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

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 $\mathrm{sym}^1 \otimes \varpi(E_2)$ 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.