---
tag: 5J25W265JA
kind: statement
role: theorem
name: Macdonald constant term conjecture
msc: [33C52, 05A19, 17B22]
uses: [G6FND7X69J, EZGTEK3HQN, 0GBZM0MT0F, 9TFC6ADC1C]
about: [G6FND7X69J]
sources:
  - url: https://en.wikipedia.org/wiki/Macdonald_polynomials
---

## Statement

Let $R$ be a reduced finite root system of rank $l$ in a Euclidean space $V$, with Weyl group $W$.
Let $d_1, \dots, d_l$ be the degrees of the basic polynomial invariants of $W$.
For $\alpha \in R$, let $e^\alpha$ be the corresponding element of the group algebra $\Q[P]$ of the weight lattice $P$ of $R$.
Let $k \ge 0$ be an integer.
Then the constant term (the coefficient of $e^0$) of
$$\prod_{\alpha \in R} \left( 1 - e^\alpha \right)^k$$
equals
$$\prod_{i=1}^{l} \binom{k\, d_i}{k}.$$

## 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|Y1X8ZZQ7VM]] and the [[Gaussian binomial coefficients|YS09Z1YNEK]].
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|NDYA85381W]], since $\prod_{j=1}^{n} \binom{jk}{k} = \frac{(nk)!}{(k!)^n}$.

## Explanations

### L1 human

Every 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.
Take the product of the factors $1 - e^\alpha$ over all roots, raise it to the power $k$, and expand.
The constant term is a product of binomial coefficients built from the degrees.

### L2 human

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|9TFC6ADC1C]] whose [[Weyl group|EZGTEK3HQN]] 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|4N53T1BZ9S]] is the type $A$ case.