Macdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona

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1 Macdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdonald,, q., Heckman Opdam q,, Macdonald., 1

2 ,,. Macdonald, q, (+ ) Weyl q Laurent., A GL, P λ (x; q, t) P λ (x q, t)., x = (x 1,..., x n ) n, x i C, n (C ) n. λ = (λ 1,..., λ n ) λ i,, λ 1 λ 2... λ n 0, 0.,, (λ 1, λ 2,...), 0. Young. λ 1, λ 2,,, 0, λ. P λ (x; q, t), n λ = (λ 1,..., λ n ), K,., λ K[x] S n, Macdonald.,. q t, Q(q, t). q t,,.,, Macdonald (A GL )..,,,. Macdonald,., Macdonald, Heckman- Opdam 1987,8,. Macdonald p-adic, Heckman- Opdam, Laurent, Heckman-Opdama Laurent 2

3 , variation, q-analogue, 1987,8 Macdonald., 90 Macdonald,.,. ( ). q, Laurent (Fourier ),, ( q ),..,.?,, 1987,8 Macdonald,, q, Macdonald.,., 90, Cherednik, Macdonald q, Hecke ( (double) Hecke ). Hecke, q., Macdonald Cherednik ( Cherednik )., Cherednik Macdonald q, KZ 1, q Cherednik,, Macdonald. Macdonald, Hecke., Macdonald,, A GL, Hecke..,. (reduced), (BC ), Macdonald Koornwinder., Gauss Jacobi., Jacobi n α, β., A 1 Knizhnik-Zamolodchikov. 3

4 GL Macdonald, A 1, Jacobi., Jacobi?,, [0, 1] x α (1 x) β, [ 1, 1], α = β Gegenbauer, (ultra spherical). A 1 Macdonald, Gegenbauer q-analogue. Gauss Jacobi, BC. q Askey Wilson, Koornwinder,,.,, q. A GL, 1. Macdonald.. 2. Hecke (Macdonald-Cherednik ). Hecke Macdonald., Macdonald q., Ruijsenaars. BC, 1., 2..,.?? A Jacobi? x α (1 x) β, (f, g) := 1 0 f(x)g(x)x α (1 x) β dx, Jacobi.? 4

5 A., [ 1, 1] ( ) P n (α,β) (t) := (α + 1) n n, n + α + β + 1 2F 1 ; 1 t n! α [0, 1] t = 1 2x. Gauss, Jacobi., α, β > 1 (P m (α,β), P n (α,β) ) := = α+β+1 1 P (α,β) m 1 2n + α + β + 1 (t)p n (α,β) (t)(1 t) α (1 + t) β dt Γ(n + α + 1)Γ(n + β + 1) δ mn 0 Γ(n + α + β + 1)n!. Jacobi Gauss, (1 t 2 ) d2 y(t) dt 2 + {β α (α + β + 2)t} d y(t) dt + n(n + α + β + 1)y(t) = 0, ( t = 1 2x ).., Jacobi. α = β Gegenbauer. A Macdonald, Gegenbauer q- (A 1 )., ( ).,,, Jacobi?, 1 0 x α 1 (1 x) β dx, (x k, x l ) = 1 0 x α+k+l (1 x) β dx 5

6 .,,,,.??.,.. n ( ),,.,.,., closed.,., f g?.., (f, g),,.,.,.,.., 30.,. 1 Macdonald (A GL ) Macdonald q, q?, q,,,,,..? 80, 90? 6

7 Macdonald ( ),.., Macdonald, ( ) (q, t)-kostka,,.,. 1.1, A GL Macdonald.., A Weyl, Laurent,., K., q t,,. K,. n x 1,..., x n n, K[x] = K[x 1,..., x n ]. n, K[x] S n, n., S n n. n n, K[x] K K. σ S n, K, x i, f = f(x 1,..., x n ) K[x] : σ(f) := f(x σ(1),..., x σ(n) ).. S n K[x] K[x]. σ f σ(f) K[x] Sn := {f K[x] σ(f) = f} K[x] 7

8 , n.,,,. n = 1.,. 2,,., N, µ = (µ 1,..., µ n ) N n, n x := (x 1,..., x n ) µ x µ := x µ 1 1 x µn n. deg x µ = µ := µ µ n., S n, S n N n (or Z n )., σ µ := (µ σ 1 (1),..., µ σ 1 (n)) inverse,.,,, inverse., σ(x µ ) = x µ 1 σ(1) xµ n σ(n) = xµ σ 1 (1) 1 x µ σ 1 (n) n = x σ µ S n K[x], compatible. (monomial) L := N n,, L + := {λ = (λ 1,..., λ n ) L λ 1 λ n 0} 2,., Bourbaki.,.. 8

