compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1

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1 compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1

2 (1) a = span(σ). () α, β Σ s α β := β α,β α α Σ. (3) α, β Σ α, β α Z. Σ Σ Σ a W (Σ) Σ Weyl Γ = {X a λ, X π Z (λ Σ)} Γ Σ ([4]) a r = λ Σ{H a λ, H πz} a r a a r a r Σ Affine Weyl W (Σ) {(s λ, nπ λ) λ Σ, n Z} λ O(a) a 1.. Affine Weyl W (Σ) P 0 a = sp 0 s W (Σ) Π Σ Σ α P 0 P 0 = {H a λ, H > 0 (λ Π), α, H < π} m : Σ R 0 1 λ Σ, s W (Σ) m(λ) = m( λ) = m(sλ) 1 Σ λ, µ Σ λ = µ m(λ) = m(µ)

3 m(λ) λ Σ H a [7] m H = m(λ) cot( λ, H )λ a, λ Σ +, λ,h π Z F (H) = m(λ) log sin( λ, H ), λ Σ +, λ,h π Z Vol(H) = exp( F (H)) > 0 m H H,Vol(H) H 1.3. Σ a H a σ = (s, X) W (Σ) H = σh Vol(H ) = Vol(H), m H = sm H 1.4. Σ a H a m H = H a austere { λ cot( λ, H )( m(λ)) λ Σ +, λ, H π Z} 1 H H asutere austere 1.6. Σ a H a austere H Σ = BC 1 = {±e 1, ±e 1 }, m(e 1 ) = m(e 1 ) H = te 1, tan t = Π { α} P 0 P0 λ, H > 0(λ Π), P0 λ, H = H P 0 { = 0(λ Π), < π ( α ), α, H = π ( α ) - - 3

4 P 0 P 0 = Π { α} P 0 H P 0 (gradf )(H) = m H 1.7. [7] P0 H P 0 H P 0 H G compact Lie (G, F ) compact M = G/F compact π : G M G Lie g g = f p a p λ a g C g(a, α) g(a, α) = {X g C [H, X] = 1 λ, H X (H a)} λ Σ m(λ) Σ = {λ a {0} g(a, α)} m(λ) = dim g(a, α) Σ a Σ g a x M F - F x F x ([4]) G = F (exp a)f x = π(exp H) (H a) F π(exp H) λ, H πz (λ Σ) H.1. F π(exp H) H.. F π(exp H) H 4

5 Harvey-Lawson[5] M austere M L A L ξ A ξ 1 1 L austere austere Harvey-Lawson[5] austere Bryant[] Euclid austere [10] compact (s- ) austere Kπ(exp H) austere H austere.3. [8] Σ compact λ, λ Σ m(λ) > m(λ) [8] M = G/F compact Rimann F - austere Leung[15] Riemann Riemann M ( ) austere.5. compact M = G/F 1.7 austere.6. [7] P0 Kπ(exp H) P 0 H F π(exp H) H P 0 F π(exp H) 5

6 P 0 H Kπ(exp H) a, ( Σ, Σ, W ) a (symmetric triad) (1) (6) (1) Σ a () Σ( a) span(σ) (3) W 1 a Σ = Σ W. (4) Σ W l = max{ α α Σ W} Σ W = {α Σ α l}. (5) α W, λ Σ W (6) α W, λ W Σ α, λ α s αλ W Σ. α, λ α s αλ Σ W. Σ a span(σ) = a a span(σ) span(σ W ) span{ Σ} = a span(σ) = a a ( Σ, Σ, W ) { Γ = X a λ, X π Z } (λ Σ) Γ 6

7 a a r a r = λ Σ,α W {H a λ, H πz, α, H π + πz} a r, a a r a r 3.. {(s λ, nπ λ λ) λ Σ, n Z} {(s α, (n+1)π α α) α W, n Z} O(a) a a ( Σ, Σ, W ) Affine Weyl W ( Σ, Σ, W ) (s λ, nπ λ) λ, H = nπ, (s λ α, (n+1)π α, H = n+1 π α α) 3.3. Affine Weyl W ( Σ, Σ, W ) P 0 a = sp 0 s W ( Σ,Σ,W ) Σ Π Σ + Π Σ + = Σ Σ +, W + = W Σ + Σ = Σ + ( Σ + ), W = W + ( W + ) Σ Π P 0 = H a P 0 0 < λ, H (λ Π), λ, H < π (λ Σ + W + ), λ, H < π (λ Σ + W + ), π < α, H < π (α W + Σ + ) 3.4. α W + P 0 = {H a α, H < π }, 0 < λ, H (λ Π)., Π { α} λ, H > 0 (λ Π), P0 λ, H = H P 0 { = 0 (λ Π ), < π ( α ), α, H = π ( α ), 7

