Multiplication kernels

TL;DR

This paper introduces multiplication kernels in the context of birational and D-modules, explores their semi-classical and quantum aspects, and discusses their role in integrable systems and separation of variables.

Contribution

It presents new concepts of multiplication kernels and their semi-classical and quantized forms, linking algebraic geometry with integrable systems and D-module theory.

Findings

Examples of multiplication kernels provided

Discussion of quantization of semi-classical kernels

Hypothetical cyclic D-module for Hitchin systems

Abstract

We introduce the notion of multiplication kernels of birational and $D$-module type and give various examples. We also introduce the notion of a semi-classical multiplication kernel associated with an integrable system and discuss its quantization. Finally, we discuss geometric and algebraic aspects of method of separation of variables, and describe hypothetically a cyclic $D$-module for the generalized multiplication kernels for Hitchin systems for groups $GL_r$.