X = E ij (1.3 L Eij = n x jν x ν=1 iν n n (1.4 (E ij = t ( ( x ij, x ij ( t ( t(l Eij = x ij. x ij g G U(g g m m=0 g X Y Y X [X, Y ] X, Y g g G U(g Ad
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- しのぶ おなか
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1 1. GL(n GL(n Lie GL(n, C Lie 1 Lie G = GL(n, R GL(n, C G G X M(n, C ϕ(x d dt ϕ(xetx t=0 = d dt ϕ(x + txx t=0 M(n, C C (i, j 1 0 E ij ( n ν, µ=1 x νµe νµ E ij = n ν=1 x νie νj (1.1 E ij = Lie n x νi x ν=1 νj (1.2 [E ij, E kl ] = δ jk E il δ li E kj g G = GL(n, C Lie gl n g M(n, C = n i, CE ij G ϕ g G π g ϕ(x = ϕ(g 1 x X M(n, C 1 t e tx L X
2 X = E ij (1.3 L Eij = n x jν x ν=1 iν n n (1.4 (E ij = t ( ( x ij, x ij ( t ( t(l Eij = x ij. x ij g G U(g g m m=0 g X Y Y X [X, Y ] X, Y g g G U(g Ad(g X M(n, C g X Ad(gX = gxg 1 2. g = M(n, C P (g g Ad(g A M(n, C A A(t = Ad(e tx A X g A U(g G g g g (2.1 X, Y = Trace XY P (g g S(g V A = g G Ad(gA. V A G S(g U(g G R U(g G U(g S(g cf. [O5] (2.2 U ϵ (g := ( k=0 k g / X Y Y X ϵ[x, Y ]; X, Y g ϵ U(g = U 1 (g S(g = U 0 (g ϵ 0 U 1 (g X ϵx U ϵ (g g ϵ 1 (2.2 ϵ U ϵ (g = U(g[ϵ] (Poincaré-Birkhoff-Witt V A C n n S n λ = (λ 1,..., λ n C S n λ n! (2.3 s j (x s j (λ (j = 1,..., n, s j (x = x i1 x ij 1 i 1 < <i j n S j (x = n i=1 xj i 2
3 n (2.4 (x i λ j (j = 1,..., n, (2.5 i=1 n (x i λ j (i = 1,..., n. λ generic (2.4 λ = (µ,..., µ, ν,..., ν n! }{{}}{{} k!(n k! k n k (2.4 (2.5 (2.6 (2.7 (x i1 µ (x in k+1 µ, (x j1 ν (x jk+1 ν (1 i 1 < < i n k+1 n, 1 j 1 < < j k+1 n, (x i µ(x i ν (i = 1,..., n µ ν (2.6 (2.7 j = 1 ( (2.3 S n [OS] cf. [OP] Heckman-Opdam [HO] (2.3 x i S n (x, λ [O2] λ = 0 S n x i 3. Generalized Verma Modules n {n 1,..., n L } n j = n n j (1 j L, n 0 = 0, (3.1 Θ = {n 1, n 2,..., n L }, ι Θ (ν = j if n j 1 < ν n j (1 ν n Θ = {n 1 < n 2 < < n L = n} n g Lie n Θ, n Θ m Θ ι Θ (i > ι Θ (j ι Θ (i < ι Θ (j ι Θ (i = ι Θ (j E ij p Θ = m Θ + n Θ m k Θ = ι CE Θ(i=ι Θ(j=k ij n = 1 j<i n CE ij, n = 1 i<j n CE ij a = n CE jj p = a + n m Θ = m 1 Θ ml Θ p Borel Θ p Θ Borel p Θ = {X g; X, Y = 0 ( Y n Θ } λ = (λ 1,..., λ L C L g affine (3.2 A Θ,λ := n λ ιθ(je jj + n Θ λ 1 I n 1 A 21 λ 2 I n 0 2 = A 31 A 32 λ 3 I n 3 ; A ij M(n i, n j; C. A L1 A L2 A L3 λ L I n L I m m M(k, l; C k l 3