9 . L + λ, n, λ m λ (x) := x µ µ S n λ. S n L,,,, L +.,, {m λ (x)} λ L +., K[x] S n = λ L + K m λ (x) K[x] = λ L K x λ.,. λ = (l, 0,..., 0) =: (l) = }{{} l m (l) (x) = x l x l n =: p l (x) ( ), λ = (1,..., 1, 0,..., 0) =: (1 }{{} l ) = l m (1 r,0,...,0)(x) = λ = (2, 1, 0,..., 0) = 1 i 1 < <i r n }{{} m (2,1,0,...,0) (x) = l x i1 x in =: e r (x) 1 i,j n i j x 2 i x j. ( ), (dominance ordering) 3. Macdonald,, (dominance ordering) 4. lex µ 1 λ 1 µ 1 + µ 2 λ 1 + λ 2 µ λ. µ µ n 1 λ λ n 1 µ µ n = λ λ n ( µ = λ ) 4, dominance ordering. 9

10 ,,., (3, 3) (4, 1, 1) 3 4, > , ( 1.2 (1) )., µ, λ L. λ = (λ 1,..., λ n ) L + µ = (µ 1,..., µ n ) S n λ., λ 1 λ, µ 1 λ, λ 1 µ 1., λ 1 + λ 2 µ 1 + µ 2. µ 1 µ n, λ 1 λ n, λ 1 + λ 2 λ 1 λ n, µ 1 + µ 2., λ L + S n S n λ µ, λ µ.., i. ε i = (0,, 0, ˇ1, 0,, 0), P := Z n = n Z ε i., µ = (µ 1,..., µ n ) µ = n µ i ε i., α ij := ε i ε j = (0,..., 0, ˇ1, 0,..., 0, i j ˇ 1, 0,..., 0) (i < j),. α i := α ii+1 = ε i ε i+1 = (0,..., 0, ˇ1, i i+1 ˇ 1, 0,..., 0) (i = 1,..., n 1)., A GL., α ij = α i + + α j

11 1.2. µ, ν P (1) µ ν µ ν. lex (2) µ ν µ ν Q + µ ν = 1 i<j n k ijα ij (k ij N).,, Q n 1 Q := Z α i,. 5 n 1 Q + := N α i (2), λ Young α ij i, j.,. (6, 4, 3, 2) (4, 4, 4, 3), (6, 4, 3, 2) (4, 4, 4, 3), α 12 α 23 α 14.,,. 1.2 Schur.. Schur, Schur,., Schur. Macdonald,, Schur, Schur, Schur. Schur, Macdonald, ( ). 5 P. 11

12 Schur, n λ = (λ 1,..., λ n ) L + s λ (x),, s λ (x) := det (xλ j+n j i ) n i,j=1 (x) K[x] Sn.. (x), Vandermonde (x) = (x 1,..., x n ) := (x i x j ) = det (x n j 1 i<j n i ) n i,j=1, det (x λ j+n j i ) n i,j=1., (x), s λ (x) K[x]., Schur, Schur,,, 6. GL n (C) GL n (C), P +, Schur. T Young λ 1 n, ( ), ( )., λ = (2, 1),, T 1 = 1 1 2, 2 1 1, T 2 = 1 2 2, 2 1 2, SSTab n (λ) λ L + Young, µ i = µ i (T ) := T i, T wt(t ) = µ := (µ 1,..., µ n ). λ = (2, 1), { SSTab 2 ((2, 1)) = T 1 = 1 1, T 2 2 = 1 2 } 2 6,,,,. 12

