  We develop the theory of Hermitian Jacobi forms of lattice index, for both
definite and indefinite Hermitian lattices. We also prove a theta decomposition
theorem for vector-valued Jacobi forms (both in the orthogonal and Hermitian
settings), with enhanced periodicity properties. This allows us to give a good
definition of orthogonal and Hermitian Jacobi forms of matrix index, when the
matrix need not be integral in any natural sense.
