where \(H\) is the Heisenberg group, and using this one defines Jacobi forms. This paper establishes an isomorphism between the space of Jacobi forms of given weight and certain vector-valued modular forms for \(G\), constructs Eisenstein series for Jacobi forms and writes Jacobi forms in terms of theta functions. The results are fully analogous to this for the symplectic group but the author has performed a public service by working out the details.