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  "body": "\n## Statement\n\nLet $R$ be a reduced finite root system of rank $l$ in a Euclidean space $V$, with Weyl group $W$.\nLet $d_1, \\dots, d_l$ be the degrees of the basic polynomial invariants of $W$.\nFor $\\alpha \\in R$, let $e^\\alpha$ be the corresponding element of the group algebra $\\Q[P]$ of the weight lattice $P$ of $R$.\nLet $k \\ge 0$ be an integer.\nThen the constant term (the coefficient of $e^0$) of\n$$\\prod_{\\alpha \\in R} \\left( 1 - e^\\alpha \\right)^k$$\nequals\n$$\\prod_{i=1}^{l} \\binom{k\\, d_i}{k}.$$\n\n## Remarks\n\nMacdonald 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|Y1X8ZZQ7VM]] and the [[Gaussian binomial coefficients|YS09Z1YNEK]].\nCherednik proved both versions for all reduced root systems in 1995, with double affine Hecke algebras.\nExample: $R = A_1 = \\set{\\alpha, -\\alpha}$, with $l = 1$ and $d_1 = 2$.\nWriting $x = e^\\alpha$, 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.\nFor $R = A_{n-1}$ the degrees are $2, 3, \\dots, n$, and the formula is the equal-parameter case of the [[Dyson conjecture|NDYA85381W]], since $\\prod_{j=1}^{n} \\binom{jk}{k} = \\frac{(nk)!}{(k!)^n}$.\n\n## Explanations\n\n### L1 human\n\nEvery root system, such as the one behind the symmetries of a square or a cube, comes with a reflection group and a list of numbers called degrees.\nTake the product of the factors $1 - e^\\alpha$ over all roots, raise it to the power $k$, and expand.\nThe constant term is a product of binomial coefficients built from the degrees.\n\n### L2 human\n\nThe constant term of a Laurent polynomial is its average over a torus.\nSo the statement evaluates a Selberg-type integral over the maximal torus of a [[compact Lie group|9TFC6ADC1C]] whose [[Weyl group|EZGTEK3HQN]] is $W$.\nThe degrees enter because $\\prod_i d_i = \\abs{W}$ and $\\sum_i (d_i - 1)$ is the number of positive roots.\nThe analogous Gaussian integrals are the Macdonald–Mehta integrals, of which the [[Mehta integral|4N53T1BZ9S]] is the type $A$ case.\n",
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