Contraction of the \mathfraksl₂ -triple associated to the (k, a) -generalized Fourier transform

2025/05/28 by Hikawa, Tatsuro
#22E45 #43A32 #47D03 #FOS: Mathematics #Functional Analysis (math.FA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2505.22607

Abstract

Ben Saïd--Kobayashi--Ørsted introduced a family of \mathfraksl₂ -triples of differential-difference operators ℍ_{k, a} , 𝔼⁺_{k, a} and 𝔼^-_{k, a} on ℝ^N ∖ \0\ indexed by a Dunkl parameter k and a deformation parameter a ≠ 0 . In the present paper, we study the behavior as the parameter a approaches 0 . In this limit, the Lie algebra \mathfrakg_{k, a} = span_ℝ \ℍ_{k, a}, 𝔼⁺_{k, a}, 𝔼^-_{k, a}\ ≅ \mathfraksl(2, ℝ) contracts to a three-dimensional commutative Lie algebra \mathfrakg_{k, 0} , and its spectral properties change. We describe the joint spectral decomposition for \mathfrakg_{k, 0} , and discuss formulas for operator semigroups with infinitesimal generators in \mathfrakg_{k, 0} . In particular, we describe the integral kernel of exp(z | x |² Δ_k) as an infinite series, which, in some low-dimensional cases, can be expressed in a closed form using the theta function.

Citations

• Deformation of the heat kernel and the Brownian motion from the perspective of the Ben Saïd--Kobayashi--Ørsted (k,a)-generalized Laguerre semigroup theory
• (k, a)-generalized Fourier transform with negative a
• Tempered homogeneous spaces IV
• Explicit formulas for the Dunkl dihedral kernel and the (κ, a)-generalized Fourier kernel
• The kernel of the radially deformed Fourier transform
• Positivity of Dunkl's intertwining operator
• Differential-difference operators associated to reflection groups

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