8 P 0 P 0 = ( ). Π { α} P ( Σ, Σ, W ) a R + = {x R x 0} m, n : Σ R + (1) m(λ) = m( λ), n(α) = n( α) m(λ) > 0 λ Σ, n(α) > 0 α W. () λ Σ, α W, s W (Σ) m(λ) = m(sλ), n(α) = n(sα) (3) σ W ( Σ), λ Σ n(λ) + m(λ) = n(σλ) + m(σλ) (4) λ Σ W, α W α,λ α α,λ α m(λ) = m(s α λ), m(λ) = n(s α λ). m(λ), n(α) λ, α ( Σ, Σ, W ) H a m H = λ Σ+ λ,h π Z m(λ) cot( λ, H )λ + n(α) tan( α, H )α. α W + α,h π Z m H H F (H) = m(λ) log sin( λ, H ) n(α) log cos( α, H ) λ Σ+ λ,h π α W + Z α,h π Z Vol(H) = exp( F (H))(> 0) H 3.6. ( Σ, Σ, W ) a H a, σ = (s, X) Affine Weyl H = σh a Vol(H ) = Vol(H), m H = sm H 3.7. ( Σ, Σ, W ) a H a m H = 0 8

9 3.8. (1) H P 0 (grad F )(H) = m H () H, H 1 P 0 (H H 1 ) d F (H + t HH dt 1 ) t=0 > Π { α} H P ( Σ, Σ, W ) a H a austere { λ cot( λ, H ) ( = m(λ)) λ Σ +, λ, H π Z} {α tan( α, H ) ( = n(α)) α W +, α, H π Z} a (1) austere () austere 3.1. H a austere (1) λ, H π Z λ (Σ W ) (W Σ) () H Γ Σ W (3) m(λ) = n(λ) λ, H π + π Z λ Σ W 4 4 compact (G, F 1, F ) compact M i = G/F i G Riemann, M i G- compact F M 1 Hermann F 1 = F Hermann 9

10 Hermann M 1 F F M 1 Â M 1 F - Â Â Hermann F i G θ i π 1 : G M 1 θ i θ i G Lie g g = f 1 p 1 = f p a p 1 p A = exp a G Â = π 1(A) Hermann ([6]) G = F AF 1 3 M 1 F - F \G/F 1 a F \G/F 1 = a/ H 1 H F π 1 (exp H 1 ) = F π 1 (exp H ) θ 1 θ = θ θ 1 (A), (B), (C) (A) G θ 1 θ G (B) ( [14]) U compact Lie, σ U G = U U θ 1 (g, h) = (h, g), θ (g, h) = (σ(g), σ(h)) 3 [1, Theorem 4.1] G compact Lie G τ Cartan σ τ [13, Theorem 6.16] G Lie g σ τ g = h k + q k + h p + p q a p q K, H k, h G A = exp a G = KAH 10

11 (C) U compact Lie, σ U G = U U θ 1 (g, h) = (h, g), θ (g, h) = (σ 1 (h), σ(g)). (B) F (θ, G) = F (σ, U) F (σ, U) M 1 = U (a, b) x = axb 1 (x U, a, b F (σ, U)) (C) Hermann σ- F (θ, U) = {(g, σ(g)) g U} M 1 = U σ- U U g x = gxσ(x) 1 (A),(B),(C) compact (G, F 1, F ) a ( Σ, Σ, W ) θ 1 θ g = (f 1 f ) (p 1 p ) (f 1 p ) (f p 1 ). α a g C g(a, α) g(a, α) = {X g C [H, X] = 1 α, H X (H a)} Σ = {α a {0} g(a, α) {0}} ɛ = ±1 g(a, α) g(a, α, ɛ) g(a, α, ɛ) = {X g(a, α) θ 1 θ X = ɛx} g(a, α) θ 1 θ - g(a, α) = g(a, α, 1) g(a, α, 1). Σ = {α Σ g(a, α, 1) {0}}, W = {α Σ g(a, α, 1) {0}} λ Σ α W m(λ) = dim C g(a, λ, 1), n(α) = dim C g(a, α, 1) 11

12 4.1. ( Σ, Σ, W ) a. 4 G 1 F 1 G 1 = F (θ 1 θ, G), F 1 = {g G 1 θ 1 (g) = g} G 1 F 1 Lie g 1 = (f 1 f ) (p 1 p ), f 1 = f 1 f compact (G 1, F 1 ) a Σ 5 Hermann (G, F 1, F ) (A),(B),(C) compact a ( Σ, Σ, W ) Hermann H a F π 1 (exp H) Affine Weyl W ( Σ, Σ, W ) F \G/F 1 P 0 P 0 5 F \G/F 1 = P0 F π 1 (exp H) H 5.1. F π 1 (exp H) 5.. P0 F π 1 (exp H) P 0 H F π 1 (exp H) H P 0 F π(exp H) 4 (G, F 1, F ) ( Σ, Σ, W ) (A) (B) (G, F 1, F ) ( Σ, Σ, W ) (C) (G, F 1, F ) G ( Σ, Σ, W ) Vogan ([13]) 5 1

