Transfer operators and Hankel transforms: horospherical limits and quantization

TL;DR

This paper explores transfer operators and Hankel transforms in the context of spherical varieties, horospherical limits, and quantization, revealing their connections to scattering theory and L-functions.

Contribution

It extends the understanding of transfer operators in horospherical limits and offers a new interpretation of Jacquet's Hankel transform through geometric quantization.

Findings

Transfer operators vary in horospherical limits with a scattering theory interpretation.

Jacquet's Hankel transform is interpreted in terms of geometric quantization.

Connections between transfer operators, scattering theory, and L-functions are established.

Abstract

Transfer operators are conjectural "operators of functoriality," which transfer test measures and (relative) characters from one homogeneous space to another. In previous work, I computed transfer operators associated to spherical varieties of rank one, and gave an interpretation of them in terms of geometric quantization. In this paper, I study how these operators vary in the horospherical limits of these varieties, where they have a conceptual interpretation related to scattering theory. I also revisit Jacquet's Hankel transform for the Kuznetsov formula, which is related to the functional equation of the standard L-function of GL(n), and provide an interpretation of it in terms of quantization.