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N], for type C, [−n, . . , −n] ≤ v ≤ [n, . . , n], for type D, [1−n, . . , 1−n] ≤ v ≤ [n−1, . . , n−1]. Under the operator w (−1) (w) ∆1♥ , such monomials are sent to 0, or to ±1 if they appear in the expansion of ∆♥ . One checks that in that last case, the coefficient is indeed the inversion weight I(w). QED For example, for type C2 , the contributing terms are x2,1 − x2,−1 h22 + x1,−2 h12 h22 − x−1,−2 h11 h12 h22 − x1,2 h21 + x−1,2 h21 h11 − x−2,1 h21 h11 h12 + x−2,−1 h21 h11 h12 h22 . One could have decided19 to denote the elements of the group by the element of the orbit of ρ♥ .

1−x1 . . xn ) )2 (λ1 −λ2 +1)(λ2 −λ3 +1) . (λn +1)sλ (x). 11) Proof. One can use induction on n, factorizing πω = πω πω , with ω = [n−1, . . , 1]. Thus one is left with computing the image under πω of the quotient of the two successive denominators appearing in the left-hand sides. For the first formula, it means computing (1 − x1 xn ) . . (1 − x1 xn−1 )(1 − x2n )(1 − x1 . . xn )−1 πω = (1 − xn e1 + · · · + (−xn )n en ))(1 − x1 . . xn )−1 πω , e1 , . . , en being the elementary symmetric functions in xn , and therefore commuting with πω .

Wj . ] → [. . − wj . . − wi . ] and I(sk w)/I(w) = hij . For example, for type C4 , one has the following chain of inversion factors : sC h22 s h s h 42 13 −[2, 3, 1, 4]←1−− −[1, 3, 2, 4] [2, 4, 1, 3]←4−−−[2, 4, 1, 3]←3−− s h sC h33 s h s h 23 34 43 ←2−− −[1, 2, 3, 4]←3−− −[1, 2, 4, 3]←4−−−[1, 2, 4, 3]←3−− −[1, 2, 3, 4] The inversions are more straightforward to read when writing the inverse elements : sC h22 s h s h 42 13 [3, 1, 4, 2]−1 ←4−−−[3, 1, 4, 2]−1 ←3−− −[3, 1, 2, 4]−1 ←1−− −[1, 3, 2, 4]−1 s h sC h33 s h s h 23 34 43 ←2−− −[1, 2, 3, 4]−1 ←3−− −[1, 2, 4, 3]−1 ←4−−−[1, 2, 4, 3]−1 ←3−− −[1, 2, 3, 4]−1 For each Weyl group of type ♥ = An−1 , Bn , Cn , Dn , Macdonald defined the following kernel18 M♥ , introducing formal parameters hji : (1 − hji xj x−1 i ) MA = 1≤i