Séminaire Lotharingien de Combinatoire 45 (2000), Article B45a

ORTHOGONAL POLYNOMIALS

ASSOCIATED WITH ROOT SYSTEMS

I. G. Macdonald

School of Mathematical Sciences, Queen Mary and Westfield College, University of London, London E1 4NS, England

Abstract. Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R, S) we construct a family of W -invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters q, t_{1}, t_{2}, …, t_{r}, where r (= 1, 2 or 3) is the number of W -orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic symmetric spaces. Also when R = S is of type A_{n}, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.

Foreword

The text that follows this Foreword is that of my 1987 preprint with the above title. It is now in many ways a period piece, and I have thought it best to reproduce it unchanged. I am grateful to Tom Koornwinder and Christian Krattenthaler for arranging for its publication in the Séminaire Lotharingien de Combinatoire.

I should add that the subject has advanced considerably in the intervening years. In particular, the conjectures in §12 below are now theorems. For a sketch of these later developments the reader may refer to my booklet “Symmetric functions and orthogonal polynomials”, University Lecture Series Vol. 12, American Mathematical Society (1998), and the references to the literature given there.

November 2000

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Introduction

The orthogonal polynomials which are the subject of this paper are Laurent polynomials in several variables. To be a little more precise, they are elements of the group algebra A of the weight lattice P of a root system R, invariant under the action of the Weyl group of R, and they depend rationally on two parameters q and t (*). They are indexed by the dominant weights and they are pairwise orthogonal with respect to a certain weight function △, to be defined later.

For particular values of the parameters q and t, these polynomials reduce to familiar objects:

(i) when q = t they are independent of q and are the Weyl characters for the root system R.

(ii) when t = 1 they are again independent of q, and are the elements of A corresponding to the orbits of the Weyl group in the weight lattice P.

(iii) when q = 0 they are (up to a scalar factor) the polynomials that give the values of zonal spherical functions on a semisimple p-adic Lie group G relative to a maximal compact subgroup K, such that the restricted root system of (G, K) is the dual root system R^{∨}. Here the value of the parameter t is the reciprocal of the cardinality of the residue field of the local field over which G is defined.

(iv) finally, when q and t both tend to 1, in such a way that (t − 1)/(q − 1) tends to a definite limit k, then (for certain values of k) our polynomials give the values of zonal spherical functions on a real (compact or non-compact) symmetric space G/K arising from finite-dimensional representations of G that have a K-fixed vector ≠ 0. Here the root system R is the restricted root system of G/K, and the parameter k is half the root multiplicity (assumed for the purposes of this description to be the same for all restricted roots).

Thus these two-parameter families of orthogonal polynomials constitute a sort of bridge between harmonic analysis on real symmetric spaces and on their p-adic analogues. It is perhaps natural to ask, in view of recent developments (quantum groups, etc.), whether there is a group-like object depending on two parameters q and t that lies behind this theory; but on this question we have nothing to say. We would only remark that such a hypothetical object would have to partake of the properties of a p-adic Lie group when q = 0, and of a real Lie group in the limiting case (q, t) → (1, 1) described in (iv) above.

All this is in fact a simplified description of the theory. The context in which we shall work is that of an “admissible pair” (R, S) of root systems: this means that R and S are finite root systems in the same vector space, having the same Weyl group W and such that S (but not necessarily R) is reduced. In this context we define parameters q_{α} and t_{α} for each root α ∈ R, such that q_{α} = q_{β} and t_{α} = t_{β} if α and β are in the same W -orbit. This is described, and the appropriate notation established, in Sections 1 and 2. The weight function △ and the accompanying scalar product are defined in Section 3. The main result of the paper is Theorem (4.1), which asserts the existence of a family of orthogonal polynomials P_{λ} associated with a given admissible pair (R, S).

(*) This is a simplified description for the purposes of this introduction.

The proof of the theorem consists in constructing a suitable self-adjoint linear operator E with distinct eigenvalues; the polynomials P_{λ} are the eigenfunctions of E, suitably normalized. In fact we need two constructions for such a linear operator. The first of these is described in Section 5, and works whenever the root system S (assumed irreducible) possesses a minuscule weight, that is to say provided that S is not of type E_{8}, F_{4} or G_{2}. The second construction, described in Section 6, is based on the premise, familiar to experts in standard monomial theory, that the next best thing to a minuscule weight is a quasi-minuscule weight, and produces an operator E with the desired properties in the cases not covered by the previous construction.

In Sections 8–11 we consider the particular cases corresponding to (i)-(iv) above. We also consider, in Section 9, the case where R is of rank 1. If R is of type A_{1}, the polynomials P_{λ} are essentially the q-ultraspherical polynomials [1], whereas if R is of type BC_{1} the P_{λ} reduce to a particular case of the orthogonal polynomials of Askey and Wilson [2]. Also, if R is of type A_{n} (n ≥ 1) the P_{λ} are essentially the symmetric functions that are the subject of Chapter VI of [11].

Finally, in Section 12 we put forward two conjectures relating to the polynomials P_{λ}. They involve a common generalization of Harish-Chandra’s c-function and its p-adic counterpart, and one of the conjectures includes as a special case the constant term conjectures of [10] and [13].

§1

Let V be a real vector space of finite dimension, endowed with a positive-definite symmetric bilinear form ⟨u, v⟩. We shall write |v| = ⟨v, v⟩^{1/2} for v ∈ V, and

v^{∨} = 2v/|v|^{2}

if v ≠ 0. If R is a root system in V, we denote by R^{∨} the dual root system {α^{∨} : α ∈ R}.

Let R and S be root systems in V (and spanning V). The pair (R, S) will be said to be admissible if R and S have the same Weyl group W, and S (but not necessarily R) is reduced.

Suppose that (R, S) is admissible. Then the set of hyperplanes in V orthogonal to the roots is the same for both R and S, and hence (as S is reduced) there exists for each α ∈ R a unique positive real number u_{α} such that

α_{∗} = u_{α}^{−1}α ∈ S,

and the mapping f : R → S defined by f(α) = α_{∗} is surjective.

Let α ∈ R, w ∈ W and let β = wα. Then w(α_{∗}) ∈ S, and is a positive scalar multiple of β, so that w(α_{∗}) = β_{∗} = (wα)_{∗}. Hence the mapping f commutes with the action of W, and u_{α} = u_{β} whenever α, β lie in the same W -orbit in R. Moreover if R is not reduced and α, 2α ∈ R, we have

(1.1)  u_{2α} = 2u_{α},

since (2α)_{∗} = α_{∗}.

From now on we shall assume that R (and therefore also S) is irreducible. Another pair (R′, S′) of root systems in V will be said to be similar to (R, S) if there exist positive real numbers a, b such that R′ = aR and S′ = bS. The effect of passing from (R, S) to a similar pair is simply to multiply each u_{α} by the same positive scalar factor.

The classification of irreducible admissible pairs (R, S) up to similarity is easily described. There are three cases to consider.

(i) R is reduced and S = R, so that u_{α} = 1 for each α ∈ R.

(ii) R is reduced, with two root-lengths, and S = R^{∨}. Then u_{α} = \frac{1}{2}|α|^{2} for each α ∈ R. We may assume that |α|^{2} = 2 for each short root α ∈ R, and then we have u_{α} = 1 if α ∈ R is short, and u_{α} = m if α is long, where m = 2 if R is of type B_{n}, C_{n} or F_{4}, and m = 3 if R is of type G_{2}.

(iii) R is not reduced, hence is of type BC_{n} (n ≥ 1). Let

(1.2)  R_{1} = \{α ∈ R : \frac{1}{2}α ∉ R\} , R_{2} = \{α ∈ R : 2α ∉ R\},

so that R_{1} and R_{2} are reduced root systems of types B_{n}, C_{n} respectively if n ≥ 2 (if n = 1 they are both of type A_{1}). Up to similarity, there are two possibilities for S when n ≥ 2, namely S = R_{1} and S = \frac{1}{2}R_{2} (which coincide when n = 1). In both cases u_{α} = 1 or 2 for each α ∈ R (it is for this reason that we chose \frac{1}{2}R_{2} rather than R_{2}).

Thus the function α ↦ u_{α} on R, when appropriately normalized, is either constant and equal to 1, or else takes just two values {1, 2} or {1, 3}. We shall assume this normalization henceforth.

Remark. The classification of irreducible admissible pairs (R, S) up to similarity is closely related to (but not identical with) the classification of irreducible affine root systems as defined in [9], or equivalently of “echelonnages” as defined in [3].

The polynomials which are the subject of this paper will involve parameters q and t_{α}, α ∈ R, such that t_{α} = t_{β} if |α| = |β|. It would be possible to regard these parameters as independent indeterminates over ℤ, but it will be more useful to think of them as real variables. So let q be a real number such that 0 ≤ q < 1, and for each α ∈ R let

q_{α} = q^{u_{α}},

so that q_{wα} = q_{α} for each w ∈ W, and the set {q_{α} : α ∈ R} is either {q} or {q, q^{2}} or {q, q^{3}}. From (1.1) we have

q_{2α} = q_{α}^{2}

if α, 2α ∈ R.

Next, for each α ∈ R let t_{α} be a real number ≥ 0, such that t_{α} = t_{β} if |α| = |β|. If α ∈ V but α ∉ R we set t_{α} = 1. Furthermore, let k_{α} = (log t_{α})/(log q_{α}) if q ≠ 0 and t_{α} ≠ 0, so that

t_{α} = q_{α}^{k_{α}}.

If α ∉ R we have k_{α} = 0.

Finally, let ℤ[t] (respectively ℤ[q, t]) denote the ring of polynomials in the t_{α} and t_{2α}^{1/2} (respectively and q) with integer coefficients, and let ℚ(q, t) denote the field of fractions of ℤ[q, t], i.e., the field of rational functions of q and the t_{α}, t_{2α}^{1/2}.

§2

Let (R, S) be an irreducible admissible pair of root systems in V. Let {α_{1}, …, α_{n}} be a basis (or set of simple roots) of R, and let R^{+} denote the set of positive roots determined by this basis. Let

Q = ∑_{i=1}^{n} ℤα_{i}, Q^{+} = ∑_{i=1}^{n} ℕα_{i}

be respectively the root lattice of R and its positive octant. Furthermore let P and P^{++} be respectively the weight lattice of R and the cone of dominant weights. We have Q ⊂ P (but Q^{+} ⊄ P^{++} if n > 1). If R is not reduced, then Q is the root lattice of R_{1} (defined in (1.2)) and P is the weight lattice of R_{2}.

We define a partial order on P by

(2.1)  λ ≥ μ if and only λ − μ ∈ Q^{+}.

Let A denote the group algebra over ℝ of the free Abelian group P. For each λ ∈ P, let e^{λ} denote the corresponding element of A, so that e^{λ} · e^{μ} = e^{λ+μ}, (e^{λ})^{−1} = e^{−λ} and e^{0} = 1, the identity element of A. The e^{λ}, λ ∈ P, form an ℝ-basis of A.

The Weyl group W of R acts on P and hence also on A: w(e^{λ}) = e^{wλ} for w ∈ W and λ ∈ P. Let A^{W} denote the subalgebra of W -invariant elements of A.

Since each W -orbit in P meets P^{++} in exactly one point, it follows that the “monomial symmetric functions”

m_{λ} = ∑_{μ∈W λ} e^{μ} (λ ∈ P^{++})

form an ℝ-basis of A^{W}. Another basis is provided by the Weyl characters: let

R_{2}^{+} = {α ∈ R^{+} : 2α ∉ R},
(2.2)  ρ = \frac{1}{2} ∑_{α∈R_{2}^{+}} α,

(2.3)  δ = ∏_{α∈R_{2}^{+}}(e^{α/2} − e^{−α/2}) = e^{ρ} ∏_{α∈R_{2}^{+}}(1 − e^{−α}).

Then wδ = ε(w)δ for each w ∈ W, where ε(w) = det(w) = ±1.

