Higher Spherical Polynomials T. Ibukiyama and D. Zagier Contents Introduction 1 Chapter I. General Theory 5 1. The generalized Legendre differential equation 5 § 2. Dimension formula and decomposition theorem 8 § 3. Scalar product and orthogonality 13 § 4. The Lie algebra of mixed Laplacians 19 § 5. The two canonical bases of a(d) 24 § P 6. Higher spherical polynomials and invariant harmonic polynomials 30 § Chapter II. Explicit Constructions 35 7. Construction of invariant harmonic polynomials for d = 4 35 § 8. Higher spherical polynomials for n = 3 38 § A. Generating function 38 B. Ratios of the various basis elements 41 C. Recursion relations 43 D. Explicit formulas for the coefficients of Pνν,d 45 E.Inhomogeneouscoordinatesandangularcoordinates 48 9. A universal generating function for the descending basis 50 § A. Existence of the power series G(n) 51 B. Proofs of the closed formula and commutation relation for 56 Lp C. Special cases and explicit formulas 59 10. Algebraic generating functions for the case n = 4 67 § A. Even values of d : experimental results 67 B. Algebraicity and non-algebraicity proofs for G(4) when s is integral 69 C. Properties of the generating function for fixed integral values of s 74 D. Generating function with respect to s 78 E. Odd values of d 80 F. Final remarks 82 Tables of higher spherical polynomials 83 Table 1. Spherical polynomials for n = 3 83 Table 2. Spherical polynomials and Gram matrices for n = 4 and n = 5 85 Table 3. Determinants of Gram matrices 92 List of Notations 95 Bibliography 96 0 HIGHER SPHERICAL POLYNOMIALS Tomoyoshi Ibukiyama and Don Zagier Abstract. In this paper, we study a new class of special functions. Specifically, we study a vector (n) space P (d) (n ∈ N, d ∈ C) of polynomials P (T ) in n × n symmetric matrices T = (tij ). For integral d ≥ n these are the polynomials on (Rd)n that are invariant with respect to the diagonal action of O(d) and harmonic on each Rd, and for general d they are the polynomial solutions of a certain system of differential equations. For n = 2 these are the classical Legendre and Gegenbauer polynomials, but for n > 2 they are new and are interesting for several reasons, including an application to the theory of Siegel modular forms and the fact that for n = 3 the associated system of differential equations is holonomic. When n = 3, each homogeneous component of P(n)(d) is one-dimensional, so there is a canonical basis. Even here the structure of the polynomials turns out to be very subtle. For n > 3 there is no obvious basis. We construct two canonical bases, dual to one another, and a generating function (generalizing the classical one for n = 2, and algebraic if n = 3 but not in general) for one of them. We also provide tables that illustrate some of the idiosyncracies of the theory. Introduction d For two natural numbers n and d and a scalar product ( , ) on R , we have a map βn from (Rd)n to the space of real symmetric n n matrices, given by Sn × βn : (x1,... ,xn) T =(tij)1 i,j n , tij =(xi,xj) . (1) 7→ ≤ ≤ d n Thus any polynomial P on n defines a polynomial P = P βn on (R ) . If d n, then we (n) S ◦ ≥ denote by (d) the space of polynomials P on n for which P is harmonic with respect to P S n xi for each i = 1,... ,n. For each multidegree a = (ae1,... ,an) (Z 0) we denote by a(d) ≥ (n) ∈ P the subspace of P (d) for which P is a homogeneous of degreee ai with respect to xi for each i = 1,... ,n.∈ (Here P the superscript “(n)” can be omitted since n is just the length of a.) Later in the paper we will generalize thee definitions of both (n)(d) and a(d) to arbitrary P P complex values of d. The elements of (n)(d) will be called higher spherical polynomials. There are several motivations for studying them:P (i) If n = 2, then (d) is one-dimensional if a = a (it is 0 otherwise) and is then P(a1,a2) 1 2 spanned by a polynomial in t11t22 and t12 which is the homogeneous version of one of the classi- cal families of orthogonal polynomials (Legendre polynomials if d = 3, Chebyshev polynomials if d =2 or d = 4, and Gegenbauer polynomial for d arbitrary). These classical polynomials occur in many places in mathematics and mathematical physics and have many nice properties: differ- ential equations, orthogonality properties, recursions, generating functions, closed formulas, etc. It is natural to try to generalize these. (ii) The polynomials in (n)(d) have an application in the theory of Siegel modular forms. This is described in detail inP [7] (similar differential operators are given in [2] and [4]) and will 1 not be pursued in this paper, but was our original reason for studying this particular space of polynomials. Briefly, the connection is as follows. If P a(d) (d n) and if ∈P ≥ F (Z)= c(T ) e2πi Tr(T Z) (Z H = Siegel upper half-space) ∈ n XT is a holomorphic Siegel modular form of weight d/2, then the function 1+ δij ∂ 2πi(t11z1+...