TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 346, Number 1, November 1994 SINGULAR POLYNOMIALS FOR FINITE REFLECTION GROUPS C. F. DUNKL, M. F. E. DE JEU, AND E. M. OPDAM Abstract. The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of G that occur in this kernel. 1. Introduction and notations Let o be a real vector space of finite dimension N, equipped with an inner product (•,•)• Let G c 0(a(-, •)) be a finite (real) reflection group. We may and will assume that (•, •) is (/-invariant. Let R be the corresponding root system, where we will assume that (a, a) = 2 for all a £ R. Choose and fix a positive system R+ in R. Let & = C[o] denote the polynomial functions ona. ^ has a natural grad- ing & = ©„>0^'n ; we let «^+ = (&n>x£sn • There is a natural representation of G in AutcJ(^), given by (g-f)(x):=f(g~lx) (g£G,f£&,x£a). The (/-action preserves the degree. Any £ £ a defines an element t\* £ Homc(a, C) = a* = £?x by £*(>/) = (£,, n) (n £ a). Let K be the complex vectorspace of C-valued (/-invariant functions on R. An element of K is called a multiplicity function (the reason for this terminology is a connection of the theory of Dunkl operators with the harmonic analysis for the Cartan motion group; the values ka of k £ K are then determined by the multiplicities of the restricted roots). To each £, £ a and A; e ATwe assign an operator T$(k) £ Endc(^a) as follows: Ti(k)f=dif+ £ ka(a,Z)£^£ (fe&), a£R+ where <% is the directional derivative corresponding to £ and ra is the reflec- tion in the hyperplane orthogonal to a. These operators were introduced in [D2]. In that paper it was shown that {T((k)\£ £ a} is a commuting family of operators, for all fixed k £ K. They are homogeneous of degree -1 and (7-equivariant: (1-1) goTi(k)og~x = Tgi(k) (t£a,g£G). Received by the editors February 4, 1994. 1991 Mathematics Subject Classification. Primary20F55,33C80,33C35, 20C30. During the preparation of this paper Dunkl was partially supported by NSF grant DMS-9103214. ©1994 American Mathematical Society 0002-9947/94 $1.00+ $.25 per Page 237 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use