The heat semigroup in the compact Heckman-Opdam setting and the Segal-Bargmann transform

Heiko Remling; Margit R\"osler

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The heat semigroup in the compact Heckman-Opdam setting and the Segal-Bargmann transform

Heiko Remling, Margit Rösler

Classical Analysis and ODEs · Submitted 15 Mar 2010

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The authors' main claim A Hilbert space of holomorphic functions is defined such that the $L^2$-heat transform becomes a unitary isomorphism.

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arXiv ID1003.2992 (version 1)

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CitationHeiko Remling, Margit R\"osler. "The heat semigroup in the compact Heckman-Opdam setting and the Segal-Bargmann transform." arXiv:1003.2992 (2010). https://pith.science/paper/HIFFUHAX

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In the first part of this paper, we study the heat equation and the heat kernel associated with the Heckman-Opdam Laplacian in the compact, Weyl-group invariant setting. In particular, this Laplacian gives rise to a Feller-Markov semigroup on a fundamental alcove of the affine Weyl group. The second part of the paper is devoted to the Segal-Bargmann transform in our context. A Hilbert space of holomorphic functions is defined such that the $L^2$-heat transform becomes a unitary isomorphism.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In the first part of this paper, we study the heat equation and the heat kernel associated with the Heckman-Opdam Laplacian in the compact, Weyl-group invariant setting.	contribution 1.00	0.89	0.05	asserted 0.96
s1	In particular, this Laplacian gives rise to a Feller-Markov semigroup on a fundamental alcove of the affine Weyl group.	result 0.90	0.54	0.06	asserted 1.00
s2	The second part of the paper is devoted to the Segal-Bargmann transform in our context.	contribution 0.97	0.81	0.05	asserted 0.96
s3	A Hilbert space of holomorphic functions is defined such that the $L^2$-heat transform becomes a unitary isomorphism.	result 0.51	0.88	0.29	asserted 1.00

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Original abstract

In the first part of this paper, we study the heat equation and the heat kernel associated with the Heckman-Opdam Laplacian in the compact, Weyl-group invariant setting. In particular, this Laplacian gives rise to a Feller-Markov semigroup on a fundamental alcove of the affine Weyl group. The second part of the paper is devoted to the Segal-Bargmann transform in our context. A Hilbert space of holomorphic functions is defined such that the $L^2$-heat transform becomes a unitary isomorphism.

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Also, note that this formula occurs in Segal-Bargmann analysis in the context of Heisenberg groups in [ref] and in the context of the compact Heckman-Opdam setting in [ref] .

The C-version Segal-Bargmann transform for finite Coxeter groups defined by the restriction principle · Version $C$ · 0.90

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Cites as background 1 paper

In the framework of Dunkl theory, the rational case has been studied by several authors, see [ref] , [ref] , [ref] .

On Segal-Bargmann analysis for finite Coxeter groups and its heat kernel · Introduction · 0.99

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