On the representations generated by Eisenstein series of weight n+3/2

We consider the Eisenstein series E(z, s; k, chi, N) of weight k = (n + 3)/2, level N > 1 and a Dirichlet character chi modulo N such that chi(2) = 1. Shimura proved that E(z, k/2; k, chi, N) is a nearly holomorphic function. We prove that E(z, k/2; k, chi, N) generates an indecomposable reducible (g, K)-module of length 2. These are new examples of indecomposable reducible (g, K)-modules generated by nearly holomorphic modular forms. (C) 2019 Elsevier Inc. All rights reserved.