## S. Ben Saïd,
T. Kobayashi, and B. Ørsted, *Laguerre
semigroup and Dunkl operators*, Compositio Mathematica **148** (2012), 1265-1336, DOI: 10.1112/S0010437X11007445.
arXiv:0907.3749 [math.RT]..

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 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 includes 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,
and 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.

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© Toshiyuki Kobayashi