For each λ ∈ P let

(2.4)  χ_{λ} = δ^{−1} ∑_{w∈W} ε(w)e^{w(λ+ρ)}.

Then χ_{λ} ∈ A^{W} for all λ ∈ P, and the χ_{λ} with λ ∈ P^{++} form an ℝ-basis of A^{W}. Moreover, we have

χ_{λ} = m_{λ} + lower terms (λ ∈ P^{++})

where by “lower terms” we mean a linear combination of the m_{μ} such that μ ∈ P^{++} and μ < λ (for the partial ordering (2.1)).

If λ ∉ P^{++}, then either χ_{λ} = 0 or else there exists μ ∈ P^{++} and w ∈ W such that μ + ρ = w(λ + ρ), and in this case χ_{λ} = ε(w)χ_{μ}.

Let f ∈ A, say

f = ∑_{λ∈P} f_{λ}e^{λ}

with only finitely many nonzero coefficients f_{λ}. We shall regard f as a function on V as follows: if x ∈ V, then

(2.5)  f(x) = ∑ f_{λ}q^{⟨λ,x⟩}.

For f as above, define

f̄ = ∑ f_{λ}e^{−λ}

so that f̄(x) = f(−x) for all x ∈ V. Also let

[f]_{1} = constant term of f = f_{0}.

Clearly we have

(2.6)  [f]_{1} = [f̄]_{1} = [wf]_{1},

for all w ∈ W.

Next, for each μ ∈ V we define T_{μ}f by

(2.7)  (T_{μ}f)(x) = f(x + μ)

so that

T_{μ}f = ∑ f_{λ} q^{⟨λ,μ⟩}e^{λ}.

Each T_{μ} is an ℝ-algebra automorphism of A, with inverse T_{−μ}. We have

(2.8)  wT_{μ}w^{−1} = T_{wμ}

for each w ∈ W, and

(2.9)  \overline{T_{μ}f} = T_{−μ}f̄.

Finally, let g = ∑ g_{λ}e^{λ} be another element of A. Then

(2.10)  [f̄T_{μ}g]_{1} = [ḡT_{μ}f]_{1}.

For both sides are equal to ∑ f_{λ}g_{λ}q^{⟨λ,μ⟩}.

§3

We shall now define a scalar product on the algebra A. For this purpose we introduce the notation

(x; q)_{∞} = ∏_{i=0}^{∞}(1 − xq^{i})

and for each k ∈ ℝ

(3.1)  (x; q)_{k} = (x; q)_{∞}/(xq^{k}; q)_{∞}.

In particular, if k ∈ ℕ we have

(3.2)  (x; q)_{k} = ∏_{i=0}^{k−1}(1 − xq^{i}).

Now let (R, S) be an irreducible admissible pair of root systems with common Weyl group W, and let

(3.3)  △ = △(q, t) = ∏_{α∈R} \frac{(t_{2α}^{1/2}e^{α}; q_{α})_{∞}}{(t_{α}t_{2α}^{1/2}e^{α}; q_{α})_{∞}}

= ∏_{α∈R}(t_{2α}^{1/2}e^{α}; q_{α})_{k_{α}}

by (3.1). If the k_{α} are all integers ≥ 0, then by (3.2) the product △ is a finite product of factors of the form 1 − q_{α}^{i}t_{2α}^{1/2}e^{α}, and is clearly W -invariant, hence is an element of A^{W}. In this case we define the scalar product of two elements f, g ∈ A to be

(3.4)  ⟨f, g⟩ = |W |^{−1}[fḡ△]_{1},

i.e., the constant term of the Laurent polynomial fḡ△, divided by the order of W.

For arbitrary values of the parameters k_{α} we proceed as follows. Let Q^{∨} be the root lattice of the dual root system R^{∨}, and let T = V/Q^{∨}. Then each e^{λ}, λ ∈ P, may be regarded as a character of the torus T by the rule e^{λ}(x^{∘}) = e^{2πi⟨λ,x⟩}, where x^{∘} ∈ T is the image of x ∈ V. By linearity, this enables us to regard each element of A as a continuous function on T.

Consider now the product

(t_{α}t_{2α}^{1/2}e^{α}(x^{∘}); q_{α})_{∞} = ∏_{r=0}^{∞} (1 − q_{α}^{k_{α}+k_{2α}+r}e^{α}(x^{∘})) ,

where α ∈ R and x^{∘} ∈ T. This product converges uniformly on T to a continuous function (since 0 ≤ q_{α} < 1) which does not vanish on T provided that k_{α} + k_{2α} ∉ −ℕ. Likewise the product (t_{2α}^{1/2}e^{α}; q_{α})_{∞} represents a continuous function on T, and therefore △ defined by (3.3) is a continuous function on T provided that

k_{α} + k_{2α} ∉ −ℕ

for all α ∈ R (where k_{2α} = 0 if 2α ∉ R). Hence △ may be expanded as a convergent Fourier series on the torus T, say

(3.5)  △ = ∑_{λ∈P} a_{λ}e^{λ},

where

a_{λ} = ∫_{T} e^{−λ}△,

the integration being with respect to normalized Haar measure on T.

We now define the scalar product of f, g ∈ A to be

(3.6)  ⟨f, g⟩ = ⟨f, g⟩_{q,t} = |W |^{−1} ∫_{T} fḡ△.

When the k_{α} are non-negative integers, this definition agrees with the previous one (3.4), since ∫_{T} e^{λ} = δ_{0λ} for λ ∈ L.

ORTHOGONAL POLYNOMIALS ASSOCIATED WITH ROOT SYSTEMS 9

Let

(3.7)  △^{+} = ∏_{α∈R^{+}}(t_{2α}^{1/2}e^{α}; q_{α})_{k_{α}},

so that △ = △^{+} · \overline{△^{+}}. From (3.6) it follows that

⟨f, g⟩ = |W |^{−1} ∫_{T} (f△^{+})(\overline{g△^{+}})

and hence that the scalar product is symmetric and positive definite.

We shall next derive another expression for the scalar product (3.6) restricted to A^{W}. For each w ∈ W let

R(w) = R^{+} ∩ −wR^{+}, t_{w} = ∏_{α∈R(w)} t_{α}, (3.8)

W (t) = ∑_{w∈W} t_{w}. (3.9)

Also let

Π = ∏_{α∈R^{+}} \frac{1 − t_{α}t_{2α}^{1/2}e^{−α}}{1 − t_{2α}^{1/2}e^{−α}}

and

(3.10)  △′ = △Π = ∏_{α∈R^{+}}(t_{2α}^{1/2}e^{α}; q_{α})_{k_{α}} (t_{2α}^{1/2}q_{α}e^{−α}; q_{α})_{k_{α}}.

From [8] we have the identity

∑_{w∈W} wΠ = W (t)

so that

(3.11)  W (t)△ = ∑_{w∈W} w△′.

Now let f, g ∈ A^{W}. Then we have

W (t)⟨f, g⟩ = \frac{W (t)}{|W |} ∫_{T} fḡ△

= |W |^{−1} ∑_{w∈W} ∫_{T} fḡ · w△′

= ∫_{T} fḡ△′ since f and g are W -invariant. Hence

(3.12)  ⟨f, g⟩ = W (t)^{−1} ∫_{T} fḡ△′

for f, g ∈ A^{W}.

Remark. If we choose to regard the parameters q and t_{α} as indeterminates over ℤ rather than as real numbers, we can expand △′ as a formal Laurent series. For this purpose let φ = ∑ m_{i}α_{i} be the highest root of R and let

x_{0} = qe^{−φ}, x_{i} = e^{α_{i}} (1 ≤ i ≤ n).

Then q = x_{0}e^{φ} = x_{0}x_{1}^{m_{1}} ⋯ x_{n}^{m_{n}} is a monomial in the x’s, and it follows that each of the products q_{α}^{i}e^{α}, q_{α}^{i+1}e^{−α}, where α ∈ R^{+} and i ≥ 0, is also a monomial in the x’s, since φ ≥ α for each root α ∈ R^{+}. Moreover the total degrees of these monomials tend to ∞ as i → ∞, and therefore △′ can be expanded as a formal power series in x_{0}, x_{1}, …, x_{n}, say

△′ = ∑_{r} b_{r}(t)x^{r},

where the sum is over all r = (r_{0}, …, r_{n}) ∈ ℕ^{n+1}, and x^{r} = x_{0}^{r_{0}} ⋯ x_{n}^{r_{n}}, and the coefficients b_{r}(t) lie in the ring ℤ[t] of polynomials in the t_{α} and t_{2α}^{1/2} with integer coefficients.

In terms of the original variables we have

x^{r} = q^{r_{0}} exp (∑_{i=1}^{n} r_{i}α_{i} − r_{0}φ) ,

and therefore for each λ ∈ Q the coefficient of e^{λ} in △′ is

(3.13)  a′_{λ}(q, t) = ∑_{r_{0}} q^{r_{0}}b_{r}(t),

where the vector r = (r_{0}, r_{1}, …, r_{n}) is determined from λ and r_{0} by the equation

(3.14)  ∑_{i=1}^{n} r_{i}α_{i} = λ + r_{0}φ,

and the sum in (3.13) is over all integers r_{0} ≥ 0 such that the r_{i} determined by (3.14) are all ≥ 0. Thus we have

△′ = ∑_{λ∈Q} a′_{λ}(q, t)e^{λ}

a formal Laurent series with coefficients in ℤ[t][[q]].

The identity (3.10) now gives

△ = W (t)^{−1} ∑_{w∈W} w△′

= W (t)^{−1} ∑_{w,λ} a′_{λ}(q, t)e^{wλ}

so that

(3.15)  △ = ∑_{λ∈Q} a_{λ}(q, t)e^{λ},

where

a_{λ}(q, t) = W (t)^{−1} ∑_{w∈W} a′_{wλ}(q, t).

The expression (3.15) is the expansion of △ as a W -invariant formal Laurent series, with coefficients in the ring of formal power series ℚ(t)[[q]], where ℚ(t) is the field of fractions of the ring ℤ[t]. This expansion is of course the same thing as the Fourier series (3.5).

If f, g ∈ A, the constant term in fḡ△ is now well-defined, being a finite linear combination of the coefficients a_{λ}(q, t), and we have ⟨f, g⟩ = |W |^{−1}[fḡ△]_{1} as in (3.4).

§4

We can now state the main result of this paper.

Theorem (4.1). For each irreducible admissible pair (R, S) of root systems there exists a unique basis (P_{λ})_{λ∈P^{++}} of A^{W} such that

(i) P_{λ} = m_{λ} +_{∑_{μ<λ} ; μ∈P^{++}}u_{λμ}(q, t)m_{μ}

with coefficients a_{λμ}(q, t) ∈ ℚ(q, t); (ii) ⟨P_{λ}, P_{μ}⟩ = 0 if λ ≠ μ.

It is clear that the P_{λ}, if they exist, are unique. If the partial order (2.1) on P^{++} were a total order, the existence of the P_{λ} would follow directly from the Gram-Schmidt orthogonalization process. However, the partial order (2.1) is not a total order (unless rank R = 1), and we should therefore have to choose a compatible total order on P^{++} before applying Gram-Schmidt. The content of (4.1) is that however we extend the partial order to a total order, we end up with the same basis of A^{W}.

Theorem (4.1) will be a consequence of the following proposition:

Proposition (4.2). For each irreducible admissible pair (R, S) there exists a linear operator E : A^{W} → A^{W} with the following three properties:

(i) E is self adjoint, i.e., ⟨Ef, g⟩ = ⟨f, Eg⟩ for all f, g ∈ A^{W}.

(ii) We have

Em_{λ} = _{∑_{μ≤λ} ; μ∈P^{++}}c_{λμ}m_{μ}

for each λ ∈ P^{++}, with coefficients c_{λμ} ∈ q^{a(λ)}ℤ[q, t], where a : P → ℚ is a homomorphism such that a(Q) ⊂ ℤ.

(iii) If λ ≠ μ, then c_{λλ} ≠ c_{μμ}; i.e., the eigenvalues of E are distinct.

