On a tamely ramified local relative Langlands conjecture via categorical representations

TL;DR

The paper advances tamely ramified relative local Langlands by constructing a spectral description for a full Iwahori Satake subcategory of $D$-modules on the loop space $LX$, under placidness and a dimension theory. It develops a relative tensor product framework for automorphic categories, proves a fully faithful functor from the automorphic side to the spectral side using monad and Künneth techniques, and establishes an equivalence on the Satake subcategories that aligns with the unramified relative Langlands duality via the integral transform. The results provide a concrete categorical realization of the tamely ramified relative local Langlands conjecture in this setting and pave the way for extending to $ ext{ell}$-adic contexts. Key methods include monad calculations, Künneth-type formulas, and the use of integral transforms to pass between automorphic and spectral categories.

Abstract

Let $G$ be a complex reductive group. For a smooth affine spherical $G$-variety $X$, assume that the unramified relative local Langlands conjecture of Ben-Zvi-Sakellaridis-Venkatesh for $X$ holds, the loop space $LX$ is an $L^+G$--placid ind--scheme, and there exists a dimension theory for $LX$, we give a spectral description of a full subcategory of Iwahori equivariant D-modules on $LX$ in terms of the relative Langlands dual of $X$, confirming a slight variant of the tamely ramified local relative Langlands conjecture proposed by Devalapurkar.

Theorems & Definitions (41)

• Conjecture 1: Unramified conjecture in the de-Rham setting
• Theorem 1: \ref{['thm:iwahori_proof']}
• Definition 1
• Proposition 1
• Example 1
• Definition 2
• Definition 3
• Definition 4: Shifted vector bundles
• Definition 5
• Definition 1