Macdonald constant term conjecture

Remarks

Macdonald stated this conjecture in 1982, together with a $q$-version: the constant term of $\prod_{\alpha \in R_+} (e^\alpha; q)_k\, (q e^{-\alpha}; q)_k$ is $\prod_{i=1}^{l} \binom{k d_i}{k}_q$, with the q-Pochhammer symbol and the Gaussian binomial coefficients. Cherednik proved both versions for all reduced root systems in 1995, with double affine Hecke algebras. Example: $R = A_1 = \set{\alpha, -\alpha}$, with $l = 1$ and $d_1 = 2$. Writing $x = e^\alpha$, the product is $(1 - x)^k (1 - x^{-1})^k = (-1)^k x^{-k} (1 - x)^{2k}$, whose constant term is $(-1)^k (-1)^k \binom{2k}{k} = \binom{2k}{k}$, as predicted. For $R = A_{n-1}$ the degrees are $2, 3, \dots, n$, and the formula is the equal-parameter case of the Dyson conjecture, since $\prod_{j=1}^{n} \binom{jk}{k} = \frac{(nk)!}{(k!)^n}$.

Going deeper

For a student

The constant term of a Laurent polynomial is its average over a torus. So the statement evaluates a Selberg-type integral over the maximal torus of a compact Lie group whose Weyl group is $W$. The degrees enter because $\prod_i d_i = \abs{W}$ and $\sum_i (d_i - 1)$ is the number of positive roots. The analogous Gaussian integrals are the Macdonald–Mehta integrals, of which the Mehta integral is the type $A$ case.