4 3.1. A Θ,λ Jordan (3.3 µ C, 1 k n J ( #{i; λ i = µ n i k}, µ µ 1 µ 0 J(m, µ = M(m, C 1 µ Jordan Θ λ. f U 0 (g = S(g ϵ = 0 f ( ( Ad(gA Θ,λ = 0 Ad(gf (AΘ,λ = 0 ( g G g G Ad(gf JΘ(λ ϵ ( g G f Ad(gJΘ(λ ϵ g G f Ann G ( M ϵ Θ (λ JΘ(λ ϵ := U ϵ ( (g X λθ (X, X p Θ MΘ(λ ϵ := U ϵ (g/jθ(λ, ϵ Ann ( M ϵ Θ(λ := { D U ϵ (g; DM ϵ Θ(λ = 0 }, I ϵ Θ(λ := Ann G ( M ϵ Θ (λ := { D U ϵ (g; Ad(gD Ann ( M ϵ Θ(λ ( g G } p Θ C Lie 1 λ Θ (3.4 λ Θ (Y + L X k := k=1 L λ k Trace(X k for X k m k Θ and Y n Θ. k=1 ϵ = 1 1 M Θ (λ = MΘ 1 (λ A Θ,λ V AΘ,λ g G Ad(gJ Θ 0 (λ g G Ad(gJ ( Θ(λ = Ann G MΘ (λ = Ann ( M Θ (λ IΘ ϵ (λ M Θ(λ m Θ λ Θ Verma Verma (3.5 M(λ Θ := U(g/J(λ Θ, J ϵ (λ Θ := X p U ϵ (g ( X λ Θ (X and J(λ Θ = J 1 (λ Θ. g Θ = {1, 2,..., n} p Θ = p M ϵ (λ Θ M ϵ (λ ϵ = 0 ϵ = 1 Verma 4
5 4. Harish-Chandra n n U 0 (g G Z 0 (g Z ϵ (g = {D U ϵ (g; Ad(gD = D ( g G} ϵ = 1 Z(g g (Ad(gE ij = t g 1 E t g E = (E ij (4.1 Z k := Trace E k (k = 1, 2,... Z k Z ϵ (g cf. Gelfand [Ge] ϵ = 0 k Harish-Chandra (4.2 γ : U ϵ (g D Γ(D U ϵ ( (a = S(a D Γ(D nu ϵ (g + U ϵ (gn U ϵ (g = U ϵ ( n U ϵ (a U ϵ (n U ϵ (a a U ϵ (a = S(a D = D + D, ϵρ := ϵ n n+1 (j 2 E jj Γ(D = γ(d Harish-Chandra (4.3 Γ : Z ϵ (g S(a Sn. Z ϵ (g Z 1,..., Z n Γ(Z k 7 [Go1] [Ca1] 100 γ ( det(e, t = n i=1( Eii t + ϵ(n i (4.4 det(e, t := det (E ij t + ϵ(n iδ ij Z ϵ (g ( t C. (4.5 det (A ij = sgn(σa σ(11 A σ(nn. σ S n Capelli E ij E ij tδ ij (4.4 ( (4.6 det (x ij det = det (E ij iδ ij. x ij 4.1. i (4.4 t k k (4.7 Z ϵ (g = C[Z 1,..., Z n ] = C[ 1,..., n ]. ii (4.1 Lie cf. 7 (4.4 o n cf. [HU] [Wa] iii λ = (λ 1,..., λ n ϵ = 0 (4.8 I ϵ λ := { D U ϵ (g; γ(ad(gd, λ = 0 ( g G } 5
6 G B(G Verma M(λ g L(λ I λ = Ann ( L(λ V {1,...,n},λ {I λ ; λ C n } g [Du] w S n w.λ = w(λ + ϵρ ϵρ ϵ = 0 generic λ I w.λ = I λ (Z 1,..., Z n λ = ρ λ ( iv Verma M ϵ (λ Ann G M ϵ (λ D γ(d, λ D Z ϵ (g gl n k γ( k, λ k = 1,..., n 5. Generalized Capelli Elements rank (5.1 A {k,n} (µ, ν = ( µik νi n k rank ( A {k,n} (µ, ν µ n k, rank ( A {k,n} (µ, ν ν k ϵ = 0 ( E ij µ n k + 1 V A{k,n} (µ,ν ( (5.2 D{i ϵ (t := det 1,...,i m}{j 1,...,j m} E ip,i q + ( ϵ(m q t δ ip i q 1 p m U ϵ (g 1 q m, Capelli { DIJ ϵ ; #I = #J = m, I, J {1,..., n} } G G ϵ [x ij, µν ] = ϵδ iµ δ jν Capelli [O4] (5.3 = ( n det x νik νjl + ϵ(m lδ ik j l 1 k m ν=1 1 l m det (x νp i q 1 p m det ( νp i q 1 q m 1 ν 1 < <ν m n 1 p m 1 q m ϵ (5.4 D ϵ IJ(µ, D ϵ I J (ν + kϵ (#I = #J = n k + 1, #I = #J = k + 1 IΘ ϵ (λ cf. (2.6 ϵ = 0 µ ν IΘ ϵ (λ (5.5 µ ν / {ϵ, 2ϵ,..., (n 1ϵ} IΘ ϵ (λ µ = ν 6
