  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 Maa\ss{} forms. We give a detailed account of the
image $\mathrm{sym}^1 \otimes \varpi(E_2)$ of the automorphic representation
$\varpi(E_2)$ 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 $\varpi(E_2)$. They found that
$\varpi(E_2)$ does not decompose as a restricted tensor product over all places
of $\mathbb{Q}$, while we discover that $\mathrm{sym}^1 \otimes \varpi(E_2)$
has a direct summand that does. This summand corresponds to a holomorphic and
modular, vector-valued analogue of $E_2$. The complement in $\mathrm{sym}^1
\otimes \varpi(E_2)$ 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.