13 , µ 1 (T 1 ) = 2, µ 2 (T 1 ) = 1, µ 1 (T 2 ) = 1, µ 2 (T 2 ) = 2, wt(t 1 ) = (2, 1), wt(t 2 ) = (1, 2),., Schur. s λ (x) = x wt(t ) T SSTab n (λ) Schur., ( ) x 3 1 x 3 2 det x 1 x 2 s (2,1) (x) = = x 2 x 1 x 1x 2 + x 1 x , T SSTab 2 ((2,1)) x wt(t ) = x wt(t 1) + x wt(t 2) = x (2,1) + x (1,2) = x 2 1x 2 + x 1 x 2 2 Schur., λ wt(t ) (T SSTab n (λ)).,, 1 1, T 1 µ 1 (T ) λ L + 1 λ , T 1 2 µ 1 (T ) + µ 2 (T ), λ L λ 1 + λ 2.. GL n GL n 1, Gelfand Tsetlin, Gelfand Tsetlin.. 7 Kostka Schur s λ (x) K[x] Sn = λ L K m + λ (x) m λ (x). s λ (x) = K λ,µ m µ (x) = K λ,µ x µ. µ L λ µ µ L + λ µ 7.,. 13

14 K λ,µ,, λ T wt(t ) µ. SSTab n (λ) wt=µ := {T SSTab n (λ) wt(t ) = µ} K λ,µ := SSTab n (λ) wt=µ., λ = (2, 1),, { SSTab 2 ((2, 1)) wt=(2,1) = T 1 = 1 1 }, SSTab 2 ((2, 1)) 2 wt=(1,2) = { T 2 = µ L., K (2,1),(2,1) = K (2,1),(1,2) = 1 0, K (2,1),µ m µ (x) = K (2,1),(2,1) m (2,1) (x) = m (2,1) (x) = x 2 1x 2 + x 1 x 2 2 = s (2,1) (x), µ L + (2,1) µ K (2,1),µ x µ = K (2,1),(2,1) x (2,1) + K (2,1),(1,2) x (1,2) = x 2 1x 2 + x 1 x 2 2 = s (2,1) (x) µ L (2,1) µ. Kostka,. }, 1.3 Macdonald A GL Macdonald,. q, t,, q (Macdonald ). 8 D x := n 1 j n j i tx i x j x i x j T q,xi., T q,xi i q-. T q,xi (f)(x) := f(x 1,..., qx i,..., x n ) 8 Jacobi,, q Macdonald. 14

15 D x T q,xi x, (x) t- T t,xi x i (x). D x T t,xi ( )(x) (x) D x = n = 1 j n j i tx i x j x i x j T t,xi ( )(x) T q,xi, (x)., (1) D x (K[x] S n ) K[x] S n. (2) D x K[x] Sn., D x (m λ (x)) = m µ (x)d µλ = m λ (x)d λ + m µ (x)d µλ., d λ := d λλ, µ L + λ µ d λ = n t n i q λ i. µ L + λ>µ,,...,. D x (Macdonald ),,. (1), f(x) K[x] S n, D x (f(x)), D x. D x (f(x)). D x (f(x)) = 1 (x) n., h(x), T t,xi ( )(x)t q,xi (f)(x) K(x) Sn 1 (x) K[x] D x (f(x)) h(x) (x),,, h(x)., g(x) h(x) = (x)g(x). D x (f(x)) = g(x),. 15

16 (2), Macdonald, 9. D x i, 1 j n j i tx i x j T q,xi = (tx i x 1 ) (tx i x i 1 )(tx i x i+1 ) (tx i x n ) x i x j (x i x 1 ) (x i x i 1 )(x i x i+1 ) (x i x n ) = t n i 1 j<i 1 tx i /x j 1 x i /x j i<j n 1 t 1 x j /x i 1 x j /x i T q,xi. x µ, Laurent, 1 j n j i, tx i x j x i x j T q,xi (x µ ) = t n i q µ i x µ = t n i q µ i x µ 1 j<i k=0 x µ 1 j<i ( 1 t x i x j 1 i<j n. 1.2 (2), µ ν µ ν = 1 tx i /x j 1 x i /x j )( xi x j ( ) kij xj x i 1 i<j n, x µ = x µ 1 1 x µ n n xν = x ν 1 1 x ν n n x ν = x µ 1 i<j n i<j n ) k i<j n l=0 1 t 1 x j /x i 1 x j /x i k ij α ij (k ij N), µ ν ( ) kij xj x i ( 1 t 1 x j x i T q,xi )( ) l xj, x µ x ν, µ ν., ( ) x µ lower., 1 j n j i tx i x j x i x j T q,xi (x µ ) = t n i q µ i x µ + (lower), i, ( n ) D x (x µ ) = t n i q µ i x µ + (lower). 9 Macdonald. x i 16

