Relative Dolbeault Geometric Langlands via the Regular Quotient

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].