Interpolation of the oscillator representation and Azumaya algebras in tensor categories

Abstract

Let $\mathfrak{C}$ be a symmetric tensor category and let $A$ be an Azumaya algebra in $\mathfrak{C}$. Assuming a certain invariant $\eta(A) \in \mathrm{Pic}(\mathfrak{C})[2]$ vanishes, and fixing a certain choice of signs, we show that there is a universal tensor functor $\Phi \colon \mathfrak{C} \to \mathfrak{D}$ for which $\Phi(A)$ splits. We apply this when $\mathfrak{C}=\underline{\mathrm{Rep}}(\mathbf{Sp}_t(\mathbf{F}_q))$ is the interpolation category of finite symplectic groups and $A$ is a certain twisted group algbera in $\mathfrak{C}$, and we show that the splitting category $\mathfrak{D}$ is S. Kriz's interpolation category of the oscillator representation. This construction has a number advantages over previous ones; e.g., it works in non-semisimple cases. It also brings some conceptual clarity to the situation: the existence of Kriz's category is tied to the non-triviality of the Brauer group of $\mathfrak{C}$.