## On Segal-Bargmann analysis for finite Coxeter groups and its heat kernel (903)

Citations: | 2 - 2 self |

### BibTeX

@MISC{Sontz903onsegal-bargmann,

author = {Stephen Bruce Sontz and Centro De Investigación En Matemáticas and A. C. (cimat},

title = {On Segal-Bargmann analysis for finite Coxeter groups and its heat kernel},

year = {903}

}

### OpenURL

### Abstract

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal-Bargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Saïd and Ørsted and independently by Soltani) and the Dunkl heat kernel, due to Rösler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal-Bargmann analysis on Euclidean space. Hall’s results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in µ-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal-Bargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall’s Version C generalized Segal-Bargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal-Bargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal-Bargmann kernels for

### Citations

1126 | Differential geometry, Lie groups and symmetric spaces - Helgason - 1978 |

459 |
Reflection groups and Coxeter groups
- Humphreys
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Citation Context ... proofs in Section 3 of the results described in the abstract. We do not present in this section all of the proofs, since these are all known results. References for this background material are [5], =-=[14]-=-, [19], [20], [23] and [26]. These may be consulted for more details and proofs. We warn the reader that some of our notation and normalizations are not standard. We let RN denote the Euclidean space ... |

391 |
Generalized Coherent States and their Applications
- Perelomov
- 1986
(Show Context)
Citation Context ...ysis, namely, the study of this space and its transform. These generalized Segal-Bargmann spaces and transforms are often quite algebraic in nature. (See the texts by Ali, et al. [1] and by Perelomov =-=[17]-=- and references therein for more details. Note that the transform is often called a coherent state transform.) However, Hall in [12] introduced a Segal-Bargmann space and transform for compact Lie gro... |

152 |
On a Hilbert space of analytic functions and an associated integral transform
- Bargmann
- 1961
(Show Context)
Citation Context ... Segal-Bargmann analysis, heat kernel analysis, Coxeter group, Dunkl operator. 1 Research partially supported by CONACYT (Mexico) project 49187. 11 Introduction Since the introduction by Bargmann in =-=[3]-=- and Segal in [22] in the early 1960’s of a certain Hilbert space of holomorphic functions and an associated integral kernel transform, there has been much research on various deformations and general... |

62 |
The Segal-Bargmann ”Coherent State” transform for compact Lie groups
- Hall
- 1994
(Show Context)
Citation Context ...braic in nature. (See the texts by Ali, et al. [1] and by Perelomov [17] and references therein for more details. Note that the transform is often called a coherent state transform.) However, Hall in =-=[12]-=- introduced a Segal-Bargmann space and transform for compact Lie groups (and other closely related differential manifolds) that has an analytic flavor, since it is based directly on the heat kernel an... |

39 | Generalized Hermite polynomials and the heat equation for Dunkl operators
- Rösler
- 1998
(Show Context)
Citation Context ... 1, µ ∈ (−1/2, ∞) and t = 1, this coincides with the definition of µ-deformed convolution given in [24]. Finally, we present the heat kernel associated with this theory. (See Rösler’s papers [19] and =-=[20]-=-.) The heat equation in this theory is ∂u ∂t = 1 2 ∆µu, where the solution is a suitably smooth function u : RN × [0, ∞) → R. Also, ∆µ is the Dunkl Laplacian introduced earlier. It is at this point th... |

37 |
Mathematical problems of relativistic physics
- Segal
- 1963
(Show Context)
Citation Context ...nalysis, heat kernel analysis, Coxeter group, Dunkl operator. 1 Research partially supported by CONACYT (Mexico) project 49187. 11 Introduction Since the introduction by Bargmann in [3] and Segal in =-=[22]-=- in the early 1960’s of a certain Hilbert space of holomorphic functions and an associated integral kernel transform, there has been much research on various deformations and generalizations of what c... |

30 |
Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva
- Rosenblum
- 1992
(Show Context)
Citation Context ...neralized Segal-Bargmann space. Also the “chaotic transform” of Soltani in [23] is a generalized Segal-Bargmann transform for us.) This generalizes a setup for dimension N = 1 studied by Rosenblum in =-=[18]-=- and by his student Marron in [15]. The author and various collaborators have also worked extensively in recent years on this formulation in dimension one and continue to do so. (See [24] and referenc... |