7 ([O5]. MΘ ϵ (λ m = 1,..., n d ϵ m(x := d ϵ m(x; Θ, λ = L ( (n x j λj ϵn +m n j 1, (6.1 d m = d m (Θ := deg x d ϵ m(x; Θ, λ = L max{n j + m n, 0}, e ϵ m(x := e ϵ m(x; Θ, λ = d ϵ m(x/d ϵ m 1(x, q ϵ (x := q ϵ (x; Θ, λ = L ( x λj ϵn j 1. d ϵ m(x MΘ ϵ (λ m, {eϵ m(x; 1 m n} MΘ ϵ (λ, q ϵ (x MΘ ϵ (λ dϵ n(x MΘ ϵ (λ { (6.2 z (l z ( z ϵ (z ϵ(l 1 if l > 0, := 1 if l ϵ = A Θ,λ U ϵ (g d 0 m(x xi n A Θ,λ m 6.3 ([O5]. d ϵ m(x = k m ν=1 (x λ m,ν N m,ν ν ν λ m,ν λ m,ν (6.3 V ϵ Θ(λ := n k m m=1 ν=1 N m,ν 1 j=0 #I=#J=m ( d j C dx IJ(x j Dϵ x=λm,ν IΘ ϵ (λ = U ϵ (gvθ ϵ(λ dϵ n(x = 0 regular ϵ = 0 Jordan L (6.4 IΘ(λ ϵ = U ϵ (gdij(λ ϵ k + ϵn k 1. k=1 #I=#J=n+1 n k 6.4. i I ϵ Θ (λ Iϵ Θ (λ d ϵ m(x; Θ, λ d ϵ m(x; Θ, λ (m = 1,..., n ϵ = 0 A Θ,λ g G Ad(gA Θ,λ ϵ = 0, λ = 0 d m (Θ d m (Θ m = 1,..., n ii ϵ = 0, λ = [Ta1] [We] (6.3 ϵ = 0, λ = 0, Θ = {n} [Ko] GL(n normal variety [KP] 7. (5.1 A {k,n} (µ, ν (x µ(x ν (E µ(e ν V A{k,n} (µ,ν µ ν n 2 Trace E kµ (n kν I{k,n} 0 (µ, ν cf. (2.7. µ = ν k (x µ(x ν ϵk Lie 7.1 ([O6]. Lie g π : g M(N, C End(C N g M(N, C M(N, C X, Y = Trace XY 2 g g 2 7
8 ( M(N, C g π F π = π (E ij 1 i N M(N, g 1 j N Z ϵ (g[x] q(x q(f π = 0 F π π q π (x g V q(fv = 0 C q(x 1 (π, V q π,v (x 7.2. i g = gl n π (7.1 q π (x = det (x E ij ϵ(n iδ ij (7.2 (7.3 q π (E = 0 (Cayley-Hamilton, L q π,m ϵ Θ (λ = (x λ j ϵn j 1. 1 i n 1 j n Z ϵ (g[x], ii q π,m ϵ Θ (λ(e N 2 L ϵ Θ (λ Lϵ Θ (λ G λ generic MΘ ϵ (λ regular IΘ ϵ (λ Lϵ Θ (λ ( k λ Θ γ( k k = 1,..., L i V g V ii g GL(n Lie π V = MΘ 0 (λ q π,m 0 Θ (λ(x A Θ,λ iii O(n Lie o n π F π =. ( Eij E ji 2 1 i n 1 j n iv Trace F k π Z ϵ (g γ(trace F k π [Go1] q π (x [Go2] q π,m ϵ (λ(x Cayley- Hamilton cf. [OO] v 6 [I1] [Um] [Sg]. vi g π q π,m ϵ Θ (λ [O6] [OO] 8. Grassmann Poisson Penrose U ϵ (g 2 Verma Verma Gap 8.1 ([O5], [OO]. λ generic regular (8.1 J ϵ Θ(λ = I ϵ Θ(λ + J ϵ (λ Θ (GAP. G GL(n SL(n GL(n, C GL(n, R, U(p, q SU (n Lie G P P Θ G/P Θ P Θ 1 λ (8.2 B(G/P Θ ; λ := { f B(G; f(gp = λ(p 1 f(g ( p P Θ } P Θ Lie 2 p Θ λ p Θ 1 (8.3 Ann ( B(G/P Θ ; λ := { D U(g; L D f = 0 ( f B(G/P ; λ } = I Θ (λ. B(G/P Θ ; λ G G G/P Θ I Θ (λ 8
9 Poisson G Lie K G (8.4 P λθ P Θ,λ : B(G/P Θ ; λ ( B(G/P ; λ Θ A(G/K; Mλ. f (P λ f(g = f(gkdk Poisson M λ λ Riemann G Z(g Ann ( M(λ Θ A(G/K; M λ G/P Θ Riemann G/K cf. [Sa] P Θ Poisson A(G/K; M λ Helgason [H1] G = SL(2, R, λ = 0 Poisson G [K ] generic λ λ = 0 Shilov Stein Hua [BV] [La] [KM], [Sh1] [Sh2] λ = 0 [Jn] [K ] P Θ λ 8 [K ] I Θ (λ (8.1 B(G/P ; λ Θ I Θ (λ B(G/P Θ ; λ λ generic λ = U(g P Θ,λ P Θ,λ I Θ (λ U(g 8.2. i Hua 7 Shilov tube 2 3 SU(m, n Shilov m = n 2 m n 3 G K m n 2 [BV] [OSh] ii Lie iii [BOS] iv Helgason Poisson [O1] Penrose G C Lie G G C P C G C V G O λ G C /P C 9 K
10 T P en : H V (O λ S G C Penrose 6 7 S Riemann G G = U(n, n Θ = {k, 2n} GL(n, C Grassmann Gr k (C n V S [Se] 6 (5.3 k + 1 n = 2 k = 1 Penrose F Grassmann Gr k (F n n F F n k F = R Gr k (R n := {k R n } ( Grassmann x 11 x 1k M o (n, k; R := X =.. M(n, k; R; rank X = k x n1 x nk = M o (n, k; R/GL(k, R. G = GL(n, R Gr k (R n Gr k (R n = GL(n, R/P k,n (= O(n/O(k O(n k { ( } g1 0 P k,n := p = ; g y g 1 GL(k, R, g 2 GL(n k, R, y M(n k, k, R 2 B(G/P k,n ; λ := { f B(G; f(xp = f(x det g 1 λ 1 det g 2 λ 2 }, p P k,n ( ( = B O(n/O(k O(n k = { f B ( M o (n, k; R ; f(xg 1 = f(x det g 1 λ1, g 1 GL(k, R } (x t x 1 X Gr 1 (F n P n 1 (F F = R Θ = {k, n} F = C Θ = {k, n} {k, n} Ann ( B(Gr k (F n ; λ 6 k + 1 n k Trace Z(g 1 9. Radon B(G/P Θ ; λ B(G/P Θ ; λ G Radon G Grassmann Radon 0 < k < l < n R k l : B ( Gr k (R n φ (R k l φ(x = φ(xydy B ( Gr l (R n GL(n, R O(l/O(k O(l k (9.1 R k l : B ( G/P k,n ; (l, 0 B ( G/P l,n ; (k, 0 G k + l < n dim Gr k (R n < dim Gr l (R n I Θ (λ 10
11 9.1 ([O4]. 0 < k < l < n k + l < n R k l M 0 (n, l; R G { Φ ( ( (x ij 1 i l B M 0 (n, l; R ; 1 j n Φ(xg = det g k Φ(x for g GL(l, R, ( det Φ(x = 0 (Capelli x iµ j ν 1 µ k+1 1 ν k+1 } for 1 i 1 < < i k+1 n, 1 j 1 < < j k+1 l. M 0 (n, l; R = {A M(n, l; R; rank A = l} 9.2. i C 9.1 [Hi] ii [Ka] [Ka], [GR] 9.3 ([O4]. G Lie P Θ Q j j = 1, 2 G/P Θ G λ µ j P Θ Q j 1 φ j G (9.2 φ 1 (q 1 xp = µ 1 (q 1 λ(pφ 1 (x (q 1 Q 1, p P Θ, φ 2 (q 2 xp = µ 2 (q 2 λ(pφ 2 (x (q 2 Q 2, p P Θ, λ = λ 2ρ PΘ (9.3 Φ φ1,φ 2 (x := K φ 1 (xkφ 2 (kdk ( = K φ(kφ 2 (x 1 kdk 9.4. i Φ φ1,φ 2 (x Q 1 Lie Q 2 Lie 6 7 I Θ (λ (9.2 φ 1, φ 2 (9.3 ii Q 1 = Q 2 = K µ j Φ φ1,φ 2 (x P Θ Lauricella F D [Kr] P = P k,n Q 2 = P l,n λ = (l, 0 µ 2 = ( k, 0 φ 2 R k l H GL(n, R H C V (H C GL(k, C, V C k l = 1 (H C, V (9.2 φ 1 ( k = 1 ( GL(1, C, C n ( H = GL(1 GL(1, C n -Gelfand [Ao], [GG] Φ(α, x = t t2 l =1 ν=1 n l αj t ν x jν ω ± 11 (α 1 + α α n = l.