+tnnzn) P F (Z) = c(T ) P (T ) e (z1,... ,zn H1) (2) 2 ∂zij Hn ∈ 1 T X is an elliptic modular form of weight d/2+ ai in zi for each i = 1,... ,n. (By considering images n g(H ) Hn, where g is a suitable element of Sp2n(R), one also gets maps from Siegel modular forms⊂ of degree n to Hilbert modular forms on totally real number fields of degree n, or to products of Hilbert modular forms on several totally real fields with degrees adding up to n.) We can understand why this should be true by taking F to be a Siegel theta series, say F (Z)= e2πi Tr( βn(x1,... ,xn) Z) n (x ,... ,xn) L 1 X ∈ where L Rd is a lattice on which the quadratic form (x,x) takes rational values. Then the function defined⊂ by (2) is the series (x1,x1) (xn,xn) 2π√ 1 zi P (x ,... ,x ) q ...q (q = e − ) ··· 1 n 1 n i x L xn L X1∈ X∈ and for P a(d) this ise a modular form of weight d/2 + ai in zi by the classical Hecke- Schoeneberg∈ theory P of theta series with harmonic polynomial coefficients. A deeper direct application of our theory to critical values of triple L function is found in [10] (see also [5]) and another direct application is given in [3]. There are various other theories of differential operators and applications of a similar kind (for example, see [8], [9], [11]). (iii) The spaces a(d) can be looked at from the point of view of representation theory. P Specifically, the map P P gives an isomorphism between a(d) and the space of “invariant harmonic polynomials,”7→ i.e., polynomials in n variables x PRd which are harmonic and ho- i ∈ mogeneous (of degree ai) ine each xi and invariant under the simultaneous action of O(d), the orthogonal group of the scalar product ( , ), on all xi; this space can be analyzed in terms of the classical representation theory of O(d). However, apart from a brief mention of this in connection with the dimension formula proved in 2, we will not use this interpretation, but § will work directly with the functions P on n, since this is more elementary, gives more explicit results, and is applicable in greater generality.S (iv) If n = 3, then the dimension of a(d) is always 0 or 1, so that the higher spherical polynomials are particularly canonical in thisP case. The higher spherical polynomials of varying multidegree give an orthogonal basis of L2 functions in three variables with respect to a certain scalar product, and the coefficients of these polynomials also turn out to be combinatorially very interesting expressions, including as special cases the classical 3j-symbol (or Clebsch-Gordan coefficients) of quantum mechanics. (v) In a sequel to this paper [12], we will show that the system of differential equations defining the higher spherical polynomials (generalized Legendre differential equations) in the case n = 3 is equivalent to an integrable Pfaffian system of rank 8, so that there are precisely eight linearly independent solutions, generalizing the Legendre functions of the first and second kinds in the classical case n = 2. In the general theory of differential equations, such systems are very rare: systems of linear partial differential equations with polynomial coefficients generally have a solution space which is either zero- or infinite-dimensional, and it is non-trivial to construct Pfaffian systems satisfying the necessary integrability conditions. The system arising from the study of higher spherical polynomials is therefore also interesting from this point of view. 2 The contents of the paper are as follows. Chapter I describes the general theory, for arbitrary n and d. In the first two sections, which are basic for everything that follows, we write down the differential equations defining the space (d) (now for any d C), prove a formula for Pn ∈ the dimensions of its homogeneous pieces a(d), and construct a specific basis P M (“monomial P ν basis”) characterized by the property that its restriction to the subspace t11 = = tnn = 0 consists of monomials. We also give a brief description of the behavior of (d) at certain··· special Pn values of d (in 2Z 0) where its dimension changes.