Moments of quadratic Dirichlet $L$-functions over rational function fields

We establish the meromorphic continuation of a multiple Dirichlet series associated to the fourth moment of quadratic Dirichlet $L$-functions, over the rational function field $\mathbb F_q(T)$ with $q$ odd, up to its natural boundary. This is the first such result in which the group of functional equations is infinite; in such cases, it is expected that the series cannot be continued everywhere but can at least be extended to a large enough region to deduce asymptotics at the central point. In this case, these asymptotics coincide with existing predictions for the fourth moment of the symplectic family of quadratic Dirichlet $L$-functions. The construction uses the Weyl group action of a particular Kac–Moody algebra; this suggests an approach to higher moments using appropriate non-affine Kac–Moody algebras.