Sums of certain arithmetic functions over 𝔽q[T] and non-unitary distributions

In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb{F}_q[T]$ to certain integrals over the ensemble of unitary matrices when $q \rightarrow \infty$. We study similar problems leading to integrals over the ensembles of symplectic and orthogonal matrices when $q \rightarrow \infty$.  This is joint work with Vivian Kuperberg.