By I. G. Macdonald

A passable and coherent thought of orthogonal polynomials in different variables, hooked up to root structures, and reckoning on or extra parameters, has built lately. This finished account of the topic presents a unified starting place for the speculation to which I.G. Macdonald has been a central contributor. the 1st 4 chapters lead as much as bankruptcy five which includes all of the major effects.

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**Example text**

8) Ti commutes with both Y µ and Y ν , hence also with Y λ . 6), suppose ﬁrst that λ is dominant. Then µ = λ +si λ is also dominant, and <µ , αi > = 0. Let w = t(λ )si t(λ ) = si t(µ ). 1). 6) for λ dominant. If now λ is not dominant, let ν = λ − πi , so that <ν , αi > = 0. 5). 6). 6) when R is of type Cn , L = Q ∨ and αi is the long simple root of R. In that case <λ , αi > is an even integer for all λ ∈ L . 2) that u j = t(π j )v −1 j for j ∈ J . 9) U j = Y j T (v j )−1 for j ∈ J , where Y j = Y π j .

In particular, let w = v j ( j∈J ). 7) ∨ T0 = T (v j )T j−1 Y α j T (v j )−1 . 5) hold in B . 3). which are 3 The braid group We need only consider the braid relations that involve T0 , (a) T0 Ti = Ti T0 if <ϕ ∨ , αi > = 0, (b) T0 Ti T0 = Ti T0 Ti if <ϕ ∨ , αi > = <ϕ, αi∨ > = 1, (c) T0 Ti T0 Ti = Ti T0 Ti T0 if <ϕ ∨ , αi > = 1, <ϕ, αi∨ > = 2. 8), hence with T0 . 7) (1) ∨ T (si sϕ ) = Ti−1 T (sϕ ) = Ti−1 T0−1 Y ϕ . Let w = si sϕ si = w−1 . 6) (2) ∨ T (w) = Ti−1 T0−1 Ti Y si ϕ . 4) gives (3) ∨ T (w) = T0 T (sϕ w)Y −wϕ .

Hence (as the labels are all ≥0) λ− − ρk is the antidominant element of the orbit W0rk (λ ). So if rk (λ ) = rk (µ ) we must have λ− − ρk = µ− − ρk , hence λ− = µ− and v(λ ) = v(µ ), whence λ = µ . 6) Let λ ∈ L . If si λ = λ for some i ∈ I , then si (rk (λ )) ∈ rk (L ). Proof Suppose that si (rk (λ )) = rk (µ ) for some µ ∈ L . 5) we have si v(λ )−1 (λ− − ρk ) = v(µ )−1 (µ− − ρk ) from which we conclude that λ− = µ− and si v(λ )−1 = v(µ )−1 . Consequently µ = v(µ )−1 µ− = si v(λ )−1 λ− = si λ = λ .