Constant Term Identities,

Orthogonal Polynomials,

and Affine Hecke Algebras

I. G. Macdonald

The main aim of this lecture is to survey a theory of orthogonal polynomials in several variables which has developed over the last ten years or so. We shall concentrate on the purely algebraic aspects of the theory, and for lack of time and competence we shall say nothing about its physical applications (completely integrable systems, KZ equations, etc.)

These polynomials include as special cases, on the one hand all the classical orthogonal polynomials in one variable (Legendre, Jacobi, Hermite, ⋯), and on the other hand polynomials that arise in the representation theory of Lie groups (characters of compact Lie groups, spherical functions on real and p-adic symmetric spaces and their quantum analogues). The underlying notion is that of a root system, to which I shall turn first.

1 Root systems

Root systems and their Weyl groups constitute the combinatorial infrastructure of much of the theory of Lie groups and Lie algebras. Thus a complex semisimple Lie algebra or a compact connected Lie group with trivial centre, is determined up to isomorphism by its root system. Moreover, and quite apart from their Lietheoretic origin, the geometry and algebra of root systems presents an apparently inexhaustible source of beautiful combinatorics.

It is time for definitions and examples. Let V be a real vector space of finite dimension, endowed with a positive definite scalar product ⟨u, v⟩. For each nonzero α ∈ V let s_{α} denote the orthogonal reflection in the hyperplane H_{α} through the origin perpendicular to α. Explicitly,

(1.1) s_{α}(v) = v − ⟨v, α^{∨}⟩α

for v ∈ V, where α^{∨} = 2α/⟨α, α⟩.

A root system R in V is a finite non-empty set of non-zero vectors (called roots) that span V and are such that for each pair α, β ∈ R we have

(1.2) ⟨α^{∨}, β⟩ ∈ ℤ, (1.3) s_{α}(β) ∈ R.

Thus each reflection s_{α}(α ∈ R) permutes R, and the group of orthogonal transformations of V generated by the s_{α} is a finite group W_{0}, called the Weyl group of R.

We may remark straightaway that the integrality condition (1.2) by itself is extremely restrictive. Let α, β ∈ R and let Θ be the angle between the vectors α and β. Then

4 cos^{2} Θ = \frac{4⟨α, β⟩^{2}}{⟨α, α⟩⟨β, β⟩} = ⟨α^{∨}, β⟩⟨α, β^{∨}⟩

is an integer, hence can only take the values 0, 1, 2, 3, 4. It follows that the only possibilities for Θ are π/m or π − (π/m), where m = 1, 2, 3, 4 or 6.

The vectors α^{∨} for α ∈ R form a root system R^{∨}, the dual of R. If α ∈ R, then also −α ∈ R (because −α = s_{α}(α)). The root system R is said to be reduced if the only scalar multiples of α in R are ±α. Furthermore, R is said to be irreducible if it is not possible to partition R into two non-empty subsets R_{1} and R_{2} such that each root in R_{1} is orthogonal to each root in R_{2} (which would imply that R_{1} and R_{2} are themselves root systems). We shall assume throughout that R is both reduced and irreducible.

For those to whom these notions are unfamiliar, some examples to bear in mind are the following. Let ε_{1}, ⋯, ε_{n} be the standard basis of ℝ^{n}(n ≥ 2), with the usual scalar product, for which ⟨ε_{i}, ε_{j}⟩ = δ_{ij}. Then the vectors

(A_{n−1}) ± ε_{i} − ε_{j}

where i ≠ j, form a root system (and V is the hyperplane in ℝ^{n} orthogonal to ε_{1} + ⋯ + ε_{n}). The Weyl group is the symmetric group S_{n}, acting on V by permuting the ε_{i}.

Moreover, each of the sets of vectors

(B_{n}) ± ε_{i} (1 ≤ i ≤ n), ±ε_{i} ± ε_{j} (1 ≤ i < j ≤ n),

(C_{n}) ±2ε_{i} (1 ≤ i ≤ n), ±ε_{i} ± ε_{j} (1 ≤ i < j ≤ n),

(D_{n}) ε_{i} ± ε_{j} (1 ≤ i < j ≤ n)

is a root system. For B_{n} and C_{n}, the Weyl group is the group of all signed permutations of the ε_{i}, of order 2^{n}n! (the hyperoctahedral group). For D_{n}, it is a subgroup of index 2 in this group. The root systems B_{n} and C_{n} are duals of each other, and A_{n−1}, D_{n} are each self-dual.

