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.
Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on R N . We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space F k (C N ) of holomorphic functions on C N with reproducing kernel equal to the Dunkl-kernel. The definition and properties of F k (C N ) extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrödinger model and in the Fock model. Further, we prove a branching decomposition of F k (C N ) as a unitary G × SL(2, R)-module and a general version of Hecke's formula for the Dunkl transform. (2000): 33C52, 43A85, 44A15
Mathematics Subject Classification
For a simple real Jordan algebra V, a family of bi-differential operators from $\mathcal{C}^\infty (V\times V)$ to $\mathcal{C}^\infty (V)$ is constructed. These operators are covariant under the rational action of the conformal group of V. They generalize the classical Rankin–Cohen brackets (case $V=\mathbb{R}$).
Let G be a connected semisimple real-rank one Lie group with finite center. We consider intertwining operators on tensor products of spherical principal series representations of G. This allows us to construct an invariant trilinear form [Formula: see text] indexed by a complex multiparameter [Formula: see text] and defined on the space of smooth functions on the product of three spheres in 𝔽n, where 𝔽 is either ℝ, ℂ, ℍ, or 𝕆 with n = 2. We then study the analytic continuation of the trilinear form with respect to (ν1, ν2, ν3), where we locate the hyperplanes containing the poles. Using a result due to Johnson and Wallach on the so-called "partial intertwining operator", we obtain an expression for the generalized Bernstein–Reznikov integral [Formula: see text] in terms of hypergeometric functions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.