[Paper Review] Differential-difference operators and radial part formulas for non-invariant elements

Hiroshi Oda|arXiv (Cornell University)|Feb 13, 2014

Advanced Algebra and Geometry11 references3 citations

TL;DR

This paper generalizes the classical radial part formula for invariant differential operators on symmetric spaces $G/K$ to non-invariant settings by introducing a category $\mathscr{C}_{\text{rad}}$ that unifies representation-theoretic notions between $G$ and a graded Hecke algebra $\mathbf{H}$, using Cherednik operators and a generalized Chevalley restriction theorem. The key contribution is a radial part formula for non-$K$-invariant elements, extending the Helgason-Fourier and Opdam-Cherednik transforms to a broader framework via natural functors with universal properties.

ABSTRACT

The classical radial part formula for the invariant differential operators and the K-invariant functions on a Riemannian symmetric space G/K is generalized to some non-invariant cases by use of Cherednik operators and a graded Hecke algebra H naturally attached to G/K. We introduce a category C_{rad} whose object is a pair of a (g_C,K)-module and an H-module satisfying some axioms which are formally the same as the generalized Chevalley restriction theorem and the generalized radial part formula. Various pairs of analogous notions in the representation theories for G and H, such as the Helgason-Fourier transform and the Opdam-Cherednik transform, are unified in terms of C_{rad}. We construct natural functors which send an H-module to a (g_C,K)-module and have some universal properties intimately related to C_{rad}.

Motivation & Objective

• To extend the classical radial part formula—valid for $K$-invariant differential operators and functions—beyond the invariant setting to non-invariant elements in $G/K$-symmetric spaces.
• To construct a category $\mathscr{C}_{\text{rad}}$ that formalizes the correspondence between $((\mathfrak{g}_{\mathbb{C}}, K)$-modules and $\mathbf{H}$-modules, mirroring the generalized Chevalley restriction theorem.
• To unify key transforms in harmonic analysis—Helgason-Fourier and Opdam-Cherednik—within a common categorical framework via natural functors.
• To establish universal functors $\Xi_{\text{rad}}$, $\Xi^{\text{min}}$, and $\Xi$ that map $\mathbf{H}$-modules to $((\mathfrak{g}_{\mathbb{C}}, K)$-modules with structural coherence.

Proposed method

• Introduces a category $\mathscr{C}_{\text{rad}}$ whose objects are pairs of $((\mathfrak{g}_{\mathbb{C}}, K)$-modules and $\mathbf{H}$-modules satisfying axioms derived from the generalized Chevalley restriction theorem.
• Uses Cherednik operators $\mathscr{T}$ on $S({\mathfrak{a}}_{\mathbb{C}})$ to define a radial part map $\gamma_0(r(\Delta)f) = \mathscr{T}(\gamma(\Delta))\gamma_0(f)$ for non-$K$-invariant $f$ and $\Delta \in U({\mathfrak{g}}_{\mathbb{C}})^K$.
• Applies the generalized radial part formula to $K$-finite functions $f$ whose $K$-types are single-petaled, a special class of $K$-types introduced by Oda.
• Constructs natural functors $\Xi_{\text{rad}}$, $\Xi^{\text{min}}$, and $\Xi$ that send $\mathbf{H}$-modules to $((\mathfrak{g}_{\mathbb{C}}, K)$-modules, exhibiting universal properties tied to $\mathscr{C}_{\text{rad}}$.
• Establishes analytic continuation of solutions to non-symmetric hypergeometric systems via holomorphic extension through regular singular points, using monodromy and path-connectedness arguments.

Experimental results

Research questions

• RQ1 How can the classical radial part formula for $K$-invariant differential operators on $G/K$ be generalized to non-$K$-invariant elements?
• RQ2 What categorical structure unifies the representation theories of $G$ and the graded Hecke algebra $\mathbf{H}$ in the non-invariant setting?
• RQ3 How do the Helgason-Fourier and Opdam-Cherednik transforms relate in a generalized framework beyond $K$-invariance?
• RQ4 What universal functors exist that map $\mathbf{H}$-modules to $((\mathfrak{g}_{\mathbb{C}}, K)$-modules while preserving radial structure?
• RQ5 Under what conditions on $f$ and $\Delta$ does the radial part formula $\gamma_0(r(\Delta)f) = \mathscr{T}(\gamma(\Delta))\gamma_0(f)$ hold for non-$K$-invariant data?

Key findings

• The radial part formula $\gamma_0(r(\Delta)f) = \mathscr{T}(\gamma(\Delta))\gamma_0(f)$ holds for any $\Delta \in U({\mathfrak{g}}_{\mathbb{C}})^K$ and any $K$-finite $f \in C^\infty(G/K)$ whose $K$-types are single-petaled.
• The category $\mathscr{C}_{\text{rad}}$ provides a formal framework that unifies the generalized Chevalley restriction theorem and radial part formulas for non-invariant elements.
• The Helgason-Fourier transform and the Opdam-Cherednik transform are unified within $\mathscr{C}_{\text{rad}}$ as dual realizations of the same underlying correspondence.
• The functors $\Xi_{\text{rad}}$, $\Xi^{\text{min}}$, and $\Xi$ are constructed as universal functors mapping $\mathbf{H}$-modules to $((\mathfrak{g}_{\mathbb{C}}, K)$-modules, with $\Xi_{\text{rad}}$ being fully faithful and $\Xi^{\text{min}}$ preserving minimal $K$-types.
• Solutions to non-symmetric hypergeometric systems extend holomorphically across singular loci of codimension ≥2, ensuring analyticity of $\Phi$ on the entire complexified Cartan subalgebra $\mathfrak{a}+iU$.