17 µ S n λ, D x (m λ (x)), 1.1 λ lower, 1.3 (1) K[x] S n = λ L K m + λ (x), λ lower, ( n ) D x (m λ (x)) = t n i q λ i m λ (x) + m µ (x)d µλ.. µ L + λ>µ Macdonald K := Q(q, t). λ L +, P λ (x) = λ µ D x P λ (x) = P λ (x)d λ, d λ = p µλ m µ (x) = m λ (x) + λ>µ p µλ m µ (x) = x λ + (lower terms), (1.1) n t n i q λ i. (1.2) P λ (x) = P λ (x; q, t). λ (A GL )Macdonald. 1.3,., λ L +, λ m µ (x) K- 10 K[x] S n λ := K m µ (x) µ L + µ λ. D x K[x] S n λ, {m µ(x)} µ, 1.3 (2),... dµ D x (..., m µ (x),..., m λ (x)) = (..., m µ (x),..., m λ * (x))... 0 d λ. λ µ, d λ d µ, D x, d λ. 10 m µ (x) m λ (x), Young,. 17

18 ,., D x, P λ (x) := µ L + µ<λ D x d µ d λ d µ m λ (x) K[x] Sn., 1.3 (2), P λ (x) = m λ (x) + p µλ m µ (x) K[x] Sn µ L + µ<λ. D x, Cayley Hamilton det (z id K[x] Sn D x ) = (z d µ ) λ µ L + µ λ (D x d λ )P λ (x) = 0. P λ (x) (1.1), (1.2), (1.1), d λ Pλ (x) = ν λ d λ p νλ m ν (x). (1.2) 1.3 (2), d λ Pλ (x) = D x Pλ (x) = ν λ p νλ D x m ν (x) = ν λ = µ λ p νλ d µν m µ (x) µ ν d µ p µλ m µ (x) + {m ν (x)} µ K[x] S n K, (d µ d ν )p µλ = µ<ν λ µ<ν λ d µν p νλ m µ (x). d µν p νλ, p λλ = 1. P λ (x) p µλ,. 18

19 , Macdonald...,. 1.5 (2 Macdonald ). φ(x 1, x 2 ) = x λ 1 1 x λ 2 2 k=0 ( ) k x2 c k x 1, D x φ(x 1, x 2 ) = φ(x 1, x 2 )(tq λ 1 + q λ 2 ). (q-)gegenbauer P (r) (x), P (1 r,0,...,0)(x) n Macdonald. 2,. Macdonald. [M1] I. G. Macdonald: Symmetric Functions and Hall Polynomials, Second Edition, Oxford University Press, bible, 6 q, t A GL, Macdonald. [M2] I. G. Macdonald: Symmetric functions and orthogonal polynomials, University Lecture Series, 12 American Mathematical Soc., [M1] [M3] 19

20 . [M3] I. G. Macdonald: Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Math. 157 Cambridge University Press, Cherednik Hecke Macdonald., A GL 1. Macdonald. 2. Hecke (Macdonald-Cherednik ).,.,, 2. ( )? Hecke,.,., 1.,,,,.,. ( ). Hecke, [M3], [M2]., [ 1] : Hecke Macdonald, , , ( ), Hecke., [ 2] : Hecke Macdonald Cherednik, 1997,. 20

21 ,., A, BC Koornwinder, Hecke, q, ,,,.., P λ (x) = P λ (x; q, t) K[x] S n n, λ L + := {λ = (λ 1,, λ n ) N n λ 1 λ n 0} n {P λ } λ L +,, q., D x, D x = D x P λ (x) = P λ (x)d λ ( n j i tx i x j x i x j ) T q,xi q. d λ, n d λ = t n i q λ i = t n 1 q λ 1 + t n 2 q λ q λ n., (t n 1 q λ 1, t n 2 q λ 2,, q λ n ) n, d λ, 1., t δ q λ. δ Macdonald, δ = (n 1, n 2,..., 0). staircase,. 3 Macdonald ( ) 3.1 q n = 2, 11 yanagida/others-j.html 21

