Fundamental solutions for partial differential equations with reflection group invariance

Abstract

The Dunkl differential-difference operator associated with a finite reflection group is used to extend the Weyl–Heisenberg algebra. The subalgebra of mastersymmetries graded by the characters of the Coxeter group W is constructed. It provides us with a new class of intertwining operators for appropriate W-invariant differential equations. These intertwining operators turn out to be suitable for constructing fundamental solutions. The remarkable example is given by the iterated wave operator with a Calogero-type potential for which the fundamental solution can be derived through the relevant intertwining operator in an explicit form. It is found that under appropriate conditions such an equation satisfies Huygens’s principle in the sense that its fundamental solution possesses a nontrivial (inner) lacuna.