27 | Markov processes related with Dunkl operator
- RÖSLER, VOIT
- 1998
(Show Context)
Citation Context ...ociated transform for Version C. (See Theorem 3.5.) Although we will not use this here, we would like to note that the Dunkl theory has close connections with probability theory as first developed in =-=[21]-=- by Rösler and Voit. They show that the Dunkl Laplacian is the generator of a strongly continuous Markov semigroup and study its associated stochastic process (a generalized Brownian motion with jump ... |

23 | Convolution operator and maximal function for the Dunkl transform
- Thangavelu, Xu
(Show Context)
Citation Context ... results described in the abstract. We do not present in this section all of the proofs, since these are all known results. References for this background material are [5], [14], [19], [20], [23] and =-=[26]-=-. These may be consulted for more details and proofs. We warn the reader that some of our notation and normalizations are not standard. We let RN denote the Euclidean space of finite dimension N ≥ 1 w... |

23 |
Do the equations of Motion Determine the Quantum Mechanical Commutation Relations?, Phys
- Wigner
- 1950
(Show Context)
Citation Context ... continue to do so. (See [24] and references therein.) We refer to this special one-dimensional case of Dunkl theory as µ-deformed quantum mechanics, since it originally appeared in a paper of Wigner =-=[27]-=- concerning a question in the theory of quantum mechanics. In this article we develop further the Segal-Bargmann analysis associated to the Dunkl theory in R N for any finite integer N ≥ 1. We prove v... |

20 | Dunkl operators: theory and applications
- Rösler
(Show Context)
Citation Context ... derivatives and of the Fourier transform in R N , based on a finite Coxeter group generated by reflections in R N . References to the original articles of Dunkl and more recent work are presented in =-=[19]-=-. However, this research was mainly focused on the configuration space R N , while Segal-Bargmann analysis also involves the phase space, which in this case is C N . More recently Soltani in [23] and ... |

12 |
Some new examples of Markov processes which enjoy the timeinversion property, Probab
- Gallardo, Yor
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Citation Context ... strongly continuous Markov semigroup and study its associated stochastic process (a generalized Brownian motion with jump discontinuities). More recent work by Gallardo and Yor on this subject is in =-=[8]-=- and [9]. Also there is some very recent related work in [6] and [7] by Demni. Another relation of this material to classical mathematics can be seen in 3the case of dimension N = 1. (I thank B. Hall... |

8 |
Segal–Bargmann transforms associated with finite Coxeter groups
- Said, Ørsted
(Show Context)
Citation Context ...ocused on the configuration space R N , while Segal-Bargmann analysis also involves the phase space, which in this case is C N . More recently Soltani in [23] and independently Ben Saïd and Ørsted in =-=[5]-=- have introduced a Hilbert space of holomorphic functions on C N and an associated Segal-Bargmann type transform in the context of the Dunkl theory on R N . (Be warned that what all of these authors c... |

6 |
Semigroups and the Bose-like oscillator
- Marron
- 1994
(Show Context)
Citation Context ...so the “chaotic transform” of Soltani in [23] is a generalized Segal-Bargmann transform for us.) This generalizes a setup for dimension N = 1 studied by Rosenblum in [18] and by his student Marron in =-=[15]-=-. The author and various collaborators have also worked extensively in recent years on this formulation in dimension one and continue to do so. (See [24] and references therein.) We refer to this spec... |

5 | The Segal–Bargmann transform for the heat equation associated with root systems
- Ólafsson, Schlichtkrull
(Show Context)
Citation Context ... Cartan decomposition of the Lie algebra G of G. (For the rest of this notation and further discussion, see [4] and [13].) This operator also is given in Equation (1.3) and analyzed in Example 1.5 in =-=[16]-=-. The point is that this formula compares favorably with the Dunkl Laplacian ∆µ (see Definition 2.3) when applied to a Coxeter group invariant function f, which in our notation is (∆µf)(x) = ∆f(x) + 2... |

4 |
Analysis on flat symmetric spaces
- Saïd, Ørsted
(Show Context)
Citation Context ...th the usual Segal-Bargmann theory in a Euclidean space of a certain class of functions (namely, the radial ones). Something similar also happens in higher dimension. Consider the setup considered in =-=[4]-=-, where G is a semisimple, connected Lie group with finite center and K is a maximal compact subgroup of G. Then the quotient space G/K is a Riemannian symmetric space of non-compact type. Then the ra... |