22 . q,, q.,. 2ϕ 1 q, Gauss q-analogue,., 2ϕ 1 ( a, b c ; q, z ) = k=0 (a; q) k (b; q) k (q; q) k (c; q) k z k ( q < 1, z < 1) (a; q) k = (1 a)(1 qa) (1 q k 1 a) (k = 0, 1,...) q-analogue.,. (a; q) = (1 q i a) i=0 q < 1, a.. (α) k = α(α + 1) (α + k 1) Pochhammer symbol, (a; q) k q- Pochhammer symbol, Askey Pochhammer (α) k Pochhammer symbol., Pochhammer symbol.,. Askey shifted factorial. q-shifted factorial (a; q) k, shifted factorial (α) k q-analogue,,. 1 qα, q R 1 q, q 0 1, 1 qα 1 q α., a = q α (a; q) k (1 q) k = (qα ; q) k (1 q) k (α) k. 2ϕ 1, a = q α, b = q β, c = q γ, 1 q, q-shifted factorial shifted, 2 ϕ 1 Gauss 2 F 1. q-analogue.,. q-analogue q, ϕ D.. ( ) a, b 1,..., b n ϕ D ; q; z 1,..., z n = (a; q) µ (b 1 ; q) µ1 (b n ; q) µn z µ 1 1 z µ n n c (c; q) µ N n µ (q; q) µ1 (q; q) µn 22

23 (Lauricella )F D q-analogue,., Macdonald, q,, n. (1) 1., ϖ r = ε ε r = (1, 1,..., 1, 0,, 0) = (1 }{{} r ) = r }{{} r α 1 = ε 1 ε 2,..., α n 1 = ε n 1 ε n, α n = ε n. α n 1, GL 1, α n., ϖ i, α j = δ i,j (i, j = 1,..., n).,.,,, 1.,,,,.,,, (1 r ). (1 r ), minimal. Macdonald, P (1 r )(x; q, t) = m (1 r )(x) = e r (x)., minimal A,. (2) 1 λ = (l, 0,, 0) = (l) = }{{} l 23

24 l. Schur l, l 1, Macdonald,. P (l) (x; q, t) = const. µ 1 + +µ n=l (t; q) µ1 (t; q) µn x µ 1 1 x µn n (q; q) µ1 (q; q) µn µ µ n = l (l, 0,..., 0), (t; q) µ1 (t; q) µn x µ 1 1 x µ n n (q; q) µ1 (q; q) µn µ 1 + +µ n =l (t;q) l (q;q) l x l 1. Macdonald 1,. P (l) (x; q, t) = (q; q) l (t; q) l µ 1 + +µ n =l (t; q) µ1 (t; q) µn x µ 1 1 x µ n n (q; q) µ1 (q; q) µn n, l, n x l 1, µ 1 = l µ 2 µ n q-shifted factorial, µ 2 µ n,. P (l) (x; q, t) = x l 1 µ 2,,µ n 0 (q l ; q) µ2 + +µ n (t; q) µ2 (t; q) µn (q 1 l t 1 ; q) µ2 + +µ n (q; q) µ2 (q; q) µn ( qx2 ϕ D, ( ) P (l) (x; q, t) = x l q l, t,..., t 1ϕ D ; q; qx 2,..., qx n q 1 l t 1 tx 1 tx 1 tx 1 ) µ2 ( qxn. 1 Macdonald. F D. 1, D x, ( 5.1).,,. n = 2, 2 ϕ 1.. D x? tx 1 ) µn 24

25 , Macdonald,, D x ( ) m n Macdonald,,. BC,,,.,?.,.. 4 Macdonald, Macdonald. K[x] S n,, Macdonald., Macdonald, Macdonald. K[x] Sn = λ L + KP λ (x), D x K[x] Sn, 1,. 4.1 ( ) q, t Macdonald. t = 1 (t = q κ, κ = 0 ) n D x = T q,xi, P λ (x; q, 1) = m λ (x) t = q (t = q κ, κ = 1 ) P λ (x; q, q) = s λ (x) Schur 25

26 κ = 1 κ = 2,. 2 q = 0 Hall Littlewood t = 0 q-whittaker q t duality Hall Littlewood q-whittaker, q t. q = 0 q-whittaker, dual Whittaker.. t = q κ, q 1 lim q 1 P λ (x; q, q κ ) = P λ (x; κ) Jack, κ = 0, κ = 1 Schur., κ = 1 zonal,. 2 zonal., t = q 1 2, zonal q-analogue. κ = 2, κ = 1 SO 2, κ = 2 Sp. Macdonald q A Heckman Opdam, Jack., Heckman Opdam A Jacobi Jack., 2 F 1, BC, A. Jack?. Jack James., GL n, SO radial part, zonal. SO. 4.2 q Macdonald D x 1, D x, q 26

27 . r = 0, 1,..., n, D (r) x = t (r 2) I {1,...,n}, I =r i I,j / I. r, T I q,x = i I T q,xi tx i x j x i x j i I T q,xi, D (r) x A I (x; t) = T I t,x (x) (x) = t (r 2) i I,j / I. D x (r) = A I (x; t) Tq,x I I {1,,n}, I =r tx i x j x i x j, D x,,,., D (0) x = 1, D (1) x = D x, D x (2),..., D (n) x = t (n 2) Tq,x1 T q,xn ( Euler ). 0 1 Euler, 2, 1, Euler..,. D x (u) := n r=0 ( u) r D (r) x 4.1 (Ruijsenaars, Macdonald). (1) : D x (r) D x (s) = D x (s) D x (r) (r, s = 0, 1,..., n), D x (u)d x (v) = D x (v)d x (u). (2) Macdnald D (r) x (r = 0, 1,..., n) : D (r) x P λ (x) = P λ (x)e r (t δ q λ ), t δ q λ = (t n 1 q λ 1,..., tq λ n 1, q λ n ), n ( D x (u) P λ (x) = P λ (x) 1 ut n i q ) λ i. 27

28 , Macdonald 1, n q, Macdonald,. D x = D x (1) 1, 1. D x (r), D x (r) D x. Macdonald D x (r) (r = 0, 1,..., n),.,., Ruijsenaars 1987.,,,,. r s,,.,,.?. Ruijsenaars, Macdonald,,. Ruijsenaars, relativistic, Ruijsenaars model, Calogero-Moser.? Ruijsenaars, Corollary., Macdonald. Macdonald, D x (r) (r = 0, 1,..., n),,, D x,. D x (r) (r = 0, 1,..., n).,. 28

29 4.3 Macdonald K = C, q, t R, 0 < q < 1, 0 < t < 1. R[x]. f, g R[x], f(x), g(x) = 1 1 n! (2π f(x 1 )g(x)w(x) dx 1 dx n 1) n T x n 1 x n, T n = {x = (x 1,..., x n ) C n x 1 = = x n = 1} n. q-analogue Jackson, Macdonald, Jackson, n.?, K = C, Hermite, f(x 1 ) f(x). Macdonald q, t Q, q, t R,, K = R., w(x) = 1 i<j n (x i /x j ; q) (tx i /x j ; q) } {{ } w + (x) (x j /x i ; q) (tx 1 i<j n j /x i ; q) }{{} w (x). x i /x j = x ε i ε j ε i ε j,., x i /x j (i < j). 1 w + (x), w (x). (x;q) x < 1, x Re(t) < 1 w(x),. w(x), D x (r) (r = 0,..., n)., Af, f = f, A g A = A,.. ϖ r = ε ε r, w + (x). T ϖ r q,x (w + (x)) w + (x) = const. 1 i r,r+1 j n tx i x j x i x j..,, 29

30 . q-shift. q,,, Cauchy,. q-shift, D x (r) S n., f(x 1 )g(x)w(x) = f(x 1 )w + (x 1 )g(x)w + (x) Tq,x ϖr, D(r) x. r = 1, D x ( 5.2)., D x P λ (x), P µ (x) = P λ (x), D x P µ (x) d λ P λ (x), P µ (x) = d µ P λ (x), P µ (x)., d λ d µ P λ (x), P µ (x) = 0., Macdonald, Macdonald., Jackson, Jackson.,,., Laurent 1,. 1, 1,,. 1, 1 = n (t; q) (qt i 1 ; q) (q; q) (t i ; q),. P λ, P λ. Kadell.,.,, Cherednik, E 8,. Cherednik Hecke,,, E 8. 30

31 ,?.,. Cherednik. 4.4 ( ), δ = (n 1, n 2,..., 0)., λ = 0 t δ q λ, t δ = (t n 1, t n 2,..., 1). δ A Macdonald, ρ ( ) ( ) P λ (t δ ) = t n(λ) 1 i<j n (t j i+1 ; q) λi λ j (t j i ; q) λi λ j (, n(λ) := ) n (i 1)λ i, 1 Gauss. Macdonald,. ( ) P λ(t δ q µ ) = P µ(t δ q λ ) (λ, µ L + ) P λ (t δ ) P µ (t δ ) P λ (x) := P λ(x) 1, P λ (t δ ) P λ (t δ q µ ) = P µ (t δ q λ ). λ, x, t δ q µ, ( ).,. x (e λx ) = λe λx, λ (e λx ) = xe λx,. dual. q. Jacobi, x Gauss n n,, Macdonald., 31

