Relative Dolbeault Geometric Langlands via the Regular Quotient

TL;DR

This paper proposes a relative geometric Langlands conjecture in the Dolbeault setting for affine homogeneous spherical varieties, generalizing Hitchin's duality, and verifies it in several classical cases using the theory of the regular quotient.

Contribution

It formulates a new conjecture relating Dolbeault period sheaves and Dirac-Higgs-like sheaves for spherical varieties, extending Hitchin's duality to broader contexts.

Findings

Conjecture verified for Friedberg-Jacquet, Jacquet-Ichino, Rankin-Selberg, and Gross-Prasad cases.

Introduces a Fourier-Mukai duality framework for Dolbeault sheaves in the spherical setting.

Utilizes the theory of the regular quotient to establish these dualities.

Abstract

Let $X = G/H$ be an affine homogeneous spherical variety with abelian regular centralizer and no type N roots. In this paper, we formulate a relative geometric Langlands conjecture in the Dolbeault setting for $M = T^*X$. More concretely, we conjecture a Fourier-Mukai duality between the Dolbeault period sheaf and a sheaf whose construction closely resembles the Dirac-Higgs bundle of a polarization of the dual symplectic representation of Ben-Zvi, Sakellaridis, and Venkatesh. These conjectures can be seen as a generalization of Hitchin's conjectural duality of branes for symmetric spaces. We verify these conjectures in several cases, including the Friedberg-Jacquet case $X = GL_{2n}/GL_n\times GL_n$, the Jacquet-Ichino case $X = PGL_2^3/PGL_2$, the Rankin-Selberg case $X = GL_n\times GL_{n+1}/GL_n$, and the Gross-Prasad case $X = SO_n\times SO_{n+1}/SO_n$. Our main tool is the theory of the regular quotient, which was described in the context of symmetric spaces in [HM24].