A reducible Weil representation of sp(4) realized by differential operators in the space of smooth functions on H2 × S1

Abstract

This work presents a novel way to obtain the associated Romanovski functions Rn,m(x) with n ≥ m in the three separate regions in terms of n and m. We obtain the raising and lowering relations with respect to the both indices, simultaneously, in the three regions. Then, a reducible Weil representation of the real Lie algebra sp(4) is realized in the space of complex-valued smooth functions on H2 × S1 by differential forms for the Cartan-Weyl basis. Its invariant subspace is the second rare instance of the highest weight irreducible representation of sp(4) all whose weight spaces are one-dimensional.