32 ,, q,. 4.5 Pieri Macdonald, Macdonald. P µ (x)p ν (x) = c λ µ,νp λ (x) λ, c λ µ,ν.,, µ, ν,,, character, Schur. c λ µ,ν Littlewood-Richardson. Cherednik?..,. 1, e r (x)p µ (x) = φ λ/µp λ (x) 1, P (l) (x) P µ (x) = µ λ L + λ µ:vertical r-strip µ λ L + λ µ:horizontal l-strip φ λ/µ P λ (x), φ λ/µ φ λ/µ. 1 Pieri., λ µ : vertical r-strip, λ µ, 0 1, r, horizontal l-strip, λ µ, 0 1, l. r l. 32

33 λ µ : vertical 3-strip µ = λ = λ µ : horizontal 3-strip µ = λ =, m + n Macdonald, x (m ) y (n ), x Macdonald y Macdonald,. P λ (x 1,..., x m, y 1,..., y n ) = µ,ν a λ µ,νp µ (x 1,..., x m )P ν (y 1,..., y n ) Schur, GL m+n GL m GL n,,., n Macdonald,. P λ (x 1,..., x n ) = ψ λ/µ = i<j µ λ L + λ µ:horizontal-strip P µ (x 1,..., x n 1 )ψ λ/µ x λ µ m (q µ i λ j +1 t j i+1 ; q) λi µ i (q µ i µ j t j i ; q) λi µ i (q µ i λ jt j i ; q) λi µ i (q µ i µ j +1 t j i ; q) λi µ i 1, n Macdonald.,. P λ (x) = n i j µ ψ (k 1) µ (k) /µ (k 1)x µ(k) k ϕ=µ (0) µ (1) µ (n) =λ k=1, horizontal strip. horizontal strip, λ T SSTab n (λ). T 1 µ (1), 2 µ (2),..., ϕ 33

34 λ ϕ = µ (0) µ (1) µ (2) µ (n) = λ, horizontal strip, 1 1.,, SSTab n (λ) x,, Schur Macdonald. Macdonald (bible). 4.6 (Cauchy ). ( 5.1). n Macdonald 1, x n y 1., m x = (x 1,..., x m ) Macdonald n y = (y 1,..., y n ) Macdonald. m n j=1 (tx i y j ; q) (x i y j ; q) = l(λ) min{m,n} b λ P λ (x 1,..., x m )P λ (y 1,..., y n ) 1,. b λ,., Macdonald, Macdonald,, Littlewood-Richardson dual. Schur Cauchy m n j=1 1 1 x i y j = l(λ) min{m,n} s λ (x 1,..., x m )s λ (y 1,..., y n ) (1) q. ( q < 1, x < 1.) (ax; q) (x; q) = k=0 (a; q) k (q; q) k x k 34

35 (2) n x = (x 1,..., x n ) 1 y n (tx i y; q) = g l (x; q, t)y l (x i y; q) g l (x; q, t). (3). D x n (tx i y; q) (x i y; q) = l=0 (t n 1 T q,y + 1 ) n tn 1 1 t 5.2. D x. ( ) (tx i y; q) (x i y; q) 5.1. (1) x Taylor. x q. (t; q) µ1 (t; q) µn (2) g l (x; q, t) = x µ 1 1 x µ n n. (q; q) µ1 (q; q) µn µ 1 + +µ n =l (3).. P (l) (x). (1) a = q α, q 1, Newton (1 x) α (α) k = x k (1) k. (2) t = q κ, q 1 n (1 x iy) κ. y n (1 x i y) κ = g l (x)y l, g l (x) = l=0 g l (x) = Res y=0 ( y l 1 n k=0 µ 1 + +µ n=l ) (1 x i y) κ dy (κ) µ1 (κ) µn (1) µ1 (1) µn x µ 1 1 x µ n n. (l = 0, 1, 2,...) (5.1), y Jordan-Pochhammer. x = (x 1,..., x n ), F D, g l (x) F D. ( ) 35

λ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T

λ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T 0 2 8 8 6 3 0 0 Young Young [F] 0.. Young λ n λ n λ = (λ,, λ l ) λ λ 2 λ l λ = ( m, 2 m 2, ) λ = n, l(λ) = l {λ n n 0} P λ = (λ, ), µ = (µ, ) n λ µ k k k λ i µ i λ µ λ = µ k i= i= i < k λ i = µ i λ k >