In this article, for irreducible admissible infinite-dimensional representations $\Pi$ and $\Pi'$ of $\mathit{GL}(2,\mathbf{C})$, we show that the local $L$-factor $L(s,\Pi \times \Pi')$ can be expressed as some local zeta integral for $\mathit{GL}(2,\mathbf{C})\times \mathit{GL}(2,\mathbf{C})$.

In 2010 Shan, Varagnolo and Vasserot introduced a family of graded algebras in order to prove a conjecture of Kashiwara and Miemietz which stated that the finite-dimensional representations of affine Hecke algebras of type $D$ categorify a module over a certain quantum group. We study these algebras, and in various cases, show how they relate to Varagnolo-Vasserot algebras and to quiver Hecke algebras which in turn allows us to deduce various homological properties.