Laguerre semigroup and Dunkl operators

Abstract

We construct a two-parameter family of actions ω k,a of the Lie algebra sl(2, R) by differential-difference operators on R N \{0}. Here k is a multiplicity function for the Dunkl operators, and a > 0 arises from the interpolation of the two sl(2, R) actions on the Weil representation of Mp(N, R) and the minimal unitary representation of O(N + 1, 2). We prove that this action ω k,a lifts to a unitary representation of the universal covering of SL(2, R), and can even be extended to a holomorphic semigroup Ω k,a . In the k ≡ 0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a = 2) and the Laguerre semigroup studied by the second author with G. Mano (a = 1). One boundary value of our semigroup Ω k,a provides us with (k, a)-generalized Fourier transforms F k,a , which include the Dunkl transform D k (a = 2) and a new unitary operator H k (a = 1), namely a Dunkl-Hankel transform. We establish the inversion formula, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for F k,a . We also find kernel functions for Ω k,a and F k,a for a = 1, 2 in terms of Bessel functions and the Dunkl intertwining operator.

Explicit formulas for the Dunkl dihedral kernel and the (κ,a)-generalized Fourier kernel

“…where L(f (t)) = F (s), Re ν > −1 and Re s > |Im a| and the Laplace convolution formula (16), we get the result. 4 Dunkl kernel associated to the dihedral group 4…”

“…In the following, we write z 1 = |z 1 |ω, z 2 = |z 2 |η ∈ C and b = |z 1 ||z 2 |. Based on the sl 2 relation of ∆ κ , |x| 2 and the Euler operator, an orthonormal basis of L 2 (R m , υ κ (x)dx) for the general Dunkl case and a series expansion of the Dunkl kernel was constructed in [3,4]. In particular, the Dunkl kernel E κ (z 1 , z 2 ) = B κ,2 (x, y) associated with the dihedral group I k has the following series expansion (see also Theorem 1)…”

“…Similarly, the Dunkl transform also has the exponential notation where µ = m+2 κ , see [3]. In [4], the authors defined a radially deformed Dunkl-type harmonic oscillator ∆ κ,a = |x| 2−a ∆ κ − |x| a , a > 0.…”

“…we take the Laplace transform with respect to t in (4). With u R = e −iπ a ( 2z a/2 aR ) 2/a , r = s 2 + ( 2 a z a/2 ) 2 , R = s + r, λ = (m − 2)/2, z = |x||y| and ξ = x, y /z, for Re s big enough, we obtain…”

“…In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition.New bounds for the kernel of the (κ, a)-generalized Fourier transform are obtained as well.Recently, a lot of attention has been given to various generalizations of the Fourier transform. This paper focusses on two in particular, namely the Dunkl transform [14,7] and the (κ, a)-generalized Fourier transform [4]. Both transforms depend on a number of parameters, and as such it is an open problem, except for certain special cases, to find concrete formulas for their integral kernels.…”