ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

0. Plenary Lectures

1991 MS Classification: Macdonald, I. G., Queen Mary and Westfield College, University of London, England Constant Term Identities, Orthogonal Polynomials and Affine Hecke Algebras

Constant term indentities The archetype of the constant term identities to be considered is the following statement, first conjectured in 1961 by F. J. Dyson but proved soon afterwards. Let x_{1}, …, x_{n} be independent indeterminates and k a positive integer. Then the constant term in the Laurent polynomial

Δ_{k}(x) = ∏_{i≠j_{i,j=1}}^{n} (1\frac{x_{i}}{x_{j}} )^{k}  (1)

is equal to (nk)!/(k!)^{n}. Later, in 1975, G. E. Andrews proposed an extrapolation of (1) involving an extra parameter q. We shall use the notion

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

Clearly, as q → 1 we have (x; q)_{k} → (1x)^{k}, and (q; q)_{k}/(1 − q)^{k} → k! Let

Δ_{k}(x; q) = ∏_{i≠j_{i,j=1}}^{n} (ε_{ij} \frac{x_{i}}{x_{j}}; q)_{k}  (2)

where ε_{ij} = 1 or q according as i < j or i > j. Then Andrews conjectured that the constant term of (2) (i. e. , involving q but none of the x_{i}) should be

(q; q)_{nk}/(q; q)_{k}^{n}.  (3)

Clearly (2) reduces to (1) as q → 1, so that Andrews’ conjecture includes Dyson’s.

From our point of view, the polynomial Δ_{k}(x; q) is to be regarded as attached to the simple Lie algebra 𝔰𝔩_{n}(ℂ) consisting of the n × n complex matrices with trace zero. In fact each finite-dimensional complex simple Lie algebra 𝔤 gives rise to an analogous Laurent polynomial and constant term identity. Orthogonal polynomials Let Λ_{n} denote the ring of symmetric polynomials in x_{1}, …, x_{n} with coefficients in the field F = ℚ(q). For each partition λ = (λ_{1}, …, λ_{n}) of length ≤ n, let m_{λ} denote the monomial symmetric function indexed by λ, that is to say the sum of all distinct monomials obtainable from x_{1}^{λ_{1}}, …, x_{n}^{λ_{n}} by permuting the x_{i}, Clearly the m_{λ} form an F -basis of Λ_{n}. We shall use the polynomial Δ_{k} = Δ_{k}(x; q) to define a scalar product on Λ_{n}, as follows: if f, g ∈ Λ_{n} then

⟨f, g⟩_{k} = constant term in fḡΔ_{k}

where ḡ is obtained from g by replacing each x_{i} by x_{i}^{−1}. One shows then that there is a unique basis (P_{λ}) of Λ_{n}, indexed by partitions of length ≤ n, such that

(a) P_{λ} = m_{λ} + ∑_{μ<λ} a_{λμ}m_{μ}

with coefficients a_{λμ} ∈ F, where μ < λ means that μ precedes λ in the lexicographic ordering; and

(b) ⟨P_{λ}, P_{μ}⟩ = 0 whenever λ ≠ μ.

Moreover, there is a closed formula for the squared norm of P_{λ}, namely

⟨P_{λ}, P_{λ}⟩_{k} = \frac{(t; t)_{n}}{(1t)^{n}} ∏_{i<j_{i,j=1}}^{n} \frac{(q^{λ_{i}−λ_{j}}t^{j−i}; q)_{k}}{(q^{λ_{i}λ_{j}}t^{j−i}; q^{−1})_{k}}  (4)

where t = q^{k}. In particular, when λ = 0 we have P_{λ} = 1, and (4) then reduces to the constant term (3) of Δ_{k}. Again, all this can be done in the context of an arbitrary finite-dimensional simple Lie algebra 𝔤 in place of 𝔰𝔩_{n}, and thus for each such 𝔤 we have a family of orthogonal polynomials P_{λ}, symmetric under the Weyl group of 𝔤 and indexed by the dominant weights. Affine Hecke algebras In the case of 𝔰𝔩_{n} the norm formula (4) can be proved directly. For arbitrary 𝔤 there is an analogous formula which was first proved in full generality by Cherednik. Cherednik’s proof uses the affine Hecke algebra H attached to 𝔤 and its action on the space spanned by the polynomials P_{λ}; this leads to a family of commuting operators on this space whose simultaneous eigenfunctions are precisely the P_{λ}. The norm formula (4) – or rather its counterpart for arbitrary 𝔤 – can then be established by induction on k.