Relative Langlands duality and Koszul duality

Relative Langlands Duality and Koszul Duality: A Technical Analysis

Overview

The paper "Relative Langlands duality and Koszul duality" (2604.14085) constructs and proves a categorical equivalence arising from a relative form of Langlands duality, supplemented and unified by a Koszul duality framework. The central object of study is the relationship between categories of $B$-equivariant $D$-modules and categories of unipotent monodromic modules for equivariant quantizations associated to dual hyperspherical varieties, under specialized gradings. The results synthesize and extend conjectures of Ben-Zvi, Sakellaridis, and Venkatesh (BZSV conjecture), together with the Koszul duality conjectures formulated by Finkelberg-Ginzburg-Travkin.

Main Conjectures and Equivalences

The primary setting considers a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$, each equipped with an equivariant quantization, $Q(X)$ and $Q(X^\vee)$ respectively. When $X$ is polarized (i.e., $X \simeq T^*_\psi(Y)$, with $Q(X) = D_\psi(Y)$), and assuming the local BZSV conjecture holds for this pair, the main results are as follows:

• Relative Langlands Duality: The derived, renormalized, equivariant category $D$0 is conjecturally equivalent to the category of $D$1-equivariant dg-modules over $D$2, with compatible $D$3 actions.
• Koszul Duality: For a Borel subgroup $D$4 and its Langlands dual $D$5, the conjecture posits that the derived category $D$6 and the category of $D$7-monodromic modules over $D$8 (with unipotent monodromy) are Koszul dual, and similarly at the level of $D$9-graded categories.

The categorical equivalence is enhanced to encompass the Hecke monoidal action—i.e., it is compatible with the action of the Hecke category on the $S$0 side and its monodromic analogue on the $S$1 side, as per the Koszul duality of Bezrukavnikov-Yun.

Framework and Proof Strategy

The paper develops a formalism whereby the Koszul duality equivalence follows from the BZSV conjectured equivalence, up to explicit verifiable statements. The technical crux involves completions and fibers of equivariant derived categories, graded over $S$2, where $S$3 is a Cartan subalgebra.

The main steps are:

1. Completion at Regular Coweights: After passing to the $S$4-graded world, the completion at a regular coweight $S$5 aligns both sides of the conjectured equivalence, involving manipulations over $S$6 and shifts by $S$7.
2. Beilinson-Bernstein Localization: The categorical structure on the quantization side is reduced to modules over $S$8 with specified central character, via localization at $S$9.
3. Identification of Fixed-Point Components: Lemmas ensure that intersections of $G\circlearrowright X$0-orbits with $G\circlearrowright X$1 decompose as unions of $G\circlearrowright X$2-orbits, and stabilizer properties guarantee correct identification of equivariant objects.
4. Explicit Equivalence Construction: The combination of completion, restriction, and shift operations precisely reconstructs the $G\circlearrowright X$3-equivariant category from the loop space category.

The equivalence is shown to be independent of the choice of regular coweight, via functors induced by tensoring with appropriately chosen Harish-Chandra bimodules or convolution with Satake objects.

Compatibility with Hecke Actions

A notable amplification is the compatibility of the categorical equivalence with Hecke monoidal actions. Specifically,

• The Koszul duality of $G\circlearrowright X$4 and $G\circlearrowright X$5, as established by Bezrukavnikov-Yun, is recovered as a special case.
• The equivalence intertwines the classical Hecke category with its monodromic counterpart, preserving the $G\circlearrowright X$6-linear structures.

Further, the Radon transform $G\circlearrowright X$7 is identified as an essential ingredient in relating various completions and localizations, and the equivalence induced by the Koszul duality is monoidal up to conjugation by the longest Weyl group element.

Extension to Twisted Cotangent Cases and Example Computations

The methodology and results are extended to twisted cotangent cases, involving Hamiltonian reductions and torsors, with the same formalism providing categorical equivalences in these settings. This flexibility is demonstrated in several concrete examples:

• Jacquet-Shalika, Mirabolic, Orthosymplectic, and Bessel Cases: Here, unconditional results are achieved by virtue of existing proofs of the BZSV conjecture, and explicit identification of the relevant spherical, hyperspherical, and twisted varieties.
• Springer-Lagrangian Isomorphism: The corrected version of the conjecture relating homology (via characteristic cycle) of Lagrangian subvarieties associated to $G\circlearrowright X$8 and $G\circlearrowright X$9 is established, with an explicit twist by the sign representation.

Technical Features and Grading Structure

A key theoretical contribution is the passage to $G^\vee\circlearrowright X^\vee$0-graded completions of derived categories. This maneuver stabilizes the setting for categorical equivalences, making them insensitive to additional gradings or completions. Furthermore, the compatibility with monoidal Hecke actions ensures that not only the abstract equivalence, but also the finer structure of categorical actions and module homology is preserved.

Practical Implications and Future Directions

The categorical equivalences proven here have broad ramifications for relative Langlands duality, geometric representation theory, and the study of equivariant $G^\vee\circlearrowright X^\vee$1-modules and their interaction with quantum Hamiltonian reductions. Practically, these results facilitate the computation of equivalences and dualities in settings relevant to the geometric Langlands program, representation theory of real and quantum groups, and the study of mirror symmetry in algebraic geometry.

Theoretically, the frameworks constructed suggest several avenues for future exploration:

• Generalizations to higher categorical settings and O(∞) representations.
• Deepening compatibility with character sheaves and endoscopic phenomena.
• Intersections with recent advancements in theta correspondences, as indicated by ongoing work.

Conclusion

This paper rigorously establishes a formal bridge between relative Langlands duality and Koszul duality for an important class of hyperspherical varieties, at a categorical level compatible with Hecke actions and gradings. The results provide explicit equivalences for equivariant $G^\vee\circlearrowright X^\vee$2-modules and monodromic quantization categories, unify multiple strands of conjectures, and corroborate them through both general formalism and concrete computational examples. The implications are both practical for computation and theoretical for foundational insights into duality phenomena in geometric representation theory.