Orthogonal polynomials on the sphere with octahedral symmetry

For any finite reflection group G acting on RN there is a family of G-invariant measures (h2 dw, where h is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for G) on the unit sphere and an associated partial differential operator (Lh f: = A(fh) -fA h; A is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to h2 do) decomposition of homogeneous polynomials, that is, if p is of degree n then I n/21 p(x) = j Ix12Jpn(-2(X) ,=0 where Lh p, = 0 and deg p, = i for each i, but with the restriction that p and p, must all be G-invariant. The main topic is the hyperoctahedral group with h (x ) = ( X IX2 *' I N)(u(X XJ i<J The special case N = 2 leads to Jacobi polynomials. A detailed study of the case N = 3 is made; an important result is the construction of a third-order differential operator that maps polynomials associated to h with indices (a, I?) to those associated with (a + 2, fi + 1). Introduction. Regular figures have long been objects of fascination. We study here the interplay between them and the theory of orthogonal polynomials. This will also touch on finite groups generated by reflections and their polynomial invariants, as well as a generalization of spherical harmonics. The underlying situation throughout will consist of spaces of homogeneous polynomials on RN that satisfy a second-order differential equation and whose restrictions to the unit sphere have an orthogonality relation with respect to a weight function, and of a finite reflection group for which all these objects are invariant. The weight (measure) h2(x) dw(x) on the unit sphere S in R3, where h(x) = (XlX2X3)a((x 23--3)) a, 2, ... (dw is surface area), and the associated polynomials will be studied most intensely. Some of the highlights of the theory are as follows: Define the differential operator Lhf:= (f?) -fAh Received by the editors April 19, 1982 and, in revised form, December 15, 1982. 1980 Mathematics Subject Classification. Primary 33A45, 33A65; Secondary 20H15. 'This research was partially supported by NSF Grant MCS 81-02581. (D 984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page