Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime

In Kuperberg and Lal´ın [Forum Math. 34 (2022), pp. 711–747], the authors studied the mean-square of certain sums of the divisor function dk(f) over the function field Fq[T] in the limit as q →∞ and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [Math. Z. 288 (2018), pp. 167–198] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields.