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$(k, a)$-generalized Fourier transform with negative $a$

Tatsuro Hikawa

TL;DR

This work addresses extending the $(k,a)$-generalized Fourier transform framework to negative deformation parameter $a$, where prior $L^2$-theory was developed only for $a>0$. It constructs a unitary intertwiner $\kappa_{-1, -(N-2+2\langle k\rangle)}$ that maps radial components between the $a$ and $-a$ theories and intertwines the associated $\mathfrak{sl}_2$-triples via an automorphism $\tau$, thereby transferring structure from the positive to the negative regime. The authors extend the $(k,a)$-Laguerre semigroup to $a<0$, define the $(k,-a)$-Fourier transform $\mathscr{F}_{k,-a}$, and provide corresponding spectral decompositions, unitarity, and intertwinement results. This yields a symmetric, unified treatment of the two minimal representations encoded by $a>0$ and $a<0$, expanding global analysis in Dunkl settings and enabling consistent analysis across deformation signs.

Abstract

The $ (k, a) $-generalized Fourier transform $ \mathscr{F}_{k, a} $ introduced by Ben Saïd--Kobayashi--Ørsted is a deformation family of the classical Fourier transform with a Dunkl parameter $ k $ and a parameter $ a > 0 $ that interpolates minimal representations of two different simple Lie groups. In the present paper, we focus on the case $ a < 0 $. As a main result, we find a unitary transform that intertwines the known case $ a > 0 $ and the new case $ a < 0 $.

$(k, a)$-generalized Fourier transform with negative $a$

TL;DR

This work addresses extending the -generalized Fourier transform framework to negative deformation parameter , where prior -theory was developed only for . It constructs a unitary intertwiner that maps radial components between the and theories and intertwines the associated -triples via an automorphism , thereby transferring structure from the positive to the negative regime. The authors extend the -Laguerre semigroup to , define the -Fourier transform , and provide corresponding spectral decompositions, unitarity, and intertwinement results. This yields a symmetric, unified treatment of the two minimal representations encoded by and , expanding global analysis in Dunkl settings and enabling consistent analysis across deformation signs.

Abstract

The -generalized Fourier transform introduced by Ben Saïd--Kobayashi--Ørsted is a deformation family of the classical Fourier transform with a Dunkl parameter and a parameter that interpolates minimal representations of two different simple Lie groups. In the present paper, we focus on the case . As a main result, we find a unitary transform that intertwines the known case and the new case .
Paper Structure (18 sections, 16 theorems, 65 equations)

This paper contains 18 sections, 16 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Result of the paper
  4. Organization of the paper
  5. Notation
  6. Review of the theory of (k, a)-generalized Fourier transforms
  7. Preliminaries on the Dunkl theory
  8. sl2-triple of differential-difference operators
  9. L2-theory for the sl2-triple
  10. Lifting to a unitary representation Ωk, a
  11. (k, a)-generalized Laguerre semigroup
  12. (k, a)-generalized Fourier transform
  13. (k, a)-generalized Fourier transform with negative a
  14. A unitary transform that intertwines the cases a > 0 and a < 0
  15. L2-theory for the sl2-triple with negative a
  16. ...and 3 more sections

Key Result

Proposition 2.2

Let $k$ be a (not necessarily non-negative) multiplicity function, $a \in \mathbb{C} \setminus \{0\}$, and $m \in \mathbb{N}$. For $p \in \mathcal{H}_k^m(S^{N - 1})$ and $f \in C^\infty(\mathbb{R}_{> 0})$, we have where $p \otimes f$ denotes the function $r\omega \mapsto p(\omega) f(r)$ on $\mathbb{R}^N \setminus \{0\}$.

Theorems & Definitions (33)

  • Proposition 2.2
  • proof
  • Proposition 2.3: BKO2012
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • Corollary 2.6: BKO2012
  • proof
  • Theorem 2.7
  • ...and 23 more