Harish-Chandra D-modules for bi-Whittaker reduction

Abstract: Let $D(G)$ be the algebra of algebraic differential operators on a complex reductive group $G$. Denote by $\mathbb{W}$ the bi-Whittaker quantum Hamiltonian reduction of $D(G)$, also known as the quantum Toda lattice. In this article we define the admissible $\mathbb{W}$-modules and the special case of Harish-Chandra modules, the latter being bi-Whittaker variants of the invariant holonomic systems of Hotta-Kashiwara. We then study their torsion properties up to completion relative to the Kazhdan filtration, which in general is unbounded in both directions, through the geometry of universal centralizer. In the case of regular infinitesimal characters, the corresponding $D(G)$-module is shown to be the minimal extension of an irregular connection on the open Bruhat cell. Certain ring-theoretic properties of the completion of $\mathbb{W}$ are also obtained.

• The paper introduces a novel D-module approach for bi-Whittaker reduction by defining Harish-Chandra modules as minimal extensions with fixed central characters.
• It employs Skryabin equivalence and Kazhdan filtration completion to demonstrate torsion-freeness and establish Auslander-regularity of the completed quantum Toda algebra.
• The study connects D-module analysis with DAHA theory, providing detailed insights into relative characters and block decomposition via affine Weyl group actions.

Summary of "Harish-Chandra D-modules for bi-Whittaker reduction" (2602.21805)

Introduction and Context

This paper provides a systematic study of the bi-Whittaker quantum Hamiltonian reduction of the algebra of algebraic differential operators $D(G)$ on a complex reductive group $G$. The central object is the quantum Toda lattice $\mathbb{W}$, realized as $$(D(G)/D(G)\mathfrak{n}_\psi)^{N_\psi \times N_\psi}$$, where $N_\psi$ incorporates bi-Whittaker conditions defined via a non-degenerate character on the nilpotent radical $\mathfrak{n}$ of a Borel subgroup. The principal aim is to define and analyze admissible $\mathbb{W}$-modules, with particular emphasis on the Harish-Chandra modules, understood as bi-Whittaker analogs of the Hotta--Kashiwara invariant holonomic systems.

Motivations arise from both representation theory and harmonic analysis. The latter focuses on systems of differential equations satisfied by relative characters such as Bessel distributions, whose analytic properties are central in automorphic forms and the study of admissible representations. The modules introduced encode these analytic features and provide a D-module theoretic interpretation of the bi-Whittaker reduction.

Main Results and Techniques

Admissible and Harish-Chandra $\mathbb{W}$-Modules

Admissible $\mathbb{W}$-modules are defined as finitely generated modules on which the center $\mathcal{Z}(\mathfrak{g})$ acts locally finitely. Harish-Chandra modules $\mathcal{G}_\nu$ correspond to fixed (generalized) central character $\nu\in \mathfrak{c}$ and parallel the usual Harish-Chandra modules but in the bi-Whittaker context, thus describing holonomic systems relevant to relative characters.

Through the Skryabin equivalence, these $\mathbb{W}$-modules are shown to encode $D(G)$-modules on which $\mathfrak{n}_\psi$ acts locally nilpotently. This identification is technically crucial for relating the analytic and algebraic perspectives.

Torsion-Freeness, Filtrations, and Universal Centralizer

A primary technical theme is the study of torsion phenomena for admissible $\mathbb{W}$-modules up to completion with respect to the Kazhdan filtration, which is generally unbounded. The geometry of the universal centralizer $\mathfrak{Z}$ provides critical information about singular supports. The filtration structure, which is non-Zariskian except in the abelian case, mandates that torsion-freeness can only be established after passage to completion.

Under these completions, the main theorem establishes that any subquotient of $\mathcal{S}$-torsion (with $\mathcal{S}$ constructed from the big Bruhat cell) in an admissible module has zero completion, reflecting a strong control of support and regularity properties after completion.

Homological Properties and Auslander Regularity

The completion of $\mathbb{W}$, denoted $\widehat{\mathbb{W}}$, is analyzed as a noncommutative Noetherian ring. The paper proves that $\widehat{\mathbb{W}}$ is Auslander-regular with global and injective dimension equal to the reductive rank $n$ of $G$. Holonomicity for $\widehat{\mathbb{W}}$-modules is established in the sense of Iwanaga, reminiscent of the Gelfand-Kirillov framework for classical D-modules. Duality properties and the behavior of Ext-functors are studied in this ring-theoretic context.

Block Decomposition and Simplicity for Regular Infinitesimal Characters

The category of admissible modules is decomposed into blocks according to the orbits of the extended affine Weyl group $\widetilde{W} = W \ltimes X^*(T)$ on $\mathfrak{t}^*$. For $\widetilde{W}$-regular infinitesimal characters, the corresponding Harish-Chandra modules $\mathcal{G}_\nu$ are proved to be simple, and further results on minimal extension and decompletion are obtained: for such regular parameters, the $D(G)$-module associated to $\mathcal{G}_\nu$ is the minimal extension of its restriction to the open Bruhat cell, and the structure of the corresponding connection is determined.

Relation to Double Affine Hecke Algebras (DAHA)

An explicit identification of the completed quantum Toda algebra with the spherical subalgebra of the degenerate nil-DAHA is revisited, following work by Ginzburg. This connection provides algebraic control and access to well-developed techniques from DAHA theory, including filtration behavior, Ore localizations, and compatibility with center actions.

Implications and Future Directions

This work places the analytic study of bi-Whittaker distributions on a robust algebraic and categorical foundation, allowing the use of noncommutative algebraic geometry and homological methods in harmonic analysis. The identification of Harish-Chandra modules as minimal extensions elucidates the structure of associated analytic distributions and provides precise support-theoretic information. The block decomposition in terms of $\widetilde{W}$-orbits refines the classical understanding of the category $\mathcal{O}$ in this bi-Whittaker setting.

On the algebraic side, the results suggest the utility of quantum Hamiltonian reduction and DAHA methods for the explicit computation of D-module solutions, with ramifications for Lusztig’s program, categorified representation theory, and the explicit description of automorphic periods.

Potential future developments include:

• A deeper investigation of the irregularity and Stokes phenomena for minimal extensions arising from bi-Whittaker reduction, including their ramifications for the analytic continuation of relative characters and periods.
• Explicit calculation of Ext-groups and higher extension data between Harish-Chandra modules in this context, possibly leading to new decomposition theorems for representations.
• Applications to the construction and analysis of types and packets for non-Archimedean and real groups, using the D-module structure to control spectral characteristics.
• Elaboration of the derived and categorified versions alluded to in the paper, aligning the theory more closely with the ongoing developments in higher categorical representation theory and the geometric Langlands program.

Conclusion

The paper delivers a comprehensive algebraic framework for the study of bi-Whittaker reductions and their associated (quantum) D-module categories. The deep integration of geometrical, homological, and filtered-algebra techniques establishes new foundations for analyzing analytic questions about relative characters and D-module solutions on reductive groups, opening paths for future research at the interface of harmonic analysis, noncommutative algebraic geometry, and geometric representation theory.