Granted the existence of E with these properties, let

E_{λ} = _{∏_{μ<λ} ; μ∈P^{++}} \frac{E − c_{μμ}}{c_{λλ} − c_{μμ}}

for λ ∈ P^{++}. Then the elements P_{λ} = E_{λ}m_{λ} of A^{W} satisfy the conditions of (4.1). Indeed, it is clear from (4.2)(ii) that the P_{λ} satisfy (4.1)(i). (The fractional exponents a(λ) cause no trouble, because a(λ) − a(μ) ∈ ℤ if λ > μ.) On the other hand, let M_{λ} be the subspace of A^{W} spanned by the m_{μ} such that μ ≤ λ; then M_{λ} is finite-dimensional and stable under E, and the minimal polynomial of E restricted to M_{λ} is ∏_{μ≤λ}(X − c_{μμ}), since the c_{μμ} are all distinct. Hence (E − c_{λλ})E_{λ} = 0 on M_{λ}, and therefore

EP_{λ} = EE_{λ}m_{λ} = c_{λλ}E_{λ}m_{λ} = c_{λλ}P_{λ}.

If now λ ≠ μ we have

c_{λλ}⟨P_{λ}, P_{μ}⟩ = ⟨EP_{λ}, P_{μ}⟩ = ⟨P_{λ}, EP_{μ}⟩
= c_{μμ}⟨P_{λ}, P_{μ}⟩

by the self-adjointness of E, and hence ⟨P_{λ}, P_{μ}⟩ = 0 by (4.2)(iii).

In the next two sections we shall construct for each irreducible admissible pair (R, S) an operator E satisfying the conditions of (4.2). Our first construction, in §5, works whenever the root system S^{∨} has a minuscule fundamental weight (equivalent conditions are that P ≠ Q, or that R is not of type E_{8}, F_{4} or G_{2}). In §6 we shall give another construction which workes in these excluded cases.

§5

In this section we shall assume that S^{∨} possesses a minuscule fundamental weight, i.e., that there exists a vector π ∈ V such that ⟨π, α_{∗}⟩ takes just two values 0 and 1 as α runs through R^{+}. We have then (2.7)

T_{π}e^{α} = q^{⟨π,α⟩}e^{α} = q_{α}^{⟨π,α_{∗}⟩}e^{α}

so that

(5.1)  T_{π}e^{α} = \{ \begin{matrix}q_{α}e^{α} & if ⟨π, α_{∗}⟩ = 1, \\ e^{α} & if ⟨π, α_{∗}⟩ = 0. \end{matrix}

Now let

Φ_{π} = (T_{π}△^{+})/△^{+}

= _{∏_{α∈R^{+}} ; ⟨π,α_{∗}⟩=1} \frac{1 − t_{α}t_{2α}^{1/2}e^{α}}{1 − t_{2α}^{1/2}e^{α}} (5.2)

by (5.1) and the definition (3.7) of △^{+}.

We define an operator E_{π} on A as follows:

(5.3)  E_{π}f = ∑_{w∈W} w(Φ_{π} · T_{π}f).

Let us first show that E_{π} is self-adjoint (on the assumption that it maps A into A, which we shall justify shortly). Since

E_{π}f = ∑_{w∈W} \frac{w(T_{π}(△^{+}f))}{w△^{+}},

and since △ = w△ = w△^{+} · \overline{w△^{+}} for each w ∈ W, we have

⟨E_{π}f, g⟩ = |W |^{−1} ∑_{w∈W} [w(T_{π}(△^{+}f)) · \overline{w(△^{+}g)}]_{1}
= [T_{π}(△^{+}f) · \overline{△^{+}g}]_{1}

by (2.6), and by (2.10) this expression is symmetrical in f and g. Hence

(5.4)  ⟨E_{π}f, g⟩ = ⟨E_{π}g, f⟩ = ⟨f, E_{π}g⟩.

To show that E_{π} maps A into A, we need to express Φ_{π} in a more convenient form. If α ∈ R^{+} and \frac{1}{2}α ∈ R^{+}, the corresponding factors in the product (5.2) combine to give

\frac{(1 − t_{α}e^{α}) (1 − t_{α/2}t_{α}^{1/2}e^{α/2})}{(1 − e^{α}) (1 − t_{α}^{1/2}e^{α/2})} = \frac{(1 + t_{α}^{1/2}e^{α/2}) (1 − t_{α/2}t_{α}^{1/2}e^{α/2})}{1 − e^{α}}

= \frac{1 + (1 − t_{α/2})t_{α}^{1/2}e^{α/2} − t_{α/2}t_{α}e^{α}}{1 − e^{α}} .

If \frac{1}{2}α ∉ R^{+} (and 2α ∉ R^{+}), this is still correct, since then t_{α/2} = 1. It follows that

(5.5)  Φ_{π} = ∏_{α∈R_{2}^{+}} \frac{1 + (1 − t_{α/2}^{⟨π,α_{∗}⟩}) t_{α}^{⟨π,α_{∗}⟩/2}e^{α/2} − (t_{α/2}t_{α})^{⟨π,α_{∗}⟩}e^{α}}{1 − e^{α}} ,

where as before R_{2}^{+} = {α ∈ R^{+} : 2α ∉ R}.

Since t_{α}^{⟨π,α_{∗}⟩} = q^{k_{α}⟨π,α⟩}, we have

(5.6)  ∏_{α∈R^{+}} t_{α}^{⟨π,α_{∗}⟩} = q^{2⟨π,ρ_{k}⟩},

where

(5.7)  ρ_{k} = \frac{1}{2} ∑_{α∈R^{+}} k_{α}α.

Hence the product (5.5) for Φ_{π} can be rewritten in the form

Φ_{π} = δ^{−1}e^{ρ}q^{2⟨π,ρ_{k}⟩}Ψ,

where δ and ρ are as in (2.2) and (2.3), and Ψ is the product

Ψ = ∏_{α∈R_{2}^{+}} (1 + (t_{α/2}^{−⟨π,α_{∗}⟩} − 1) t_{α}^{−⟨π,α_{∗}⟩/2}e^{−α/2} − (t_{α/2}t_{α})^{−⟨π,α∗⟩}e^{−α}) .

If we multiply out this product, we shall obtain

(5.8)  Φ_{π} = δ^{−1}q^{2⟨π,ρ_{k}⟩} ∑_{X} φ_{X}(t)e^{ρ−σ(X)},

summed over all subsets X of R^{+} such that

(5.9)  α ∈ X ⇒ 2α ∉ X,

with the following notation:

σ(x) = ∑_{α∈X} α,
(5.10)  φ_{X}(t) = ∏_{α∈X} φ_{α}(t),
(5.11)  φ_{α}(t) = \{ \begin{matrix}−(t_{α/2}t_{α})^{−⟨π,α_{∗}⟩} & if 2α ∉ R, \\ (t_{α}^{−⟨π,α_{∗}⟩} − 1) t_{2α}^{−⟨π,α_{∗}⟩/2} & if 2α ∈ R. \end{matrix}

We can now calculate E_{π}e^{μ}, where μ ∈ P. Since w(T_{π}e^{μ}) = q^{⟨π,μ⟩}e^{wμ}, and since wδ = ε(w)δ for each w ∈ W, where ε(w) = det(w) = ±1, we shall obtain from (5.8)

E_{π}e^{μ} = δ^{−1}q^{⟨π,2ρ_{k}+μ⟩} ∑_{X} φ_{X}(t) ∑_{w∈W} ε(w)e^{w(μ+ρ−σ(X))}

= q^{⟨π,2ρ_{k}+μ⟩} ∑_{X} φ_{X}(t)χ_{μ−σ(X)}

from which it follows that E_{π} maps A into A^{W}.

Now let λ ∈ P^{++}. Then we have

E_{π}m_{λ} = ∑_{μ∈W λ} E_{π}e^{μ}

= ∑_{X} φ_{X}(t) ∑_{μ∈W λ} q^{⟨π,2ρ_{k}+μ⟩}χ_{μ−σ(X)}. (5.12)

In this sum, either χ_{μ−σ(X)} = 0 or else there exists w ∈ W and ν ∈ P^{++} such that

(5.13)  ν + ρ = w(μ + ρ − σ(X))

in which case χ_{μ−σ(X)} = ε(w)χ_{ν}. But ρ − σ(X) is of the form

ρ − σ(X) = \frac{1}{2} ∑_{α∈R_{2}^{+}} ε_{α}α,

where each coefficient ε_{α} is ±1 or 0, hence w(ρ − σ(X)) is of the same form, and therefore

(5.14)  w(ρ − σ(X)) = ρ − σ(Y)

for same subset Y of R^{+} such that α ∈ Y ⇒ 2α ∉ Y. From (5.13) and (5.14) it follows that

(5.15)  ν = wμ − σ(Y) ≤ wμ ≤ λ

and hence that E_{π}m_{λ} is a linear combination of the χ_{ν} such that ν ∈ P^{++} and ν ≤ λ. Hence we have

E_{π}m_{λ} = _{∑_{ν≤λ} ; ν∈P^{++}}b_{λν}χ_{ν},

where from (5.12) the coefficient b_{λν} is given by

b_{λν} = ∑ ε(w)q^{⟨π,2ρ_{k}+μ⟩}φ_{X}(t)

summed over triples (X, μ, w) where X ⊂ R^{+}, μ ∈ W λ and w ∈ W satisfy (5.9) and (5.13). From (5.6) and the definition (5.10) of φ_{X}(t) it follows that q^{⟨π,2ρ_{k}⟩}φ_{X}(t) ∈ ℤ[t], the ring of polynomials over ℤ generated by the t_{α} and t_{2α}^{1/2}. As to the scalar product ⟨π, μ⟩, we have

⟨π, μ⟩ = ⟨π, w_{0}λ⟩ + ⟨π, θ⟩

where w_{0} is the longest element of W and θ = μ − w_{0}λ ∈ Q^{+}. Now ⟨π, α_{∗}⟩ = 0 or 1 for each α ∈ R^{+}, and hence ⟨π, α⟩ = 0 or u_{α} for α ∈ R^{+}. Hence ⟨π, θ⟩ is a non-negative integer and therefore b_{λν} ∈ q^{⟨π,w_{0}λ⟩}ℤ[q, t]. The exponent ⟨π, w_{0}λ⟩ need not be an integer, but it is a rational number.

From this it follows that

(5.16)  E_{π}m_{λ} = _{∑_{ν≤λ} ; ν∈P^{++}}c_{λν}(π)m_{ν}

with coefficients c_{λν}(π) ∈ q^{⟨π,w_{0}λ⟩}ℤ[q, t].

We must now calculate the leading coefficient c_{λλ}(π) in (5.16). From (5.15) it follows that ν = λ if and only if Y is empty and wμ = λ, that is to say if and only if μ = w^{−1}λ and w(ρ − σ(X)) = ρ, or equivalently σ(X) = ρ − w^{−1}ρ. But this implies [8] that X = R_{2}(w) = R_{2}^{+} ∩ −wR_{2}^{+}. Hence the coefficient of m_{λ} in (5.16) is

c_{λλ}(π) = ∑_{w∈W} ε(w)φ_{R_{2}(w)}q^{⟨π,2ρ_{k}+w^{−1}λ⟩}.

Since ε(w) = (−1)^{|R_{2}(w)|}, we obtain from (5.10) and (5.11)

ε(w)φ_{R_{2}(w)}(t) = ∏_{α∈R_{2}(w)}(t_{α/2}t_{α})^{−⟨π,α_{∗}⟩}

= ∏_{α∈R(w)} t_{α}^{−⟨π,α_{∗}⟩}

= q^{⟨π,w^{−1}ρ_{k}−ρ_{k}⟩}

since t_{α}^{−⟨π,α_{∗}⟩} = q^{−⟨π,k_{α}α⟩}. Hence

(5.17)  c_{λλ}(π) = q^{⟨π,ρ_{k}⟩} ∑_{w∈W} q^{⟨wπ,λ+ρ_{k}⟩}
= q^{⟨π,ρ_{k}⟩}m̃_{π}(λ + ρ_{k})

where m̃_{π} = ∑_{w∈W} e^{wπ}.

