Laguerre semigroup and Dunkl operators

被引:84
作者
Ben Said, Salem [1 ]
Kobayashi, Toshiyuki [2 ]
Orsted, Bent [3 ]
机构
[1] Univ Nancy 1, Inst Elie Cartan, F-54506 Vandoeuvre Les Nancy, France
[2] Univ Tokyo, Grad Sch Math Sci, IPMU, Meguro Ku, Tokyo 1538914, Japan
[3] Univ Aarhus, Dept Math Sci, DK-8000 Aarhus C, Denmark
基金
日本学术振兴会;
关键词
Dunkl operators; generalized Fourier transform; Coxeter groups; Schrodinger model; holomorphic semigroup; Weil representation; Hermite semigroup; Hankel transforms; Heisenberg inequality; rational Cherednik algebra; minimal representation; DISCRETE DECOMPOSABILITY; MINIMAL REPRESENTATION; REDUCTIVE SUBGROUPS; ORTHOGONAL POLYNOMIALS; UNCERTAINTY PRINCIPLE; REFLECTION GROUPS; RESTRICTION; RESPECT; FORMULA; A(Q)(LAMBDA);
D O I
10.1112/S0010437X11007445
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We construct a two-parameter family of actions omega(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 omega(k,a) lifts to a unitary representation of the universal covering of SL(2, R), and can even be extended to a holomorphic semigroup Omega(k,a). In the k equivalent to 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 Omega(k,a) provides us with (k, a)-generalized Fourier transforms F-k,F-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,F-a. We also find kernel functions for Omega(k,a) and F-k,F-a for a = 1, 2 in terms of Bessel functions and the Dunkl intertwining operator.
引用
收藏
页码:1265 / 1336
页数:72
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