4 | Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel
- Soltani
(Show Context)
Citation Context ...d in [15]. However, this research was mainly focused on the configuration space R N , while Segal-Bargmann analysis also involves the phase space, which in this case is C N . More recently Soltani in =-=[19]-=- and independently Ben Saïd and Ørsted in [4] have introduced a Hilbert space of holomorphic functions on C N and an associated Segal-Bargmann type transform in the context of the Dunkl theory on R N ... |

3 | Radial Dunkl processes associated with Dihedral systems
- Demni
(Show Context)
Citation Context ...tic process (a generalized Brownian motion with jump discontinuities). More recent work by Gallardo and Yor on this subject is in [10] and [11]. Also there is some very recent related work in [7] and =-=[8]-=- by Demni. Another relation of this material to classical mathematics can be seen in the case of dimension N = 1. In that case the Dunkl Laplacian (see Def. 2.3 here or Eq. (2.5.1) in [18]) for even f... |

2 | The µ-deformed Segal-Bargmann transform is a Hall type transform, in preparation
- Sontz
(Show Context)
Citation Context ...Rosenblum in [18] and by his student Marron in [15]. The author and various collaborators have also worked extensively in recent years on this formulation in dimension one and continue to do so. (See =-=[24]-=- and references therein.) We refer to this special one-dimensional case of Dunkl theory as µ-deformed quantum mechanics, since it originally appeared in a paper of Wigner [27] concerning a question in... |

2 | the µ-deformed Segal-Bargmann space gets two measures, to appear in
- Sontz, How
- 2008
(Show Context)
Citation Context ...ace for a given t > 0 is realized by using two measures on the phase space or, equivalently, as a closed subspace of the space of L 2 holomorphic functions on C × {−1, 1} for some measure on it. (See =-=[25]-=-.) Under some rather restrictive hypotheses, Asai has shown in [2] for dimension N = 1 that the Segal-Bargmann space associated to a probability measure on the configuration space R can be realized as... |

1 |
Hilbert space of analytic functions associated with the modified Bessel function and related orthogonal polynomials
- Asai
(Show Context)
Citation Context ...e space or, equivalently, as a closed subspace of the space of L 2 holomorphic functions on C × {−1, 1} for some measure on it. (See [25].) Under some rather restrictive hypotheses, Asai has shown in =-=[2]-=- for dimension N = 1 that the Segal-Bargmann space associated to a probability measure on the configuration space R can be realized as the L 2 space of holomorphic functions on the phase space C for a... |

1 |
Hypercyclic and chaotic convolution operators associated with the Dunkl operator
- Betancor, Sifi, et al.
- 2005
(Show Context)
Citation Context ... pointwise in x ∈ R N , but we do not wish to go into details. Now that we have defined a translation operator, a natural next step is to define an associated convolution operator as has been done in =-=[6]-=- and [24] in dimension N = 1. The case of arbitrary finite dimension is treated in [26]. 18Definition 2.6 For functions φ, ψ : RN → C and x ∈ RN we define their Dunkl convolution product ∗µ,t by ∫ (φ... |

1 |
Note on radial Dunkl processes, arXiv: 0812.4269v2 [math.PR
- Demni
(Show Context)
Citation Context ... stochastic process (a generalized Brownian motion with jump discontinuities). More recent work by Gallardo and Yor on this subject is in [10] and [11]. Also there is some very recent related work in =-=[7]-=- and [8] by Demni. Another relation of this material to classical mathematics can be seen in the case of dimension N = 1. In that case the Dunkl Laplacian (see Def. 2.3 here or Eq. (2.5.1) in [18]) fo... |

1 |
Some Remarkable Properties of the Dunkl
- Gallardo, Yor
- 2006
(Show Context)
Citation Context ... continuous Markov semigroup and study its associated stochastic process (a generalized Brownian motion with jump discontinuities). More recent work by Gallardo and Yor on this subject is in [10] and =-=[11]-=-. Also there is some very recent related work in [7] and [8] by Demni. Another relation of this material to classical mathematics can be seen in the case of dimension N = 1. In that case the Dunkl Lap... |