Multiplication Kernels for the Analytic Langlands Program in Genus Zero

TL;DR

This paper provides an explicit proof of a multiplication kernel formula that intertwines Hecke and Gaudin operators within the analytic Langlands program for genus zero, clarifying its relation to other key objects.

Contribution

It offers the first explicit proof of Gaiotto's multiplication kernel formula in the genus-zero analytic Langlands setting, connecting it to broader Langlands objects.

Findings

Explicit formula for the multiplication kernel $K_3$

Proof of how $K_3$ intertwines Hecke and Gaudin operators

Discussion of the kernel's relation to other Langlands objects

Abstract

We provide an explicit proof of a recent result of Gaiotto arXiv:2110.02255 which gives an explicit formula for a so-called "multiplication kernel'' $K_3(x, y, z; t)$ intertwining the action of Hecke operators and Gaudin operators in three sets of variables. This function $K_3$ arises naturally in the context of the analytic formulation of the geometric Langlands program in the genus-zero case arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. We also discuss how the kernel $K_3$ relates to other objects typically considered in the analytic Langlands program.