Differential-difference operators associated to reflection groups
1989/01/01 by Charles F. Dunkl · 49 citations
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #Advanced Mathematical Modeling in Engineering #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.1090/s0002-9947-1989-0951883-8
Abstract
There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf Rn</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A commutative set of differential-difference operators, each homogeneous of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf R2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.
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