Restricting Hecke–Siegel operators to Jacobi modular forms

We compute the action of Hecke operators on Jacobi forms of “Siegel degree” n and m×m index M, provided 1⩽j⩽n−m. We find they are restrictions of Hecke operators on Siegel modular forms, and we compute their action on Fourier coefficients. Then we restrict the Hecke–Siegel operators T(p), Tj(p2) (n−m