It remains to examine whether the eigenvalues c_{λλ}(π) of E_{π} are all distinct as λ runs through P^{++}, for a suitable choice of the minuscule weight π. It will appear that this is so in all cases except D_{n}, n ≥ 4 (and of course excepting E_{8}, F_{4} and G_{2}, where there is no minuscule weight).

Let

p_{r}(x) = ∑_{w∈W}⟨x, wπ⟩^{r} (x ∈ V).

The p_{r}, r ≥ 1, are W -invariant polynomial functions on V.

(5.18). Suppose that S is not of type D_{n} (n ≥ 4), and that if S is of type A_{n} the minuscule weight π is the fundamental weight corresponding to an end node of the Dynkin diagram. Then the p_{r} generate the ℝ-algebra of W -invariant polynomial functions on V, and hence separate the W -orbits in V.

This is easily verified for S of type A, B or C. For E_{6} and E_{7} see [12].

Assume now that the hypotheses of (5.18) are satisfied, and that λ, μ ∈ P^{++} are such that c_{λλ}(π) = c_{μμ}(π), i.e., that

∑_{w∈W} q^{⟨λ+ρ_{k},wπ⟩} = ∑_{w∈W} q^{⟨μ+ρ_{k},wπ⟩}.

By operating on both sides with (q∂/∂q)^{r} and then setting q = 1 and t_{α} = 1 for each α ∈ R, we obtain p_{r}(λ) = p_{r}(μ) for all r ≥ 1. Hence by (5.18) λ and μ are in the same W -orbit, and therefore λ = μ. It follows that the eigenvalues c_{λλ}(π) of E_{π} are all distinct.

There remains the case where S(= R) is of type D_{n}. Let ε, …, ε_{n} be an orthonormal basis of V; we may then assume that R^{+} consists of the vectors ε_{i} ± ε_{j} with i < j. Then P^{++} consists of the vectors λ = ∑ λ_{i}ε_{i} for which the λ_{i} are all integers or all half-integers, and λ_{1} ≥ ⋯ ≥ λ_{n−1} ≥ |λ_{n}|. The fundamental weights

π_{1} = \frac{1}{2}(ε_{1} + ⋯ + ε_{n}), π_{2} = π_{1} − ε_{n}

are both minuscule, and the W -orbit of π_{1} (respectively π_{2}) consists of all sums \frac{1}{2} ∑ ±ε_{i} containing an even (respectively odd) number of minus signs. Hence the formula (5.17) gives

c_{λλ}(π_{1}) ± c_{λλ}(π_{2}) = n! ∏_{i=1}^{n} (q^{λ_{i}}t^{n−i} ± q^{−λ_{i}})

where t = t_{α}, α ∈ R.

Choose an integer N > \frac{1}{2}n(n − 1). Then the eigenvalues of the operator

(5.19)  E = \frac{1}{n!} ((t^{N} + 1) E_{π_{1}} + (t^{N} − 1) E_{π_{2}})

are from above

c_{λ} = t^{N} ∏_{i=1}^{n}(q^{λ_{i}}t^{n−i} + q^{−λ_{i}}) + ∏_{i=1}^{n}(q^{λ_{i}}t^{n−i} − q^{−λ_{i}}).

Now suppose that λ, μ ∈ P^{++} are such that c_{λ} = c_{μ}. From our choice of N it follows that

(5.20)  ∏_{i=1}^{n}(q^{λ_{i}}t^{n−i} + q^{−λ_{i}}) = ∏_{i=1}^{n}(q^{μ_{i}}t^{n−i} + q^{−μ_{i}}),

(5.21)  ∏_{i=1}^{n}(q^{λ_{i}}t^{n−i} − q^{−λ_{i}}) = ∏_{i=1}^{n}(q^{μ_{i}}t^{n−i} − q^{−μ_{i}}).

From (5.20) we conclude that λ_{i} = μ_{i} (1 ≤ i ≤ n − 1) and λ_{n} = ±μ_{n}. If λ_{n} ≠ 0, then (5.21) shows that λ_{n} = μ_{n}, and hence λ = μ. So the eigenvalues c_{λ} of E are all distinct.

To recapitulate, let E : A^{W} → A^{W} be the operator E_{π} defined by (5.3) when R is not of type D_{n}, and by (5.19) when R is of type D_{n}. By (5.4), (5.16) and the discussion above, it follows the E satisfies the three conditions of (4.2).

§6

In the cases where there is no minuscule weight available, another construction is needed. We shall assume in this section that R is reduced, so that either S = R or S = R^{∨}, from the classification in §1.

Let φ ∈ R^{+} be such that φ_{∗} is the highest root of S, and let π = (φ_{∗})^{∨} = u_{φ}φ^{∨}. For each α ∈ R^{+} the Cauchy-Schwarz inequality gives

0 ≤ ⟨π, α_{∗}⟩ = \frac{2⟨φ_{∗}, α_{∗}⟩}{|φ_{∗}|^{2}} ≤ \frac{2|α_{∗}|}{|φ_{∗}|} ≤ 2

with equality if and only if α_{∗} = φ_{∗}. Since ⟨π, α_{∗}⟩ is an integer, it follows that for each α ∈ R^{+}

(6.1)  ⟨π, α_{∗}⟩ = \{ \begin{matrix}0 or 1 & if α ≠ φ, \\ 2 & if α = φ. \end{matrix}

Thus π just fails to be a minuscule weight.

Remark. In fact φ_{∗} = φ, so that u_{φ} = 1 and q_{φ} = q. This is clear if S = R, whereas if S = R^{∨} (and R has two root-lengths) φ is the highest short root of R, so that u_{φ} = 1 in this case also. Hence π = φ^{∨}.

ORTHOGONAL POLYNOMIALS ASSOCIATED WITH ROOT SYSTEMS 19

Let

Φ_{π} = (T_{π}△^{+})/△^{+}

as in §5, and define an operator F_{π} on A^{W} as follows:

F_{π}f = ∑_{w∈W} w(Φ_{π} · U_{π}f)

where U_{π} = T_{π} − 1.

Let us first show that F_{π} is self-adjoint. If f, g ∈ A^{W} we have

⟨F_{π}f, g⟩ = |W |^{−1} ∑_{w∈W} [w((T_{π}△^{+})(U_{π}f)) · \overline{w(△^{+}g)}]_{1}
= [(T_{π}△^{+})(T_{π}f − f)\overline{△^{+}g}]_{1}
= [T_{π}(△^{+}f) · \overline{△^{+}g}]_{1} − [(T_{π}△^{+})\overline{△^{+}} · fḡ]_{1}
= A(f, g) − B(f, g),

say. We have A(f, g) = A(g, f) by (2.10). As to B(f, g), let G = (T_{π}△^{+})\overline{△^{+}} and let w_{0} be the longest element of the Weyl group W. Then w_{0}π = −π, and w_{0}△^{+} = \overline{△^{+}}, so that

w_{0}G = (T_{−π}\overline{△^{+}})△^{+} = \overline{T_{π}△^{+}}△^{+} = \overline{G}

and therefore

B(f, g) = [Gfḡ]_{1} = [(w_{0}G)fḡ]_{1}
= [\overline{G}fḡ]_{1} = [Gf̄g]_{1} = B(g, f).

It follows that

(6.2)  ⟨F_{π}f, g⟩ = ⟨F_{π}g, f⟩ = ⟨f, F_{π}g⟩

and hence that F_{π} is self-adjoint.

From (6.1) we have, since q_{φ} = q,

Φ_{π} = \frac{1 − t_{φ}e^{φ}}{1 − e^{φ}} · \frac{1 − qt_{φ}e^{φ}}{1 − qe^{φ}}_{∏_{α∈R^{+}} ; α≠φ} \frac{1 − t_{α}^{⟨π,α_{∗}⟩}e^{α}}{1 − e^{α}}

= q^{2⟨π,ρ_{k}⟩} \frac{(1 − t_{φ}^{−1}e^{−φ})(1 − q^{−1}t_{φ}^{−1}e^{−φ})}{(1 − e^{−φ})(1 − q^{−1}e^{−φ})}_{∏_{α∈R^{+}} ; α≠φ} \frac{1 − t_{α}^{−⟨π,α_{∗}⟩}e^{−α}}{1 − e^{−α}}

= δ^{−1}e^{ρ}q^{2⟨π,ρ_{k}⟩} \frac{(1 − t_{φ}^{−1}e^{−φ})(1 − q^{−1}t_{φ}^{−1}e^{−φ})}{1 − q^{−1}e^{−φ}}_{∏_{α∈R^{+}} ; α≠φ} (1 − t_{α}^{−⟨π,α_{∗}⟩}e^{−α})

and therefore

(6.3)  Φ_{π} = δ^{−1}q^{2⟨π,ρ_{k}⟩} \frac{(1 − t_{φ}^{−1}e^{−φ})(1 − q^{−1}t_{φ}^{−1}e^{−φ})}{1 − q^{−1}e^{−φ}} ∑_{X} ψ_{X}(t)e^{ρ−σ(X)}

summed over all subsets X of R^{+} such that φ ∉ X, where

σ(X) = ∑_{α∈X} α,

and

(6.4)  ψ_{X}(t) = (−1)^{|X|} ∏_{α∈X} t_{α}^{−⟨π,α_{∗}⟩}.

Now let λ ∈ P^{++} and μ ∈ W λ. Since T_{π}e^{μ} = q^{⟨μ,π⟩}e^{μ} = q^{⟨μ,φ^{∨}⟩}e^{μ}, we have

(6.5)  U_{π}m_{λ} = ∑_{μ∈W λ} (q^{⟨μ,φ^{∨}⟩} − 1) e^{μ}.

Any μ ∈ W λ such that ⟨μ, φ^{∨}⟩ = 0 will contribute nothing to this sum. The remaining elements of the orbit W λ fall into pairs {μ, w_{φ}μ} where ⟨μ, φ^{∨}⟩ > 0 and w_{φ} is the reflection associated with φ. Hence we may rewrite (6.5) in the form

U_{π}m_{λ} = _{∑_{μ∈W λ} ; ⟨μ,φ^{∨}⟩>0} (q^{⟨μ,φ^{∨}⟩} − 1) e^{μ} (1 − (qe^{φ})^{−⟨μ,φ^{∨}⟩})

from which it follows that

(6.6)  \frac{U_{π}m_{λ}}{1 − q^{−1}e^{−φ}} = _{∑_{μ∈W λ} ; ⟨μ,φ^{∨}⟩>0} (q^{⟨μ,φ^{∨}⟩} − 1) ∑_{j=0}^{⟨μ,φ^{∨}⟩−1} q^{−j}e^{μ−jφ}.

From (6.3) and (6.6) we obtain

(6.7) Φ_{π} · U_{π}m_{λ} = δ^{−1}q^{2⟨π,ρ_{k}⟩} ∑_{X,μ} ψ_{X}(t)e^{ρ−σ(X)}(1 − t_{φ}^{−1}e^{−φ})

× (1 − q^{−1}t_{φ}^{−1}e^{−φ}) (q^{⟨μ,φ^{∨}⟩} − 1) ∏_{j=0}^{⟨μ,φ^{∨}⟩−1} q^{−j}e^{μ−jφ}.

This is a sum of terms of the form aδ^{−1}e^{η}, η ∈ P. Since ∑_{w∈W} w(δ^{−1}e^{η}) ∈ A^{W}, it

follows from (6.7) that F_{π}m_{λ} ∈ A^{W}, and hence that F_{π} maps A^{W} into A^{W}.

Moreover, the terms aδ^{−1}e^{η} that occur in (6.7) are such that

η = ρ − σ(X) + μ − jφ

where 0 ≤ j ≤ ⟨μ, φ^{∨}⟩ + 1 and ⟨μ, φ^{∨}⟩ ≥ 1 and X ⊂ R^{+} − {φ}. If η is not regular (i.e., if W_{η} ≠ 1, where W_{η} is the subgroup of W that fixes η) it will contribute nothing to (6.7). If on the other hand η is regular, then we have wη = ξ + ρ for some ξ ∈ P^{++} and some w ∈ W, so that

(6.8)  ξ + ρ = w(ρ − σ(X)) + w(μ − jφ).

There are two cases to consider.

(i) Suppose that wφ = α ∈ R^{+}. Since ρ − σ(X) is of the form \frac{1}{2} ∑_{α∈R^{+}} ε_{α}α, where

each ε_{α} is ±1, it follows that w(ρ − σ(X)) is of the same form, hence that

w(ρ − σ(X)) = ρ − σ(Y)

for some subset Y of R^{+}. Hence

(6.9)  ξ = wμ − jα − σ(Y) ≤ wμ ≤ λ

and therefore each such term aδ^{−1}e^{η} in (6.7) contributes ε(w)aχ_{ξ}, where ξ ≤ λ, to F_{π}m_{λ}.

Moreover we have equality in (6.9) if and only if j = 0, the subset Y is empty, and wμ = λ, i.e., μ = w^{−1}λ and σ(X) = ρ − w^{−1}ρ, so that X = R(w) = R^{+} ∩ −wR^{+}. The coefficient a of δ^{−1}e^{η} = δ^{−1}e^{w^{−1}(λ+ρ)} in (6.7) is then

(6.10)  a = ψ_{R(w)}(t)q^{2⟨π,ρ_{k}⟩} (q^{⟨λ,wπ⟩} − 1)

(since q_{φ}^{⟨μ,φ^{∨}⟩} = q^{⟨w^{−1}λ,π⟩} = q^{⟨λ,wπ⟩}). From (6.4) we have

ψ_{R(w)}(t) = ε(w) ∏_{α∈R(w)} t_{α}^{−⟨π,α_{∗}⟩}

and since t_{α}^{−⟨π,α_{∗}⟩} = q^{−⟨π,k_{α}α⟩} it follows that

(6.11)  ψ_{R(w)}(t) = ε(w)q^{−⟨π,ρ_{k}−w^{−1}ρ_{k}⟩}.

From (6.10) and (6.11) the coefficient of δ^{−1}e^{w^{−1}(λ+ρ)} in (6.7) is therefore

(6.12)  a = ε(w)q^{⟨π,ρ_{k}⟩} (q^{⟨wπ,λ+ρ_{k}⟩} − q^{⟨wπ,ρ_{k}⟩})

and the corresponding contribution to F_{π}m_{λ} is

(6.13)  q^{⟨π,ρ_{k}⟩} (q^{⟨wπ,λ+ρ_{k}⟩} − q^{⟨wπ,ρ_{k}⟩}) χ_{λ}.

(ii) Suppose now that wφ = −α, where α ∈ R^{+}, and let ν = w_{φ}μ = μ − ⟨μ, φ^{∨}⟩φ. Then

μ − jφ = ν − φ + j′φ,

where j′ = ⟨μ, φ^{∨}⟩ + 1 − j, so that 0 ≤ j′ ≤ ⟨μ, φ^{∨}⟩ + 1. Hence (6.8) now takes the form

ξ + ρ = w(ρ − σ(X) − φ) + w(ν + j′φ).

Since φ ∉ X, we have w(ρ − σ(X) − φ) = ρ − σ(Y) for some Y ⊂ R^{+}, and therefore

(6.14)  ξ = wν − σ(Y) − j′α ≤ wν ≤ λ.

So again each term aδ^{−1}e^{η} in (6.7) contributes ε(w)aχ_{ξ}, where ξ ≤ λ, to F_{π}m_{λ}.

Moreover, we have equality in (6.14) if and only if j′ = 0, the subset Y is empty, and wν = λ, i.e., ν = w^{−1}λ and σ(X)+φ = ρ−w^{−1}ρ, so that X ∪{φ} = R(w) and μ = w_{φ}w^{−1}λ and j = ⟨μ, φ^{∨}⟩ + 1. Hence the coefficient of δ^{−1}e^{η} = δ^{−1}e^{w^{−1}(λ+ρ)} in Φ_{π} · U_{π}m_{λ} is now

a = ψ_{R(w)−{φ}}(t)q^{2⟨π,ρ_{k}⟩}t_{φ}^{−2} · q^{−⟨μ,φ^{∨}⟩} (q^{⟨μ,φ^{∨}⟩} − 1) .

Since −⟨μ, φ^{∨}⟩ = −⟨w_{φ}w^{−1}λ, φ^{∨}⟩ = ⟨w^{−1}λ, φ^{∨}⟩ = ⟨λ, wφ^{∨}⟩ we have

a = −t_{φ}^{−2}ψ_{R(w)−{φ}}(t)q^{2⟨π,ρ_{k}⟩} (q^{⟨λ,wπ⟩} − 1) .

Moreover, from (6.4),

−t_{φ}^{−2}ψ_{R(w)−{φ}}(t) = ε(w)t_{φ}^{−2}_{∏_{α∈R(w)} ; α≠φ}t_{α}^{−⟨π,α_{∗}⟩}

= ε(w) ∏_{α∈R(w)} t_{α}^{−⟨π,α_{∗}⟩}

= ε(w)q^{−⟨π,ρ_{k}−w^{−1}ρ_{k}⟩}

as in (6.11). So finally the coefficient of δ^{−1}e^{w^{−1}(λ+ρ)} in (6.7) is given by the same expression (6.12) as before, and the corresponding contribution to F_{π}m_{λ} is again given by (6.13).

To recapitulate, these calculations show that F_{π}m_{λ} is a linear combination of the Weyl characters χ_{ξ} such that ξ ∈ P^{++} and ξ ≤ λ, and an inspection of (6.7) shows that the coefficient of χ_{ξ} in F_{π}m_{λ} lies in q^{−⟨λ,φ^{∨}⟩}ℤ[q, t]. Moreover the coefficient of χ_{λ} is

(6.15) q^{⟨π,ρ_{k}⟩} ∑_{w∈W} (q^{⟨wπ,λ+ρ_{k}⟩} − q^{⟨wπ,ρ_{k}⟩}) = q^{⟨π,ρ_{k}⟩}(m̃_{π}(λ + ρ_{k}) − m̃_{π}(ρ_{k}))

where m̃_{π} = ∑_{w∈W} e^{wπ} = |W_{π}|m_{π}. Hence we have

(6.16)  F_{π}m_{λ} = _{∑_{μ≤λ} ; μ∈P^{++}}c_{λμ}(π)m_{μ}

with c_{λλ}(π) given by (6.15), and c_{λμ}(π) ∈ q^{−⟨λ,φ^{∨}⟩}ℤ[q, t].

To complete the proof of (4.2), it remains to establish that the c_{λλ}(π), λ ∈ P^{++}, are all distinct when R is of type E_{8}, F_{4}, or G_{2}. For this purpose we argue as in the last part of §5: if λ, μ ∈ P^{++} are such that c_{λλ}(π) = c_{μμ}(π), then by operating with (q∂/∂q)^{r} and then setting q = 1 we shall obtain

∑_{w∈W}⟨wπ, λ⟩^{r} = ∑_{w∈W}⟨wπ, μ⟩^{r}

for each r ≥ 1. But in each case π is either the highest root or the highest short root of R, and it is known [12] that when R is of type E_{8}, F_{4} or G_{2} the polynomial functions on V

p_{r}(x) = ∑_{w∈W}⟨wπ, x⟩^{r} (r ≥ 1)

generate the ℝ-algebra of W -invariant polynomial functions in V, and therefore separate the W -orbits in V. Hence λ and μ lie in the same W -orbit, and so λ = μ.

By (6.2), (6.16) and the above discussion, it follows that the linear operator F_{π} on A^{W} satisfies the three conditions of (4.2) when R is of type E_{8}, F_{4} or G_{2}. This completes the proof of (4.2) and hence of Theorem (4.1).

§7

Let us say that a linear operator L : A^{W} → A^{W} is triangular if

Lm_{λ} = α_{λ}m_{λ} + lower terms

for each λ ∈ P^{++}, that is to say if the matrix of L relative to the basis (m_{λ}) of A^{W} is triangular.

(7.1). If L is triangular and self-adjoint, the P_{λ} are eigenfunctions of L.

Proof. Since L is triangular, we have

LP_{λ} = _{∑_{μ≤λ} ; μ∈P^{++}}α_{λμ}P_{μ}

with coefficients a_{λμ} given by

a_{λμ}|P_{μ}|^{2} = ⟨LP_{λ}, P_{μ}⟩ = ⟨P_{λ}, LP_{μ}⟩

since L is self-adjoint. But

LP_{μ} = _{∑_{ν≤μ} ; ν∈P^{++}}a_{μν}P_{ν},

so that ⟨P_{λ}, LP_{μ}⟩ = 0 unless μ = λ. Hence a_{λμ} = 0 unless μ = λ, which proves (7.1).

From (7.1) it follows that all self-adjoint triangular linear operators on A^{W} are simultaneously diagonalized by the P_{λ}, and therefore commute with each other.

Consider in particular the case in which R is of type A_{n}. Then all the fundamental weights π_{i} (1 ≤ i ≤ n) are minuscule, and hence the construction of §5 furnishes n self-adjoint triangular linear operators E_{π_{i}} (1 ≤ i ≤ n) on A^{W}, which by the above remark commute with each other. Moreover the eigenvalues of E_{π_{i}} are (up to a scalar factor) m_{π_{i}}(λ + ρ_{k}). Since the m_{π_{i}} generate the algebra A^{W}, it follows that for each f ∈ A^{W} there is a unique linear operator L_{f} ∈ ℝ[E_{π_{1}}, …, E_{π_{n}}] such that

(7.2)  L_{f}P_{λ} = f(λ + ρ_{k})P_{λ}

for all λ ∈ P^{++}. Moreover, since each E_{π_{i}} is a linear combination of the translation operators T_{wπ_{i}}, w ∈ W, it follows that L_{f} is of the form

(7.3)  L_{f} = ∑_{μ∈P} a_{μ,f}(q, t)T_{μ}

with coefficients a_{μ,f}(q, t) which are rational functions of q and t.

For an arbitrary admissible pair (R, S) of root systems, we can of course define L_{f} for each f ∈ A^{W} by (7.2). But except in the case A_{n} just mentioned, I do not know whether L_{f} is expressible in the form (7.3).

§8

In the following sections we shall consider some particular cases.

(i) Suppose first that k_{α} = 0 for each α ∈ R, i.e., t_{α} = 1. Then △ = 1, so that the scalar product is now

⟨f, g⟩ = |W |^{−1}[fḡ]_{1}.

It follows that

(8.1)  P_{λ} = m_{λ}

for all λ ∈ P^{++}, and hence that

(8.2)  ⟨P_{λ}, P_{μ}⟩ = |W_{λ}|^{−1}δ_{λμ}

where W_{λ} is the subgroup of W that fixes λ.

(ii) Next, suppose that R is reduced and that k_{α} = 1 for each α ∈ R, so that t_{α} = q_{α}. Then

△ = ∏_{α∈R}(1 − e^{α}) = δδ̄

and the scalar product is now

⟨f, g⟩ = |W |^{−1}[fδ · \overline{gδ}]_{1}.

Hence in this case we have

(8.3)  P_{λ} = χ_{λ}

for all λ ∈ P^{++}, and

(8.4)  ⟨P_{λ}, P_{μ}⟩ = δ_{λμ}.
§9

In this section we shall consider the cases where R has rank 1, hence is of type A_{1} or BC_{1}.

(i) Suppose first that R is of type BC_{1}. In this case it terms out that the polynomials P_{λ} are a particular case of the orthogonal polynomials defined by Askey and Wilson [2]. They define

(9.1) p_{n}(x; a, b, c, d | q)

= a^{−n}(ab; q)_{n} (ac; q)_{n} (ad; q)_{n} ·_{4}φ_{3}[ \begin{matrix}q^{−n}, q^{n−1}abcd, ae^{iθ}, ae^{−iθ} \\ ab, ac, ad \end{matrix}; q, q]

where x = cos θ, the parameters a, b, c, d and q lie in the interval (−1, 1) of ℝ, and _{4}φ_{3} is the usual notation for a q-hypergeometric series. Thus p_{n} is a Laurent polynomial in e^{iθ} with leading term (abcdq^{n−1}; q)_{n}e^{niθ}, and in fact is (despite its appearance) symmetrical in all four parameters a, b, c, d. For our purposes it is more convenient to take e^{iθ} rather than cos θ as the argument, and we prefer to have the leading coefficient equal to 1; so we define

(9.2)  p̃_{n}(e^{iθ}; a, b, c, d | q) = (abcdq^{n−1}; q)_{n}^{−1} p_{n}(cos θ; a, b, c, d | q).

Then Theorem (2.2) of [2] takes the form

(9.3)  \frac{1}{2π} ∫_{0}^{π} p̃_{m} p̃_{n} |f(e^{iθ})|^{2} dθ = δ_{mn}h̃_{n}

where

(9.4)  f(u) = (u^{2}; q)_{∞}/(au; q)_{∞}(bu; q)_{∞}(cu; q)_{∞}(du; q)_{∞}

and

(9.5)  h̃_{n} = \frac{(Aq^{2n−1}; q)_{∞} (Aq^{2n}; q)_{∞}}{(Aq^{n−1}; q)_{∞} (q^{n+1}; q)_{∞}} Π (abq^{n}; q)_{∞}^{−1}

where A = abcd and Π means the product of the six terms such as (abq^{n}; q)_{∞}^{−1}.

Now let R be a root system of type BC_{1}, with R^{+} = {α, 2α} and S = {±α}. Then q_{α} = q and q_{2α} = q^{2}. We shall write k_{1}, k_{2}, t_{1}, t_{2} for k_{α}, k_{2α}, t_{α}, t_{2α} respectively, so that t_{1} = q^{k_{1}} and t_{2} = q^{2k_{2}}. We have P^{++} = ℕα.

Let

(9.6)  (a, b, c, d) = (q^{1/2}, −q^{1/2}, t_{1}t_{2}^{1/2}, −t_{2}^{1/2}).

With this choice of parameters, f(u) takes the form

f(u) = \frac{(t_{2}^{1/2}u; q)_{∞} (u^{2}; q^{2})_{∞}}{(t_{1}t_{2}^{1/2}u; q)_{∞} (t_{2}u^{2}; q^{2})_{∞}}

so that △ = f(e^{α})f(e^{−α}). It now follows from (9.1), (9.2) and (9.3) that for n ≥ 0

(9.7)  P_{nα} = p̃_{n}(e^{α}; q^{1/2}, −q^{1/2}, t_{1}t_{2}^{1/2}, −t_{2}^{1/2} | q)

and that |P_{nα}|^{2} is given by (9.5) when the parameters a, b, c, d are as in (9.6). After some reduction we find that

(9.8)  |P_{nα}|^{2} = \frac{(q^{2n}t_{1}t_{2}; q)_{k_{1}} (q^{2n}t_{1}^{2}t_{2}; q^{2})_{k_{2}}}{(q^{2n+1}t_{2}; q)_{k_{1}} (q^{2n+2}; q^{2})_{k_{2}}}

For later reference we shall calculate P_{nα}(ρ_{k}^{∗}), where ρ_{k}^{∗} = \frac{1}{2}(k_{1} + k_{2})α^{∨}, so that e^{α}(ρ_{k}^{∗}) = q^{k_{1}+k_{2}} = t_{1}t_{2}^{1/2}. The original formula (9.1) for p_{n} shows that when e^{iθ} is replaced by a, the series _{4}φ_{3} reduces to its first term, which is 1. By the symmetry of the parameters a, b, c, d it follows that when e^{iθ} is replaced by c in p_{n}, the result is

c^{−n}(ac; q)_{n} (bc; q)_{n} (cd; q)_{n}.

In view of (9.6) and (9.7), this observation enables us to calculate P_{nα}(ρ_{k}^{∗}); after some reduction, we obtain

(9.9)  P_{nα}(ρ_{k}^{∗}) = (t_{1}t_{2}^{1/2})^{−n} \frac{(q^{2n}t_{1}t_{2}; q)_{k_{1}} (q^{2n}t_{1}^{2}t_{2}; q^{2})_{k_{2}}}{(t_{1}t_{2}; q)_{k_{1}} (t_{1}^{2}t_{2}; q^{2})_{k_{2}}} .

(ii) Suppose now that R is of type A_{1}, with positive root α, so that P^{++} = \frac{1}{2}ℕα. In this case the P_{λ} are essentially the q-ultraspherical polynomials of Askey and Ismail [1]. If λ = \frac{1}{2}nα, where n ≥ 0, we have

(9.10)  P_{λ} = \frac{(q; q)_{n}}{(t; q)_{n}} φ_{n}(e^{α/2})

where t = t_{α} and

(9.11)  φ_{n}(x) = ∑_{i+j=n} \frac{(t; q)_{i} (t; q)_{j}}{(q; q)_{i} (q; q)_{j}} x^{i−j}

(so that φ_{n}(e^{iθ}) = C_{n}(cos θ; t | q) in the notation of [1]). The φ_{n} have the generating function

(9.12)  F (x, u) = ∑_{n≥0} φ_{n}(x)u^{n} = 1/(xu; q)_{k}(x^{−1}u; q)_{k}

(where k = k_{α}, so that t = q^{k}), as follows from (9.11) and the q-binomial theorem.

One way of establishing (9.10) is to verify that the φ_{n} defined by (9.11) are eigenfunctions of the operator E_{π} of §5, which in the present situation takes the form

(9.13)  (E_{π}f)(x) = \frac{tx − x^{−1}}{x − x^{−1}} f(q^{1/2}x) + \frac{x − tx^{−1}}{x − x^{−1}} f(q^{1/2}x^{−1})

for a Laurent polynomial f(x) ∈ ℝ[x, x^{−1}]. More precisely, we have to verify that

E_{π}φ_{n} = (tq^{n/2} + q^{−n/2})φ_{n}

or equivalently (9.12) that

(E_{π}F)(x, u) = tF (x, q^{1/2}u) + F (x, q^{−1/2}u).

But this is straightforward to verify from (9.12) and (9.13).

From [1] we have

(9.14)  |P_{nα/2}|^{2} = (q^{n}t; q)_{k}/(q^{n+1}; q)_{k}

and

(9.15)  P_{nα/2}(\frac{1}{2}kα^{∨}) = t^{−n/2}(q^{n}t; q)_{k}/(t; q)_{k};

this latter formula is equivalent to φ_{n}(t^{1/2}) = t^{−n/2}(t^{2}; q)_{n}/(q; q)_{n}, which follows from (9.12) since

F (t^{1/2}, ut^{1/2}) = 1/(u; q)_{k}(tu; q)_{k}

= 1/(u; q)_{2k}

= ∑_{n≥0} \frac{(t^{2}; q)_{n}}{(q; q)_{n}} u^{n}.

§10

We consider next the case where q = 0, the t_{α} being arbitrary. Then we have

△ = ∏_{α∈R} \frac{1 − t_{2α}^{1/2}e^{α}}{1 − t_{2α}^{1/2}t_{α}e^{α}} .

In this case there is an explicit formula for P_{λ}:

(10.1)  P_{λ} = W_{λ}(t)^{−1} ∑_{w∈W} w (e^{λ} ∏_{α∈R^{+}} \frac{1 − t_{α}t_{2α}^{1/2}e^{−α}}{1 − t_{2α}^{1/2}e^{−α}} )

where W_{λ} is the subgroup of W that fixes λ ∈ P^{++}, and

W_{λ}(t) = ∑_{w∈W_{λ}} t_{w}

with t_{w} as defined in (3.8).

To prove (10.1), we shall first show that P_{λ} = m_{λ} + lower terms, and then that ⟨P_{λ}, P_{μ}⟩ = 0 if λ ≠ μ.

Let

Φ_{λ} = e^{λ} ∏_{α∈R^{+}} \frac{1 − t_{α}t_{2α}^{1/2}e^{−α}}{1 − t_{2α}^{1/2}e^{−α}}

= δ^{−1}e^{λ+ρ} ∏_{α∈R_{2}^{+}} (1 + (1 − t_{α/2})t_{α}^{1/2}e^{−α/2} − t_{α/2}t_{α}e^{−α})

as in §5, where R_{2}^{+} = {α ∈ R^{+} : 2α ∉ R} and δ, ρ are as defined in (2.2), (2.3). On multiplying out this product we shall obtain

(10.2)  Φ_{λ} = δ^{−1} ∑_{X} φ_{X}(t)e^{λ+ρ−σ(X)}

summed over all subsets X of R^{+} such that α ∈ X ⇒ 2α ∉ X, where σ(X) = ∑_{α∈X} α

and

(10.3)  φ_{X}(t) = ∏_{α∈X} φ_{α}(t),

φ_{α}(t) = \{ \begin{matrix}−t_{α/2}t_{α} & if 2α ∉ R, \\ (1 − t_{α})t_{2α}^{1/2} & if 2α ∈ R. \end{matrix}

Let

Q_{λ} = ∑_{w∈W} wΦ_{λ}.

Then it follows from (10.2) that

(10.4)  Q_{λ} = ∑_{X} φ_{X}(t)χ_{λ−σ(X)}

summed over subsets X ⊂ R^{+} as above. If χ_{λ−σ(X)} ≠ 0, there exists w ∈ W and μ ∈ P^{++} such that

μ + ρ = w(λ + ρ − σ(X))

and we have χ_{λ−σ(X)} = ε(w)χ_{μ}. Now (5.14) w(ρ − σ(X)) = ρ − σ(Y) for some subset Y of R^{+} such that α ∈ Y ⇒ 2α ∉ Y. Hence μ = wλ − σ(Y) ≤ wλ ≤ λ, and so it follows from (10.4) that Q_{λ} is a linear combination of the χ_{μ} such that μ ∈ P^{++} and μ ≤ λ.

Moreover, we have μ = λ if and only if wλ = λ and σ(Y) = 0, that is to say if and only if w ∈ W_{λ} and σ(X) = ρ − w^{−1}ρ, which implies that X = R_{2}(w) and hence (10.3)

φ_{X}(t) = ∏_{α∈R_{2}(w)}(−t_{α/2}t_{α})

= ε(w) ∏_{α∈R(w)} t_{α} = ε(w)t_{w}.

Hence the coefficient of χ_{λ} in Q_{λ} is

∑_{w∈W_{λ}} t_{w} = W_{λ}(t)

and therefore P_{λ} as defined by (10.1) is of the form m_{λ} + lower terms.

It remains to prove that ⟨P_{λ}, P_{μ}⟩ = 0 if λ ≠ μ. We may assume that λ ≰ μ. We have

Q_{λ}△ = ∑_{w∈W} wΨ_{λ}

where

Ψ_{λ} = △Φ_{λ} = e^{λ}△^{+} = e^{λ} ∏_{α∈R^{+}} \frac{1 − t_{2α}^{1/2}e^{α}}{1 − t_{α}t_{2α}^{1/2}e^{α}}

= e^{λ} ∏_{α∈R^{+}} (1 + (t_{α} − 1) ∑_{r≥1} t_{α}^{r−1}t_{2α}^{r/2}e^{rα})

= ∑_{μ∈Q^{+}} a_{μ}e^{λ+μ}

say, with a_{0} = 1. Hence

Q_{λ}△ = ∑_{μ∈Q^{+}} a_{μ} ∑_{w∈W} e^{w(λ+μ)}.

If λ + μ = w_{1}π, where π ∈ P^{++} and w_{1} ∈ W, then we have π ≥ w_{1}π = λ + μ ≥ λ, with equality only if μ = 0. Hence

Q_{λ}△ = |W_{λ}|m_{λ} + higher terms;

and since λ ≰ μ this sum has no terms in common with

P_{μ} = m_{μ} + lower terms.

Hence ⟨Q_{λ}, P_{μ}⟩ = |W |^{−1}[P̄_{μ}Q_{λ}△]_{1} = 0. This completes the proof of (10.1). Moreover, these calulations show that

⟨Q_{λ}, P_{λ}⟩ = \frac{|W_{λ}|}{|W |}[m_{λ}\overline{m_{λ}}]_{1} = 1

so that

(10.5)  |P_{λ}|^{2} = W_{λ}(t)^{−1}.

The formula (10.1) is essentially the formula of [7], Theorem (4.1.2) for the zonal spherical function on a p-adic Lie group. More precisely, let G be a simplyconnected group of p-adic type, as defined in [7], and let K be a special maximal compact subgroup of G, such that the root system Σ_{1} of [7], (3.1) is the dual R^{∨} of R. The root structure of G attaches a positive integer q_{α^{∨}} to each root α^{∨} ∈ R^{∨}, and we take t_{α} = q_{α^{∨}}^{−1}. The double cosets of K in G are indexed by the elements of Q^{++} = P^{++} ∩ Q, and the zonal spherical functions ω_{S} on G relative to K are parametrized by the ℂ-algebra homomorphisms s : ℂ[Q]^{W} → ℂ.

For each λ ∈ Q^{++} let g_{λ} be a representative of the corresponding double coset of K in G. Then the formula for the zonal spherical function is

ω_{s}(g_{λ}^{−1}) = u_{λ}(t)s(P_{λ})

with P_{λ} as in (10.1) and

u_{λ}(t) = \frac{W_{λ}(t)}{W (t)} ∏_{α∈R^{+}} t_{α}^{⟨λ,α^{∨}⟩/2}.

Moreover, △ is essentially the Plancherel measure on the space of positive definite zonel spherical functions on G relative to K.

§11

In this section we shall consider the “limiting case” as q → 1, the parameters k_{α} remaining fixed. We shall assume that

(11.1)  k_{α} ≥ 0

for all α ∈ R.

Let

△_{k} = ∏_{α∈R}(1 − e^{α})^{k_{α}}

considered (as in §3) as a continuous function on the torus T = V/Q^{∨}. For f, g ∈ A we define

(11.2)  ⟨f, g⟩_{k} = |W |^{−1} ∫_{T} fḡ△_{k}

using △_{k} in place of △(q, t). As before, this scalar product on A is symmetric and positive definite.

Suppose that, in addition to (11.1), we have

(11.3)  k_{α} + 2k_{2α} ≥ 1

for all α ∈ R (so that k_{α} ≥ 1 if 2α ∉ R). Then

(11.4)  lim_{q→1} △(q, t) = △_{k}

uniformly on T.

This is a consequence of the following fact [6]: if r, s ∈ ℝ and z ∈ ℂ then

lim_{q→1} \frac{(q^{r}z; q)_{∞}}{(q^{s}z; q)_{∞}} = (1 − z)^{s−r}

uniformly on the disc |z| ≤ 1, provided that r ≤ s and r + s ≥ 1. If we take q = q_{α}, r = k_{2α}, s = k_{α} + k_{2α} we obtain

lim_{q→1}(t_{2α}^{1/2}e^{α}; q_{α})_{k_{α}} = (1 − e^{α})^{k_{α}}

uniformly on T, provided that (11.1) and (11.3) hold. Taking the product over all α ∈ R, we obtain (11.4).

Until further notice we shall assume (11.3) as well as (11.1).

Let

f(q) = ∑_{λ∈P} f_{λ}(q)e^{λ}

be an element of A depending on q ∈ (0, 1). If f_{λ}(q) → f_{λ} as q → 1 for each λ ∈ P, we shall write

lim_{q→1} f(q) = f

where f = ∑ f_{λ}e^{λ}.

Suppose also that

lim_{q→1} g(q) = g

in A. Then

(11.5)  lim_{q→1}⟨f(q), g(q)⟩_{q,t} = ⟨f, g⟩_{k}.

By linearity it is enough to prove this when f(q) = f_{λ}(q)e^{λ}, and g(q) = g_{μ}(q)e^{μ}. We have then

⟨f(q), g(q)⟩_{q,t} = |W |^{−1}f_{λ}(q)g_{μ}(q) ∫_{T} e^{λ−μ}△(q, t)

which by (11.4) tends to the limit

|W |^{−1}f_{λ}g_{μ} ∫_{T} e^{λ−μ}△_{k} = ⟨f, g⟩_{k}

as q → 1.

Consider now the behaviour of

(11.6)  P_{λ}(q, t) = _{∑_{μ≤λ} ; μ∈P^{++}}u_{λμ}(q, t)m_{μ}  (λ ∈ P^{++})

as q → 1. We claim that

(11.7)  lim_{q→1} P_{λ}(q, t) exists for each λ ∈ P^{++} :

in other words, that each of the coefficients u_{λμ}(q, t) tends to a finite limit as q → 1.

We shall prove this by induction on λ. When λ = 0 there is nothing to prove, since P_{0}(q, t) = 1; so assume that λ ≠ 0 and that

(11.8)  P_{μ}(k) = lim_{q→1} P_{μ}(q, t)

exists for all μ ∈ P^{++} such that μ < λ.

The equations (11.6) can be inverted to give say

(11.9)  m_{λ} = _{∑_{μ≤λ} ; μ∈P^{++}}v_{λμ}(q, t)P_{μ}(q, t)

with v_{λλ} = 1. The v’s are cofactors of the (unipotent) matrix formed by the u’s, hence are polynomials in the u’s, and conversely the u’s are polynomials in the v’s. From (11.9) and the orthogonality of the P ’s we have

v_{λμ}(q, t) = \frac{⟨m_{λ}, P_{μ}(q, t)⟩_{q,t}}{|P_{μ}(q, t)|_{q,t}^{2}} .

Hence, by (11.5) and (11.8), we have

lim_{q→1} v_{λμ}(q, t) = \frac{⟨m_{λ}, P_{μ}(k)⟩_{k}}{|P_{μ}(k)|_{k}^{2}}

whenever μ < λ. But, as we have just remarked, u_{λμ}(q, t) is a polynomial in the v’s with integer coefficients. Hence

u_{λμ}(k) = lim_{q→1} u_{λμ}(q, t)

exists for each μ < λ, and (11.7) is proved.

Since u_{λμ}(q, t) ∈ ℂ(q, t), hence is a rational function of q and the q^{k_{α}}, its limit as q → 1 may be computed by differentiating its numerator and denominator sufficiently often, and then setting q = 1. This shows that u_{λμ}(k) is rational function of the k_{α}.

We now define

(11.10)  P_{λ}(k) = ∑_{μ≤λ} u_{λμ}(k)m_{μ}

= lim_{q→1} P_{λ}(q, t).

By (11.5) we have

(11.11)  ⟨P_{λ}(k), P_{μ}(k)⟩_{k} = 0

if λ ≠ μ, and the properties (11.10) and (11.11) characterize the polynomials P_{λ}(k).

The existence of these polynomials has been established by Heckman and Opdam [4, 14] by other methods.

Define linear operators □ and D_{α}(α ∈ R) on A by

□e^{λ} = |λ|^{2}e^{λ}, D_{α}e^{λ} = ⟨λ, α⟩e^{λ}

and as in [14] let

(11.12)  L(k) = □ + \frac{1}{2} ∑_{α∈R} k_{α} \frac{1 + e^{−α}}{1 − e^{−α}} D_{α}.

An equivalent definition is

(11.13)  L(k)f = δ_{k}^{−1/2}□ (δ_{k}^{1/2}f) − (δ_{k}^{−1/2}□δ_{k}^{1/2}) f

where

δ_{k} = ∏_{α∈R^{+}}(e^{α/2} − e^{−α/2})^{2k_{α}}.

From (11.12), it is clear that wL(k)w^{−1} = L(k) for all w ∈ W, and that if f ∈ A then g = δL(k)f ∈ A, where δ is as defined in §2. If f ∈ A^{W}, it follows that g is skew-symmetric with respect to W, and hence L(k)f = δ^{−1}g ∈ A^{W}. Thus L(k) maps A^{W} into A^{W}, and a simple calculation shows that

L(k)m_{λ} = ⟨λ, λ + 2ρ_{k}⟩m_{λ} + lower terms

for λ ∈ P^{++}, where ρ_{k} is given by (5.7). Moreover, since △ = δ_{k}^{1/2}\overline{δ_{k}^{1/2}}, it follows from (11.13) that L(k) is self-adjoint for the scalar product (11.2). Hence as in (7.1) we conclude that the P_{λ} are eigenfunctions of L(k), and more precisely that

(11.14)  L(k)P_{λ}(k) = ⟨λ, λ + 2ρ_{k}⟩P_{λ}(k)

for λ ∈ P^{++}.

From (11.12) and (11.14) we derive the following recurrence relation for the coefficients u_{λμ}(k) in (11.10):

(11.15) (|λ + ρ_{k}|^{2} − |μ + ρ_{k}|^{2})u_{λμ}(k) = 2 ∑_{α∈R^{+}} ∑_{r≥1} k_{α}⟨μ + rα, α⟩u_{λ,μ+rα}(k).

Here λ, μ ∈ P^{++}, μ < λ, and u_{λμ} = u_{λ,wμ} for all w ∈ W. Let ν = λ − μ ∈ Q^{+}, then

|λ + ρ_{k}|^{2} − |μ + ρ_{k}|^{2} = ⟨ν, λ + μ + 2ρ_{k}⟩.

We have ⟨ν, λ + μ⟩ > 0; also

⟨ρ_{k}, α_{i}^{∨}⟩ = k_{α_{i}} + 2k_{2α_{i}}

for a simple root α_{i} ∈ R, from which it follows that ⟨ρ_{k}, ν⟩ is a positive linear combination of the k_{α}. Hence if the k_{α} are all ≥ 0, as we are assuming throughout (11.1), we have

|λ + ρ_{k}|^{2} − |μ + ρ_{k}|^{2} > 0

whenever λ, μ ∈ P^{++} and λ > μ. It follows now from the recurrence formula (11.15) by induction on ν = λ−μ that the coefficients u_{λμ}(k) are positive, and more precisely that they are rational functions of the k’s in which both numerator and denominator are polynomials in the k’s with positive integral coefficients. So finally we can drop the restriction (11.3); the polynomials P_{λ} are well-defined provided that (11.1) holds.

Now let G/K be a non-compact symmetric space, and G = KAN an Iwasawa decomposition of the semisimple Lie group G. Let 𝔤, 𝔨, 𝔞 be the Lie algebras of G, K, A respectively, let 𝔞^{∗} be the vector space dual to 𝔞, and Σ ⊂ 𝔞^{∗} the restricted root system of G/K. For each β ∈ Σ let m_{β} denote the multiplicity of β. We shall take

(11.16)  V = 𝔞^{∗}, R = 2Σ, k_{α} = \frac{1}{2}m_{α/2}

for each α ∈ R. Then L(k) is the radial part of the Laplacian on G/K.

Let 𝔥 ⊃ 𝔞 be a Cartan subalgebra of 𝔤; then 𝔥 = 𝔞 ⊕ (𝔥 ∩ 𝔨), and we put 𝔥_{ℝ} = 𝔞 ⊕ i(𝔥 ∩ 𝔨). Let M be a finite-dimensional irreducible representation space for G, with highest weight λ ∈ 𝔥_{ℝ}^{∗}. Then ([5], p. 535) M is spherical, i.e., has a nonzero vector fixed by K, if and only if λ vanishes on i(𝔥 ∩ 𝔨) and (regarded as an element of V = 𝔞^{∗}) we have ⟨λ, α^{∨}⟩ ∈ ℕ for all α ∈ R^{+}, i.e., λ ∈ P^{++}.

The recursion formula (11.15) conincides with Harish-Chandra’s recursion formula ([5], p. 427) for the coefficients of the zonal spherical function ω_{λ} defined by M. Hence for X ∈ 𝔞^{+}, the positive Weyl chamber in 𝔞, we have

(11.17)  ω_{λ}(exp X) = P_{λ}(X)/P_{λ}(0)

where each e^{λ} (λ ∈ P) is now to be regarded as a function on 𝔞 = V^{∗} by the rule e^{λ}(X) = e^{λ(X)}, the latter e being the classical exponential function.

If G/K is compact, the formula (11.17) is the same, except that now X ∈ i𝔞 in place of 𝔞^{+}.

Note the contrast with the p-adic situation of §10. There the restricted root system was R^{∨}, and the weights λ ∈ P^{++} indexed the double cosets of K in G. Here, on the other hand, the restricted root system is (similar to) R, and the λ ∈ P^{++} index the finite-dimensional spherical representations.

§12

We shall conclude with some conjectures. In order to state them concisely, we introduce the q-gamma function, defined for 0 < q < 1 by

Γ_{q}(x) = \frac{(q; q)_{x−1}}{(1 − q)^{x−1}} = \frac{(q; q)_{∞} (1 − q)^{1−x}}{(q^{x}; q)_{∞}} ,

where x ∈ ℝ but x ≠ 0, −1, −2, …, at which points Γ_{q} has simple poles. We have

Γ_{q}(x + 1) = \frac{1 − q^{x}}{1 − q} Γ_{q}(x)

and hence

(12.1)  lim_{k→0} \frac{Γ_{q}(kx)}{Γ_{q}(ky)} = lim_{k→0} \frac{1 − q^{ky}}{1 − q^{kx}} \frac{Γ_{q}(kx + 1)}{Γ_{q}(ky + 1)} = \frac{y}{x}.

As q → 1, Γ_{q}(x) → Γ(x) for each x, and therefore we shall write Γ_{1}(x) for the ordinary gamma function.

It will be convenient also to introduce

(12.2)  Γ_{q}^{∗}(x) = 1/Γ_{q}(1 − x).

When q = 1 we have

(12.3)  Γ_{1}^{∗}(x) = \frac{sin πx}{π}Γ_{1}(x)

but there is no particularly simple relationship between Γ_{q}^{∗}(x) and Γ_{q}(x) for general values of q.

Now define, for λ ∈ V and α ∈ R^{+},

(12.4)  c_{α}(λ; q_{α}) = \frac{Γ_{q_{α}}(⟨λ, α^{∨}⟩ + \frac{1}{2}k_{α/2})}{Γ_{q_{α}}(⟨λ, α^{∨}⟩ + \frac{1}{2}k_{α/2} + k_{α})}

(where as usual k_{α/2} = 0 if \frac{1}{2}α ∉ R), and

(12.5)  c(λ) = c(λ; q, t) = ∏_{α∈R^{+}} c_{α}(λ; q_{α}).

Also define c_{α}^{∗}(λ; q_{α}) and c^{∗}(λ) by using Γ^{∗} in place of Γ in (12.4) and (12.5).

We can now state Conjecture (12.6). For all λ ∈ P^{++}

|P_{λ}|^{2} = \frac{c^{∗}(−λ − ρ_{k})}{c(λ + ρ_{k})}.

Suppose in particular that the k_{α} are non-negative integers. Then (12.6) takes the form

(12.6′)  |P_{λ}|^{2} = ∏_{α∈R^{+}} ∏_{i=0}^{k_{α}−1} \frac{1 − q_{α}^{⟨λ+ρ_{k},α^{∨}⟩+ \frac{1}{2} k_{α/2}+i}}{1 − q_{α}^{⟨λ+ρ_{k},α^{∨}⟩− \frac{1}{2} k_{α/2}−i}} .

Even when λ = 0 (so that P_{λ} = 1) there is something to be proved here. Indeed, when R is reduced and λ = 0, the conjecture (12.6′) reduces to the constant term conjectures of [10] and [13]. For conjecture A′ of [13] asserts that the constant term of the product

(12.7)  ∏_{α∈R^{+}} ∏_{i=0}^{k_{α}−1}(1 − q_{α}^{i}e^{α})(1 − q_{α}^{i+1}e^{−α})

should be equal to

(12.8)  ∏_{α∈R} \frac{(q_{α}; q_{α})_{|⟨ρ_{k},α^{∨}⟩+k_{α}|}}{(q_{α}; q_{α})_{|⟨ρ_{k},α^{∨}⟩|}} .

Now the product (12.7) is precisely the product △′ defined in (3.10), and by (3.12) the constant term of △′ is W (t)|1|^{2}, which by (12.6′) is equal to

(12.9)  W (t) ∏_{α∈R^{+}} \frac{(q_{α}; q_{α})_{⟨ρ_{k},α^{∨}⟩+k_{α}−1} (q_{α}; q_{α})_{⟨ρ_{k},α^{∨}⟩−k_{α}}}{(q_{α}; q_{α})_{⟨ρ_{k},α^{∨}⟩−1} (q_{α}; q_{α})_{⟨ρ_{k},α^{∨}⟩}} .

On the other hand we have

W (t) = ∏_{α∈R^{+}} \frac{1 − q_{α}^{⟨ρ_{k},α^{∨}⟩+k_{α}}}{1 − q_{α}^{⟨ρ_{k},α^{∨}⟩}} ,

by ([8], 2.4 nr) applied to the root system S^{∨}. Hence (12.9) is equal to (12.8), which proves our assertion.

If however R is of type BC_{n}, there are two choices for S when n ≥ 2, so that our conjecture (12.6′) when λ = 0 contains two distinct constant-term conjectures related to the root system BC_{n}. Neither of these is obviously equivalent to Morris’s Conjecture A for BC_{n} ([13], 3.4).

Recall next that each f ∈ A is regarded as a function on V, by the rule e^{λ}(x) = q^{⟨λ,x⟩} (x ∈ V, λ ∈ P). Also let

α^{∗} = (α_{∗})^{∨} = u_{α}α^{∨}

for each α ∈ R, and

ρ_{k}^{∗} = \frac{1}{2} ∑_{α∈R^{+}} k_{α}α^{∗}.

Conjecture (12.10). For λ ∈ P^{++},

P_{λ}(ρ_{k}^{∗}) = q^{−⟨λ,ρ_{k}^{∗}⟩}c(ρ_{k})/c(λ + ρ_{k}).

Both conjectures (12.6) and (12.10) are true in each of the situations considered in Sections 8–11:

(i) When R is reduced and the k_{α} are all equal to 1, we have P_{λ} = χ_{λ} (8.3), and (12.6) reduces to |χ_{λ}|^{2} = 1, i.e., to (8.4). As to (12.10), we have ρ_{k}^{∗} = \frac{1}{2} ∑_{α∈R} α^{∗} = ρ^{∗}

say, and

P_{λ}(ρ_{k}^{∗}) = χ_{λ}(ρ^{∗}) = \frac{∑_{w∈W} ε(w) q^{⟨w(λ+ρ),ρ^{∗}⟩}}{∑_{w∈W} ε(w) q^{⟨wρ,ρ^{∗}⟩}} .

By Weyl’s denominator formula for the root system S^{∨}, this factorizes to give

q^{−⟨λ,ρ^{∗}⟩} ∏_{α∈R^{+}} \frac{1 − q_{α}^{⟨λ+ρ,α^{∨}⟩}}{1 − q_{α}^{⟨ρ,α^{∨}⟩}}

in agreement with (12.10).

(ii) When the k_{α} are all zero we have P_{λ} = m_{λ} and |P_{λ}|^{2} = |W_{λ}|^{−1}. On the other hand, it follows from our definitions that when the k_{α} are all zero we have c_{α}^{∗}(−λ − ρ_{k}; q_{α}) = 1 for all α ∈ R^{+} and c_{α}(λ + ρ_{k}; q_{α}) = 1 for all α ∈ R^{+} such that ⟨λ, α^{∨}⟩ ≠ 0. When ⟨λ, α^{∨}⟩ = 0 we have to interpret c_{α}(λ + ρ_{k}; q_{α}) by means of the limit relation (12.1); with k_{α} = k for all α this leads to

lim_{k→0} \frac{c^{∗}(−λ − ρ_{k})}{c(λ + ρ_{k})} = _{∏_{α∈R^{+}} ; ⟨λ,α^{∨}⟩=0} \frac{⟨ρ, α^{∨}⟩ + e_{α}}{⟨ρ, α^{∨}⟩ + e_{α} + 1}

where e_{α} = \frac{1}{2} if \frac{1}{2}α ∈ R, and e_{α} = 0 otherwise; and this product is equal to |W_{λ}|^{−1}. This checks (12.6) in this case, and (12.10) is analogous (both sides are equal to |W |/|W_{λ}|).

(iii) When R is of rank 1 (§9) the formulas (9.8), (9.9), (9.14) and (9.15) show that both conjectures are true.

(iv) When R is of type A_{n}, the polynomials P_{λ} are essentially the same as the symmetric functions P_{λ}(x; q, t) studied in [11], Chapter VI. Both conjectures are true in this case, and are proved in loc. cit.

(v) In the situation of §10 we must express everything in terms of the t_{α} before setting q = 0. We have

q_{α}^{⟨λ+ρ_{k},α^{∨}⟩} = q_{α}^{⟨λ,α^{∨}⟩}q^{⟨ρ_{k},α^{∗}⟩}

and ⟨ρ_{k}, α^{∗}⟩ = \frac{1}{2} ∑_{β∈R^{+}} k_{β}u_{β}⟨β_{∗}, α^{∗}⟩, so that

q^{⟨ρ_{k},α^{∗}⟩} = ∏_{β∈R^{+}} t_{β}^{⟨β_{∗},α^{∗}⟩/2} = t^{ht(α^{∗})}

in the notation of [8]. It follows that c^{∗}(−λ − ρ_{k}) = 1 when q = 0, and that

c(λ + ρ_{k}) = _{∏_{α∈R^{+}} ; ⟨λ,α^{∨}⟩=0} \frac{1 − t_{α/2}t_{α}t^{ht(α^{∗})}}{1 − t_{α/2}t^{ht(α^{∗})}}

which from the results of [8] is easily seen to be equal to the polynomial W_{λ}(t). Hence in the present situation the right-hand side of (12.6) is equal to W_{λ}(t)^{−1}, which by (10.5) is equal to |P_{λ}|^{2}.

Next consider (12.10). We have

e^{α}(ρ_{k}^{∗}) = q^{⟨α,ρ_{k}^{∗}⟩} = ∏_{β∈R^{+}} t_{β}^{⟨β^{∨},α⟩/2} = t^{ht(β)}

and in the formula (10.1) for P_{λ}, when we evaluate at ρ_{k}^{∗}, all the terms will vanish except that corresponding to w_{0}, the longest element of W. Consequently we obtain

(e^{λ}P_{λ})(ρ_{k}^{∗}) = W (t)/W_{λ}(t)

which from above is also equal to c(ρ_{k})/c(λ + ρ_{k}), thus verifying (12.10) in this case.

(vi) Finally, in the “limiting case” q → 1 considered in §11, Heckman [4] has proved that |P_{λ}|_{k}^{2}/|1|_{k}^{2} has the value predicted by (12.6), and recently Opdam [15] has evaluated |1|_{k}^{2}. These results confirm that (12.6) is true in the limiting case.

As to (12.10), it follows from the work of Harish-Chandra ([5] Chapter V) that in the symmetric space situation (11.16) the zonal spherical function ω_{λ} for λ ∈ P^{++} is given by

ω_{λ}(exp X) = \frac{c(λ + ρ_{k})}{c(ρ_{k})}P_{λ}(X) (X ∈ 𝔞^{+}).

Hence by comparison with (11.17) we have

(12.11)  P_{λ}(0) = c(ρ_{k})/c(λ + ρ_{k})

which proves (12.10) in this case. Recently Opdam ([15], Cor. 5.2) has proved (12.11) for arbitrary values of the k_{α}.

References

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School of Mathematical Sciences, Queen Mary and Westfield College, University of London, London E1 4NS, England