12 l Φ x ij = α j Φ for 1 i n H, x ij n ν=1 x νi Φ = kδ ij Φ for 1 i, j l GL(l, R, x νj 2 Φ x i1 j 1 x i2 j 2 = 2 Φ x i2 j 1 x i1 j 2 for 1 i 1 < i 2 n, 1 j 1 < j 2 l Capelli i n = 4, l = 2 Gauss ii 9.5 G [Ta2] iii Penrose [Se] References [Ao] K. Aomoto, On the structure of integrals of power products of linear functions, Sc. Papers Coll. Gen. Education, Univ. of Tokyo 27(1977, [BOS] S. Ben Saïd, T. Oshima and N. Shimeno, Fatou s theorems and Hardy-type spaces for eigenfunctions of the invariant differential operators on symmetric spaces, Intern. Math. Research Notice 16(2003, [BV] N. Berline and M. Vergne, Equations du Hua et noyau de Poisson, Lect. Note. in Math. 880(1981, [Ca1] A. Capelli, Üeber die Zurückführung der Cayley schen Operation Ω auf gewöhnliche Polar- Operationen, Math. Ann. 29(1887, [Ca2], Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37(1890, [DP] C. DeConcini and C. Procesi, Symmetric functions, conjugacy classes and the flag variety, Invent. Math. 64(1981, [Du] M. Duflo, Sur la classification des idéaux primitifs dans l algèbre enveloppante d une algèbre de Lie semi-simple, Ann. of Math. 105(1977, [Ge] I. M. Gelfand, Center of the infinitesimal group ring, Mat. Sb., Nov. Ser. 26(68(1950, ; English transl. in Collected Papers, Vol. II, pp [GG] I. M. Gelfand and S. I. Gelfand, Generalized hypergeometric equations, Soviet Math. Dokl 33(1986, [Go1] M. D. Gould, A trace formula for semi-simple Lie algebras, Ann. Inst. Henri Poincaré, Sect. A 32(1980, [Go2], Characteristic identities for semi-simple Lie algebras, J. Austral. Math. Soc. Ser. B 26(1985, [GR] E. L. Grinberg and B. Lubin, Radon inversion and Grassmannians via Gårding-Gindikin fractional integrals, to appear in Ann. of Math. [HO] G. J. Heckman and E. M. Opdam, Root system and hypergeometric functions. I, Comp. Math. 64(1987, [H1] S. Helgason, A duality for symmetric spaces with applications to group representations II, Advances in Math., 22(1976, [H2], The Radon transform (2nd ed., Birkhauser, Boston, [Hi] T. Higuchi, Generalized Capelli Operator Grassmann Radon,, 2004 [HU] R. Howe and T. Umeda, The Capelli identity, the double commutation theorem, and multiplicity free actions, Math. Ann. 290(1991, [I1] M. Itoh, Explicit Newton s formula for gl n, J. Alg. 208(1998, [I2], The Capelli elements for the orthogonal Lie algebras, J. Lie Theory 10(2000, [IU] M. Itoh and T. Umeda, On the central elements in the universal enveloping algebra of the orthogonal Lie algebras, Compositio Math. 127(2001, [Jn] K. D. Johnson, Generalized Hua operators and parabolic subgroups, Ann. of Math. 120(1984, [Ka] T. Kakehi, Integral geometry on Grassmannian manifolds and calculus of invariant differential operators, J. Funct. Anal., 168(1999,
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