The Howe Duality for the Dunkl Version of the Dirac Operator

Abstract

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x| 2 . A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. Ørsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. Kähler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs.

A Dirac–Dunkl Equation on S 2 and the Bannai–Ito Algebra

“…In fact, the relations (6) hold in any dimension and for any choice of the reflection group with different values of the constant γ [3,23]. Let P N (R 3 ) denote the space of homogeneous polynomials of degree N in R 3 , where N is a nonnegative integer.…”

The Z2n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

“…Suppose that (20) holds at level n and consider the set B = B ∪{x} with x ∈ A and |A ∩ B| = n. Consider y ∈ A ∩ B; one can write Γ B as in (18). Since x, y ∈ A, Γ A commutes with Γ {x,y} and one can write {Γ A , Γ B } as in (19), the only difference with (19) being that x, y ∈ A. Upon applying the induction hypothesis, one finds (20) with B replaced by B after a straightforward calculation.…”

The metaplectic Howe duality and polynomial solutions for the symplectic Dirac operator

“…In fact Ker(Ds) is, as a vector space, isomorphic to P ol(R 2n+2 ) (see Corollary 5.5) and we leave the question of its representation theoretic content open. In [18], the authors studied the specific deformation of Howe duality and Fischer decomposition for the Dirac operator acting on spinor valued polynomials, coming from the Dunkl deformation of the Dirac operator. It is an interesting question to develop the Dunkl version of the symplectic Dirac operator in the context of symplectic reflection algebras (see [11]).…”