The Bernstein-Gelfand Tensor Product Functor and the Weight-2 Eisenstein Series

Abstract

The Bernstein-Gelfand tensor product functors are endofunctors of the category of Harish-Chandra modules provided by tensor products with finite dimensional modules. We provide an automorphic analogue of these tensor product functors, implemented by vector-valued automorphic representations that are trivial at all finite places. They naturally explain the role of vector-valued modular forms in recent work by Bringmann-Kudla on Harish-Chandra modules associated with harmonic weak Maaß forms. We give a detailed account of the image sym 1 ⊗ (E 2 ) of the automorphic representation (E 2 ) generated by the Eisenstein series of weight 2 under one of those tensor product functors. This builds upon work by Roy-Schmidt-Yi, who recently determined the structure of (E 2 ). They found that (E 2 ) does not decompose as a restricted tensor product over all places of Q, while we discover that sym 1 ⊗ (E 2 ) has a direct summand that does. This summand corresponds to a holomorphic and modular, vector-valued analogue of E 2 . The complement in sym 1 ⊗ (E 2 ) arises from one of the vector-valued examples in the work of Bringmann-Kudla. Our approach allows us to determine its structure at the finite places.