Constant Term identities, Orthogonal Polynomials … 305

In fact, the root systems A_{n}(n ≥ 1), B_{n}(n ≥ 2), C_{n}(n ≥ 3) and D_{n}(n ≥ 4) almost exhaust the catalogue of reduced irreducible root systems (up to isomorphism). Apart from these, there are just five others, the “exceptional” root systems, denoted by E_{6}, E_{7}, E_{8}, F_{4} and G_{2}. (In each case the numerical suffix is the dimension of the space V spanned by R, which is also called the rank of R.)

Let R be any (reduced, irreducible) root system in V and consider the complement

X = V − ⋃_{α∈R}H_{α}

of the union of the reflecting hyperplanes H_{α}, α ∈ R. The connected components of X are open simplicial cones which are permuted simply transitively by the Weyl group W_{0}. Let Γ be one of these components, chosen once and for all; it is bounded by n = dim V hyperplanes H_{α_{i}}(1 ≤ i ≤ n), and

Γ = {x ∈ V : ⟨α_{i}, x⟩ > 0 (1 ≤ i ≤ n)}.

The α_{i} are the simple roots determined by Γ, and each root α ∈ R is of the form

(1.4) α = ∑_{1}^{r} r_{i}α_{i}

with integral coefficients r_{i} all of the same sign. A root α ∈ R is positive (resp. negative) relative to Γ if ⟨α, x⟩ > 0(resp. < 0) for all x ∈ Γ. Equivalently, α ∈ R is positive (resp. negative) if the coefficients r_{i} in (1.4) are all ≥ 0(resp. ≤ 0). Let R^{+} (resp. R^{−}) denote the set of positive (resp. negative) roots. Then R^{−} = −R^{+}, and R = R^{+} ∪ R^{−}. Moreover, there is a unique root φ ∈ R^{+}, called the highest root, for which the sum of the coefficients ∑ r_{i} in (1.4) is maximal. In A_{n−1}, for example, we may take the simple roots to be α_{i} = ε_{i} − ε_{i+1}(1 ≤ i ≤ n − 1); the positive roots are then ε_{i} − ε_{j} with i < j, and the highest root is ε_{1} − ε_{n}.

The abelian group Q generated by R, whose elements are the integral linear combinations of the roots, is a lattice in V (i. e. a free abelian group of rank n = dim V) called the root lattice. Clearly the simple roots α_{1}, ⋯ α_{n} form a basis of Q. We denote by Q^{+} the subsemigroup of Q consisting of all sums ∑ r_{i}α_{i} where the coefficients are non negative integers.

Next, the set P of all λ ∈ V such that ⟨λ, α^{∨}⟩ ∈ ℤ for all α ∈ R is another lattice, called the weight lattice. It has a basis consisting of the fundamental weights π_{1}, ⋯, π_{n}, defined by the equations ⟨π_{i}, α_{j}^{∨}⟩ = δ_{ij}. We denote by P^{+} the set of dominant weights (i. e. λ ∈ P such that ⟨λ, α^{∨}⟩ ≥ 0 for all α ∈ R^{+}). We have P ⊃ Q (by (1.2)) but P^{+} ⊅ Q^{+} (unless n = 1, i. e. R = A_{1}). The quotient P/Q is a finite group, since both P and G are lattices of the same rank n. Clearly, both P and Q are stable under the action of the Weyl group W_{0}. Each W_{0}-orbit in P contains exactly one dominant weight, i. e. P^{+} is a fundamental region for the action of W_{0} on P.

Finally, the Weyl group W_{0} acts on V and therefore also on the algebra S(V) of polynomial functions on V. It can be shown that the subring S(V)^{W_{0}} of W_{0}-invariant polynomial functions in generated by n = dim V algebraically independent homogeneous polynomial functions, of degrees say d_{1}, ⋯, d_{n}. The functions themselves are not uniquely determined, but their degrees are: they are called the degrees of W_{0}. For example, if R is A_{n−1}, so that W_{0} is the symmetric group S_{n}, we may take as generators of S(V)^{W_{0}} the power sums

x_{1}^{r} + ⋯ + x_{n}^{r} (2 ≤ r ≤ n)

where x_{1}, ⋯, x_{n} are coordinates in ℝ^{n}. Thus in this case the degrees are 2, 3, ⋯, n.

2 Constant term identities

Let F be a field of characteristic zero and let A = F [P] be the group algebra over F of the weight lattice P. Since the group operation in P is addition, we shall use an exponential notation in A, and denote by e^{λ} the element of A corresponding to λ ∈ P. These “formal exponentials” e^{λ} form an F -basis of A, such that e^{λ} · e^{μ} = e^{λ+μ} and (e^{λ})^{−1} = e^{−λ}. In particular, e^{0} = 1 is the identity element of A. The ring A is an algebra of Laurent polynomials, namely A = F [u_{1}^{±1}, ⋯ u_{n}^{±1}] where u_{i} = e^{π_{i}} (π_{i} the fundamental weights).

If

f = ∑_{λ∈P} f_{λ}e^{λ}

is an element of A, with coefficients f_{λ} ∈ F, the constant term of f is f_{0}, the coefficient of e^{0} = 1 in f. We can now state two constant term identities that generalize those of Dyson and Andrews described in the abstract to this lecture. As before, R is a reduced irreducible root system and k a non negative integer.

(2.1) The constant term in

∏_{α∈R}(1 − e^{α})^{k}

is equal to

∏_{i=1}^{n} \binom{kd_{i}}{k}

where d_{1}, ⋯ d_{n} are the degrees of the Weyl group of R.

When R is A_{n−1}, the roots are α = ε_{i} − ε_{j} where i ≠ j, so that e^{α} = x_{i}x_{j}^{−1} where x_{i} = e^{ε_{i}}. Moreover, as we have seen, the degrees of the Weyl group in this case are 2, 3, ⋯, n; and

\binom{2k}{k}\binom{3k}{k} ⋯ \binom{nk}{k} = \frac{(nk)!}{k!^{n}}

Thus we recover Dyson’s original conjecture [5].

Next, in order to state the generalization of Andrew’s conjecture we introduce

the q-analogue of the binomial coefficient \binom{r}{s}, namely the Gaussian polynomial

[ \begin{matrix}r \\ s \end{matrix} ] = \frac{(1 − q^{r})(1 − q^{r−1}) ⋯ (1 − q^{r−s+1})}{(1 − q)(1 − q^{2}) ⋯ (1 − q^{s})}

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which tends to \binom{r}{s} as q → 1.

(2.2) The constant term in

∏_{α∈R^{+}}  ∏_{i=0}^{k−1} (1 − q^{i}e^{α}) (1 − q^{i+1}e^{−α})

is equal to

∏_{i=1}^{n} [ \begin{matrix}kd_{i} \\ k \end{matrix} ] .

When R is A_{n−1}, the positive roots are α = ε_{i} − ε_{j} with i < j, so that we recover Andrews’ conjecture. Clearly, also, (2.2) reduces to (2.1) when we let q → 1.

When these conjectures and others like them were first put forward ([12], [18]), they appeared as isolated curiosities, and it was not clear what, if anything, lay behind them. Later [13] it became clear that they could be considered as a special case of a conjectured norm fomula for orthogonal polynomials, as we shall explain in the next section.

The identity (2.1) was first proved uniformly for all R by Opdam [20], using the technique of shift operators developed by Heckman and Opdam in the context of their theory of hypergeometric functions and Jacobi polynomials [8]. The qversion (2.2) took longer to resolve, and was finally proved in full generality by Cherednik [3], although by that time all the root systems with the exception of E_{6}, E_{7} and E_{8} has been dealt with one by one ([2], [9], [6], [7]).

3 Orthogonal polynomials

As in §2, let A be the group algebra F [P] where F is a field of characteristic 0. The Weyl group W_{0} acts on P and therefore also on A : w(e^{λ}) = e^{wλ}(λ ∈ P, w ∈ W_{0}). Let A_{0} denote the subalgebra of W_{0}-invariants.

Since each W_{0}-orbit in P meets P^{+} exactly once, it follows that the orbit-sums

(3.1) m_{λ} = ∑_{μ∈W_{0}λ} e^{μ}

where λ ∈ P^{+} and W_{0}λ is the W_{0}-orbit of λ, form an F -basis of A. Another basis of A_{0} is obtained as follows. Let

(3.2 p = \frac{1}{2} ∑_{α∈R^{+}} α

and let

(3.3) δ = ∏_{α∈R^{+}}(e^{α/2} − e^{−α/2}).

In fact, p ∈ P^{+} and δ ∈ A : we have

(3.4) δ = ∑_{w∈W_{0}} ε(w)e^{wp}

where ε(w) = det(w) = ±1. Thus δ is skew-symmetric for W_{0}, i. e. we have wδ = ε(w)δ for each w ∈ W_{0}. For each λ ∈ P^{+}, the sum

∑_{w∈W_{0}} ε(w)e^{w(λ+p)}

is likewise skew-symmetric, and is divisible by δ in A. The quotient

(3.5) X_{λ} = δ^{−1} ∑_{w∈W_{0}} ε(w)e^{w(λ+p)}

is an element of A_{0} called the Weyl character corresponding to λ. In terms of the orbit-sums we have

(3.6) X_{λ} = m_{λ} + ∑_{μ<λ} K_{λμ}m_{μ}

where the coefficients K_{λμ} are integers (indeed positive integers) and μ < λ means that λ − μ ∈ Q^{+} and λ ≠ μ.

From (3.6) it follows that the χ_{λ} form another F -basis of A_{0}. From now on we shall take F to be the field ℚ(q, t) of rational functions in two indeterminates q, t. Let

(3.7) Δ = Δ(q, t) = ∏_{α∈R^{+}} ∏_{r=0}^{∞} \frac{(1 − q^{r}e^{α})(1 − q^{r+1}e^{−α})}{(1 − q^{r}te^{α})(1 − q^{r+1}te^{−α})}.

Suppose first that t = q^{k} where k is a non-negative integer. Then Δ is a finite product, namely the polynomial whose constant term was the subject of (2.2). (In the general case, Δ can be expanded as a formal power series in the n+1 variables u_{0}, u_{1}, ⋯, u_{n}, where u_{i} = e^{α_{i}}(1 ≤ i ≤ n) and u_{0} = qe^{−φ}, φ the highest root of R.

We shall use Δ to define a scalar product on A, as follows. If f ∈ A, say

f = ∑_{λ∈P} f_{λ}e^{λ},

let

(3.8) f^{∗} = ∑_{λ∈P} f_{λ}^{∗}e^{−λ}

where f_{λ}^{∗} is the image of f_{λ} under the automorphism (q, t) ↦ (q^{−1}, t^{−1}) of F. We now define, for f, g ∈ A,

(3.9) (f, g) = constant term in fg^{∗}Δ.

We can now state

(3.10) There is a unique F -basis (P_{λ})_{λ∈P^{+}} of A_{0} such that

(i) P_{λ} = m_{λ} + ∑_{μ<λ} u_{λμ}m_{μ} with coefficients u_{λμ} ∈ F;

(ii) (P_{λ}, P_{μ}) = 0 if λ ≠ μ.

It is easy to see that the P_{λ}, if they exist, are uniquely determined by (i) and (ii). Their existence, however, requires proof. If the partial order λ > μ on P^{+} were a total ordering, existence would follow directly from the Gram-Schmidt orthogonalization process. But it is not a total ordering (unless R = A_{1}) and we should therefore have to extend it to a total ordering before applying the Gram-Schmidt mechanism. Thus the content of (3.10) is that however we extend the partial order λ > μ to a total order, we always obtain the same basis.

We shall not reproduce the original proof ([13] [16]) of (3.10) here, since if will arise more naturally later in the context of affine Hecke algebras. Instead, let us look at some special cases:

(1) When t = 1, we have Δ = 1 and P_{λ} is the orbit-sum m_{λ} (3.1).

(2) When t = q, P_{λ} is the Weyl character X_{λ} (3.5).

(3) When q → 0, t being arbitrary, the P_{λ} (suitably normalized) occur as the values of spherical functions on a p-adic symmetric space, when t^{−1} is a prime power.

(4) Let t = q^{k} and fix k (which need not be an integer) and let q → 1, so that t → 1 also. In the limit we have Δ = ∏_{α∈R}(1 − e^{α})^{k}. In this limiting case the polynomials P_{λ} are the “Jacobi polynomials” of Heckman and Opdam [8]. For particular values of k these polynomials occur as values of spherical functions, but this time on a real symmetric space.

(5) Finally, when R is A_{n−1}, the P_{λ} are the symmetric functions of ([15], chapter VI), restricted to n variables x_{1}, ⋯, x_{n} such that x_{1} ⋯ x_{n} = 1.

To conclude this section, we shall record some properties of the polynomials P_{λ}. For simplicity of statement, we shall assume that t = q^{k} where k is a positive integer.

a.) Norms The squared norm of P_{λ} is given by the formula

(3.11) (P_{λ}, P_{λ}) = W_{0}(t) ∏_{α∈R^{+}} ∏_{i=0}^{k−1} \frac{1 − q^{⟨λ+kp,α^{∨}⟩+i}}{1 − q^{⟨λ+kp,α^{∨}⟩−i}}

where p is given by (3.2) and W_{0}(t) is the Poincaré polynomial of the Weyl group W_{0}:

W_{0}(t) = ∑_{w∈W_{0}} t^{ℓ(w)}

Documenta Mathematica · Extra Volume ICM 1998 · I · 303–317

where l(w) is the length of w, i. e. the number of α ∈ R^{+} such that wα ∈ R^{−}.

Notice that when λ = 0 we have P_{λ} = 1, so that in this case (3.11) gives the constant term of Δ, i. e. it gives the constant term identity (2.2) (though a little work is required to recast it in that form). The formula (3.11) was originally conjectured in [13], and verified there in some cases. In the limiting case q → 1, it was first proved for all root systems R by Opdam [20], and then in full generality by Cherednik [3]. We shall indicate a proof later, in §5.

b.) Specialization Let P^{∨} be the weight lattice of the dual root system R^{∨}: it consists of all λ ∈ V such that ⟨λ, α⟩ ∈ ℤ for all α ∈ R. It will be convenient to regard each f ∈ A as a function on P^{∨}, as follows: if μ ∈ P^{∨} and f = ∑ f_{λ}e^{λ}, then

f(μ) = ∑ f_{λ}q^{⟨λ,μ⟩}.

Then we have

(3.12) P_{λ}(kp^{∨}) = q^{−⟨λ,kp^{∨}⟩} ∏_{α∈R^{+}} ∏_{i=0}^{k−1} \frac{1 − q^{⟨λ+kp,α^{∨}⟩+i}}{1 − q^{⟨kp,α^{∨}⟩+i}}

where

p^{∨} = \frac{1}{2} ∑_{α∈R^{+}} α^{∨}

(warning: p^{∨} ≠ 2p/⟨p, p⟩).

When k = 1 and q → 1, this reduces to Weyl’s formula for the dimension of an irreducible representation of a compact Lie group. The formula (3.12) was originally conjectured in [13]. As with (3.11), it was first proved for all R in the limiting case q → 1 by Opdam [20], and then in full generality by Cherednik [4].

c.) Symmetry For λ ∈ P let

P̃_{λ} = P_{λ}/P_{λ}(kp^{∨}).

Then we have

(3.13)  P̃_{λ}(μ + kp^{∨}) = P̃_{μ}(λ + kp)

for all λ ∈ P^{+} and μ ∈ (P^{∨})^{+}, and on the right-hand side of (3.13), P_{μ} is an orthogonal polynomial for R^{∨}, so that P̃_{μ} = P_{μ}/P_{μ}(kp). When R is of type A_{n−1}, (3.13) is due to Koornwinder ([15], chapter VI, §6). The general case is due to Cherednik [4].

4 The affine root system and the extended affine Weyl group

The root systems and Weyl groups of §1 have affine counterparts, to which we now turn. As before, R is a reduced, irreducible root system spanning a real vector space V of dimension n ≥ 1. Let Q^{∨}, P^{∨} respectively denote the root lattice and the weight lattice of the dual root system R^{∨}.

We shall regard each α ∈ R as a linear function on V: α(x) = ⟨α, x⟩ for x ∈ V. Let c denote the constant function 1 on V. Then

(4.1) S = S(R) = {α + nc : α ∈ R, n ∈ ℤ}

is the affine root system associated with R. The elements of S are affine-linear functions on V, called affine roots, and we shall denote them by italic letters, a, b, . . ..

For each a ∈ S, let H_{a} denote the affine hyperplane in V on which a vanishes, and let s_{a} denote the orthogonal reflection in this hyperplane. The affine Weyl group W_{S} is the group of affine isometries of V generated by these reflections. For each α ∈ R, the mapping s_{α} ∘ s_{α+c} takes x ∈ V to x + α^{∨}, so that

τ(α^{∨}) = s_{α} ∘ s_{α+c}

is translation by α^{∨}. It follows that W_{S} contains a subgroup of translations isomorphic to Q^{∨}, and we have

(4.2) W_{S} = W_{0} ⋉ τ(Q^{∨})

(semidirect product).

The extended affine Weyl group is

(4.3) W = W_{0} ⋉ τ(P^{∨}).

It acts on V as a discrete group of isometries, and hence by transposition on functions on V. As such, it permutes the affine roots a ∈ S.

As in §1, let R^{+} be a system of positive roots in R and α_{1}, ⋯, α_{n} the simple roots, φ the highest root. Correspondingly, the affine roots a_{0}, a_{1}, ⋯, a_{n}, where a_{0} = −φ + c and α_{i} = α_{i} (1 ≤ i ≤ n) form a set of simple roots for S : each a ∈ S is of the form

(4.4) a = ∑_{i=0}^{n} r_{i}a_{i}

where the r_{i} are integers, all of the same sign. Let

C = {x ∈ V : a_{i}(x) > 0 (0 ≤ i ≤ n)}

so that C is an open n-simplex bounded by the hyperplanes H_{a_{i}} (0 ≤ i ≤ n). The group W_{S} is generated by the reflections s_{i} = s_{a_{i}}(0 ≤ i ≤ n), subject to the relations

(4.5) s_{i}^{2} = 1, (4.6) s_{i}s_{j}s_{i} ⋯ = s_{j}s_{i}s_{j} ⋯

whenever i ≠ j and s_{i}s_{j} has finite order m_{ij} in W_{S}, there being m_{ij} terms on either side of (4.6). In other words, W_{S} is a Coxeter group on the generators s_{0}.s_{1}, ⋯, s_{n}.

The connected components of V − ⋃_{a∈S}H_{a} are open simplexes, each congruent

to C, and each component is of the form wC for a unique element w ∈ W_{S}. Thus, for example, when R is of type A_{2} we obtain the familiar tessellation of the Euclidean plane by congruent equilateral triangles.

An affine root a ∈ S is positive (resp. negative) relative to C if a(x) > 0 (resp. a(x) < 0) for x ∈ C. Equivalently, a ∈ S is positive or negative according as the coefficients r_{i} in (4.4) are all ≥ 0 or all ≤ 0. Let S^{+} (resp. S^{−}) denote the set of positive (resp. negative) affine roots. Then S^{−} = −S^{+}, and S = S^{+} ∪ S^{−}.

Explicitly, the positive affine roots are α + rc where r ≥ 0 if α ∈ R^{+}, and r ≥ 1 if α ∈ R^{−}. It follows that the product Δ (3.7) may be written in the form

(4.7) Δ = ∏_{a∈S^{+}} \frac{1 − e^{a}}{1 − te^{a}}

where for a = α + rc ∈ S, e^{a} = e^{α+rc} = q^{r}e^{α} (i. e. we define e^{c} = q).

We shall now define a length function on the extended group W. If w ∈ W, let

ℓ(w) = card(S^{+} ∩ wS^{−}),

the number of positive affine roots made negative by w. Equivalently, ℓ(w) is the number of hyperplanes H_{a}, a ∈ S, that separate C from wC.

Now W, unlike W_{S}, is not in general a Coxeter group (unless P^{∨} = Q^{∨}) and may contain elements ≠ 1 of length zero. Let

Ω = {w ∈ W : ℓ(w) = 0}

The elements of Ω stabilize the simplex C, and hence permute the simple affine roots. For each w ∈ W there is a unique w′ ∈ W_{S} such that wC = w′C, and hence w factorizes uniquely as w = w′v, with w′ ∈ W_{S} and v ∈ Ω. Consequently we have

(4.8) W = W_{S} ⋊ Ω

(semidirect product). From (4.2), (4.3) and (4.8) it follows that Ω ≅ W/W_{S} ≅ P^{∨}/Q^{∨}, hence is a finite abelian group.

Next, the braid group B of W is the group with generators T (w), w ∈ W, and relations

T (v)T (w) = T (vw)

whenever ℓ(vw) = ℓ(v) + ℓ(w). We shall denote T (s_{i}) by T_{i} (0 ≤ i ≤ n) and T (ω) (ω ∈ Ω) simply by ω. Then B is generated by T_{0}, T_{1}, ⋯, T_{n} and Ω subject to the following relations:

(a) the counterparts of (4.6), namely the braid relations

(4.9) T_{i}T_{j}T_{i} ⋯ = T_{j}T_{i}T_{j} ⋯

where i ≠ j and there are m_{ij} terms on either side;

(b) the relations

(4.10) ωT_{i}ω^{−1} = T_{j}

for ω ∈ Ω, where ω(a_{i}) = a_{j}.

Let λ ∈ (P^{∨})^{+} be a dominant weight for R^{∨}, and define

Y^{λ} = T (τ(λ))

where τ(λ) is translation by λ. If λ and μ are both dominant, we have

(4.11) Y^{λ} · Y^{μ} = Y^{λ+μ}

in B. If now λ is any element in P^{∨}, we can write λ = μ − ν where μ, ν are both dominant, and we define

(4.12) Y^{λ} = Y^{μ}(Y^{ν})^{−1}.

In view of (4.10), this definition is unambiguous. The elements Y^{λ}, λ ∈ P^{∨}, form a commutative subgroup of B, isomorphic to P^{∨}.

5 The affine Hecke algebra

The Hecke algebra H of W is the quotient of the group algebra F [B] of the braid group by the ideal generated by the elements (T_{i} − t^{1/2})(T_{i} + t^{−1/2}) (0 ≤ i ≤ n). (The field F should now include t^{1/2} as well as q and t.) For each w ∈ W, we denote the image of T (w) in H by the same symbol T (w): these elements form an F -basis of H. Thus H is generated over F by T_{0}, T_{1}, ⋯, T_{n} and Ω subject to the relations (4.9), (4.10), together with the Hecke relations

(5.1) (T_{i} − t^{1/2})(T_{i} + t^{−1/2}) = 0.

When t = 1, H is the group algebra of W.

The following proposition is due to Cherednik [3].

(5.2) The Hecke algebra H acts on A = F [P] as follows:

T_{i}e^{μ} = t^{1/2}e^{s_{i}μ} + (t^{1/2} − t^{−1/2}) (1 − e^{a_{i}})^{−1} (e^{μ} − e^{s_{i}μ}) ,
ωe^{μ} = e^{ωμ}.

where 0 ≤ i ≤ n and ω ∈ Ω. Moreover, this representation is faithful.

A proof of (5.2) is sketched in [14]. (In the formulas above, recall that e^{a_{0}} = e^{−φ+c} = qe^{−φ}.)

The elements Y^{λ}, λ ∈ P^{∨}, span a commutative subalgebra of H, isomorphic to A^{∨} = F [P^{∨}]. If u ∈ A^{∨}, say

u = ∑ u_{λ}e^{λ}

let

u(Y) = ∑ u_{λ}Y^{λ} ∈ H.

(5.3) For each w ∈ W, the adjoint of T (w) for the scalar product (3.9) on A is T (w)^{−1}, i. e., we have

(T (w)f, g) = (f, T (w)^{−1}g)

for all f, g ∈ A. In particular, the adjoint of Y^{λ} is Y^{−λ}, and the adjoint of u(Y), where u ∈ A^{∨}, is u^{∗}(Y) (3.8).

It is enough to show that the adjoint of T_{i} (resp. ω ∈ Ω) is T_{i}^{−1} (resp. ω^{−1}), and this may be verified directly from the definitions.

From (5.2) we have an action of A^{∨} on A, with u ∈ A acting as u(Y). One shows that A_{0} = A^{W_{0}} is stable under the action of A_{0}^{∨} = (A^{∨})^{W_{0}}, so that we have an action of A_{0}^{∨} on A_{0}. It turns out (see, e.g. [16] chapter III) that this action is diagonalized by the polynomials P_{λ}(λ ∈ P^{+}), and more precisely that

(5.4) u(Y)P_{λ} = u(−λ − kp)P_{λ}

for all u ∈ A^{∨}. The pairwise orthogonality of the P_{λ} then follows immediately from (5.3) and (5.4).

Likewise, the action of A^{∨} on A can be diagonalized, and this gives rise to a family of non-symmetric orthogonal polynomials:

(5.5) There is a unique F -basis (E_{λ})_{λ∈P} of A such that

(i) E_{λ} = e^{λ}+ lower terms,

(ii) (E_{λ}, E_{μ}) = 0 if λ ≠ μ.

(By “lower terms” is meant a linear combination of exponentials e^{μ} where μ < λ in a certain partial ordering on P.)

The polynomials E_{λ} are simultaneous eigenfunctions of all operators u(Y), u ∈ A^{∨}. (See [19] or [16], Ch. III.)

Consider now the operators

U^{+} = ∑_{w∈W_{0}} t^{ℓ(w)/2}T (w),
U^{−} = ∑_{w∈W_{0}} ε(w)t^{−ℓ(w)/2}T (w),

on A. The operator U^{+} maps A onto A_{0}, and in particular if λ ∈ P^{+} then U^{+}E_{λ} is a scalar multiple of P_{λ}.

Next consider, again for λ ∈ P^{+},

Q_{λ} = U^{−}E_{λ}.

If λ is not regular (i. e. if ⟨λ, α_{i}⟩ = 0 for some i) then Q_{λ} = 0.

Both P_{λ} and Q_{λ} are linear combinations of the E_{μ}, μ ∈ W_{0}λ, with coefficients that can be computed explicitly. Hence both (P_{λ}, P_{λ}) and (Q_{λ}, Q_{λ}) can be expressed in terms of (E_{λ}, E_{λ}). In this way we obtain [14]

(5.6) \frac{(Q_{λ}, Q_{λ})}{(P_{λ}, P_{λ})} = q^{−Nk} ∏_{α∈R^{+}} \frac{1 − q^{⟨λ+kp,α^{∨}⟩+k}}{1 − q^{⟨λ+kp,α^{∨}⟩−k}}

where as usual t = q^{k}, and N = card(R^{+}).

To conclude, we shall sketch a proof of Cherednik’s norm formula (3.1). The proof will be by induction on k, the cases k = 0 and k = 1 being trivial. From now on we shall write P_{λ,k} and Q_{λ,k} in place of P_{λ} and Q_{λ}, to stress the dependence on the parameter k, and likewise for the scalar product: (f, g)_{k} in place of (f, g). Let

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

Then the P ’s and Q’s are related as follows:

(5.7) For all λ ∈ P^{+}, we have

P_{λ,k+1} = π_{k}^{−1}Q_{λ+p,k}.

Taking λ = 0, it follows that Q_{p,k} = π_{k}. The formula (5.7) may be regarded as a generalization of Weyl’s character formula (3.5), which is the case k = 0.

From (5.7) we obtain

(5.8) \frac{(P_{λ,k+1}, P_{λ,k+1})_{k+1}}{(Q_{λ+p,k}, Q_{λ+p,k})_{k}} = q^{Nk} \frac{W_{0}(q^{k+1})}{W_{0}(q^{k})}.

Coupled with (5.6) (with λ replaced by λ + p) this gives

\frac{(P_{λ,k+1}, P_{λ,k+1})_{k+1}}{(P_{λ+p,k}, P_{κ+p,k})_{k}} = \frac{W_{0}(q^{k+1})}{W_{0}(q^{k})} ∏_{α∈R^{+}} \frac{1 − q^{⟨λ+(k+1)p,α^{∨}⟩+k}}{1 − q^{⟨λ+(k+1)p,α^{∨}⟩−k}}

and (3.11) follows by induction on k.

For simplicity of exposition we have restricted ourselves in this survey to affine root systems of the type S(R) (4.1). The general picture is that one can attach to any irreducible affine root system S, reduced or not, families of orthogonal polynomials P_{λ}, Q_{λ} and E_{λ} as above. These depend (apart from q) on as many parameters t_{i} as there are orbits in S under the affine Weyl group W_{S}, and the whole theory can be developed in this more general context. For an irreducible S, the maximum number of orbits is 5, and is attained by the (non-reduced) affine root systems denoted by C^{∨}C_{n}(n ≥ 2) in the tables at the end of [11]. Correspondingly, we have orthogonal polynomials P_{λ}, Q_{λ} and E_{λ} depending on q and five parameters t_{i}. These P_{λ} are the orthogonal polynomials defined by Koornwinder [10], which are therefore amenable to the Hecke algebra techniques described here. A full account will (eventually) appear in the book [17].

References

[1] G. E. Andrews, Problems and prospects for basic hypergeometric functions, in Theory and Applications of Special Functions, ed R. Askey, Academic Press, New York, 1975.

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[3] I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. Math. 141 (1995), 191–216.

[4] I. Cherednik, Macdonald’s evaluation conjectures and difference Fourier transform, Inv. Math. 122 (1995), 119–145.

[5] F. J. Dyson, Statistical theory of the energy levels of complex systems I, J. Math. Phys. 3 (1962), 140–156.

[6] F. Garvan and G. Gonnet, Macdonald’s constant term conjectures for exceptional root systems, Bull AMS 24 (1991), 343–347.

[7] L. Habsieger, La q-conjecture de Macdonald-Morris pour G_{2}, C. R. Acad. Sci. Paris, Sér. 1, Math. 303 (1986), 211–213.

[8] G. J. Heckman and E. M. Opdam, Root system and hypergeometric functions I, II, Comp. Math. 64 (1987), 329–373.

[9] K. Kadell, A proof of the q-Macdonald-Morris conjecture for BC_{n}, Memoirs of the AMS 108 (1994).

[10] T. H. Koornwinder, Askey–Wilson polynomials for root systems of type BC, in Hypergeometric functions on domains of positivity, Jack polynomials and application, ed. D. St. P. Richards, Contemp. Math. 138, AMS (1992).

[11] I. G. Macdonald, Affine root systems and Dedekind’s η-function, Inv. Math. 15 (1972), 91–143.

[12] I. G. Macdonald, Some conjectures for root system, SIAM J. Math. Anal. 13 (1982), 988–1007.

[13] I. G. Macdonald, Orthogonal polynomials associated with root systems, preprint (1987).

[14] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Sém. Bourbaki 797 (1995).

[15] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press (1995).

[16] I. G. Macdonald, Symmetric functions and orthogonal polynomials, AMS Univ. Lecture Series, 12 (1998).

[17] I. G. Macdonald, Affine root systems and affine Hecke algebras, in preparation.

[18] W. G. Morris, Constant term identities for finite and affine root systems: conjectures and theorems. Thesis, Madison (1982).

[19] E. M. Opdam, Root systems and hypergeometric functions III, IV, Comp. Math. 67 (1988), 21–49, 191–209.

[20] E. M. Opdam, Some applications of hypergeometric shift operators, Inv. Math. 98 (1989), 1–18.

I. G. Macdonald 8 Blandford Avenue Oxford OX2 8DY U. K.

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Documenta Mathematica · Extra Volume ICM 1998 · I