Relative Dolbeault Geometric Langlands via the Regular Quotient

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 $M=T^{\ast }X$. 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 $X=G{L}_{2n}/G{L}_n\times G{L}_n$, the Jacquet-Ichino case $X=PG{L}_2^3/PG{L}_2$, the Rankin-Selberg case $X=G{L}_n\times G{L}_{n+1}/G{L}_n$, and the Gross-Prasad case $X=S{O}_n\times S{O}_{n+1}/S{O}_n$. Our main tool is the theory of the regular quotient, which was described in the context of symmetric spaces in [HM24].

v2: Added assumptions on flatness of regular centralizers, and corrected 2 critical errors in main statements: the placement of the exterior algebra in the Dirac-Higgs bundle in Conjecture 1.9 and the statement of our main global result, Conjecture 1.10, which is a weaker version of Conjecture 1.9 that is proven in examples. Sections 2, 4, and 5 have been heavily re-written