Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian

Let Ω be a G-invariant convex domain in N including 0, where G is a complex Coxeter group associated with reduced root system R N. We consider holomorphic functions f defined in Ω which are Dunkl polyharmonic, that is, (Δh)nf=0 for some integer n. Here Δh=∑j=1N j2 is the complex Dunkl Laplacian, and j is the complex Dunkl operator attached to the Coxeter group G, jf(z)=( f/ zj)(z)+∑v∈R+κv((f(z)-f(σvz))/ z,v )vj, where κv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any complex Dunkl polyharmonic function f has a decomposition of the form f(z)=f0(z)+(∑n=1Nzj2)f1(z)+ +(∑n=1Nzj2)n-1fn-1(z), for all z∈Ω, where fj are complex Dunkl harmonic functions, that is, Δhfj=0.