13 P 0 H F π 1 (exp H) Hermann austere 5.3. F (exp H) austere H austere [10] M 1 Riemann L M 1 x L ξ T x L M 1 σ ξ L σ ξ (x) = x, (dσ ξ ) x ξ = ξ, σ ξ (L) = L (dσ ξ ) 1 x A ξ (dσ ξ ) x = A ξ austere austere s- austere Hermann austere 6 g compact Lie J g {0} (adj) 3 = adj G = Int(g) M = G J g G-, G K K = {k G k J = J} K Lie k k = {X g [J, X] = 0} g m m = Im adj g = k m ( ). 13

14 g e πadj (+1)- ( 1)- k m J m K- M = G/K compact Hermite compact Hermite L M L M τ L M Lagrange compact (compact Hermite ) ([, Lemma 4.1]) compact Hermite compact [16], [18] G I τ I τ : G G; g τgτ 1 G I τ F (I τ ) (G, F (I τ )) compact g g = l p J k p ([, Theorem 4.3]) p a J a a (G, F (I τ )) R 6.1 L al (a G) a = exp H (H a) 6.1. [11] L al (a = exp H) H L al = M a = W (R)J W (R)J L W (R) M a = W (R)J ([1]) M a L L W (R) ([19]) S L x, y S s x (y) = y s x x L -number # L # L Chen- ([3]) 14

15 6. L 1, L M L L i i g L 1 L M compact Hermite I τ1 I τ (τ 1 τ G ) compact I τ1 I τ p 1 p a J a compact (G, F (I τ1 ), F (I τ )) ( Σ, Σ, W ) a p i p i compact (G, F (I τi )) a i R i L 1 al (a G) a = exp H (H a) 6.. L 1 al (a = exp H) H L 1 al = W ( Σ)J = W (R 1 )J a = W (R )J a W ( Σ)J W ( Σ) 6.3. L 1 al (a = exp H) a = a 1 Σ = R 1 L 1 al = W (R 1 )J L 1 [1] R. Bott, The geometry and representation theory of compact Lie groups, Representation theory of Lie groups, (1970), 65 90, London Math. Soc. Lecture Note Ser. 34. [] R. L. Bryant, Some remarks on the geometry of austere manifolds, Bol. Soc. Bras. Mat., 1 () (1991),

16 [3] B.-Y.-Chen and T. Nagano, A Riemannian geometric invariant and its applications to Borel and Serre, Trans. Amer. ath. Soc. 308 (1988), [4] D.Hirohashi, O. Ikawa and H. Tasaki, Orbits of isotropy groups of compact symmetric spaces, Tokyo J. Math. 4 (001) [5] R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math., 148 (198), [6] E. Heintze, R. S. Palais, C. Therng and G. Thorbergsson, Hyperpolar actions on symmetric spaces, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp [7] D. Hirohashi, H. Tasaki, H. Song and R. Takagi, Minimal orbits of the isotropy groups of symmetric spaces of compact type, Differntial geometry and its applications 13 (000) [8] O. Ikawa, The geometry of symmetric triad and orbit spaces of Hermann actions J. Math. Soc. Japan 63 (011), [9] O. Ikawa, A note on symmetric triad and Hermann action, Proceedings of the workshop on differential geometry and submanifolds and its related topics, Saga, August 4 6, 01, 0 9. [10] O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds and austere submanifolds, J. Math. Soc. Japan 61 No. (009) pp [11] O. Ikawa, M. Tanaka and H. Tasaki, The fixed point set of a holomorphic isometry, the intersection of two real forms in a Hermitian symmetric space of compact type and symmetric triads, in perparation. [1] M. F.-Jensen, Spherical functions on a real semisimple Lie group, Journal of functional analysis 30, (1978). [13] Anthony W. Knapp, Lie groups beyond an introduction, Birkhäuser. 16

17 [14] N. Koike, Examples of certain kind of minimal orbits of Hermann actions, Hokkaido Math. J. 43 (014), 1 4. [15] D. S. P. Leung, The reflection principle for minimal submanifolds of Riemannian symmetric spaces, J. Differential Geometry, 8 (1973) [16] D. S. P. Leung, Reflective submanifolds. IV, Classification of real forms of Hermitian symmeric spaces, J. Differential Geom., 14 (1979), [17] T. Matsuki, Classification of two involutions on compact semisimple Lie groups and root systems, J, Lie Theory, 1 (00), [18] M. Takeuchi, Stability of cretain minimal submanifolds of compact Hermitian symmetric spaces, Tohoku Math. Journ. 36 (1984), [19] M. Takeuchi, Two-number of symmetric R-spaces, Nagoya Math. J. 115 (1989), [0] H. Tamaru, The local orbit types of symmetric spaces under the actions of the isotropy subgroups, Differential Geom. Appl. 11 (1999), no. 1, [1] M. S. Tanaka and H. Tasaki, The intersection of two real forms in Hermitian symmetric spaces of compact type, J. Math. Soc. Japan 64 (01), [] M. S. Tanaka and H. Tasaki, Antipodal sets of symmetric R-spaces, Osaka J. Math. 50 (013), [3] M. S. Tanaka and H. Tasaki, The intersection of two real forms in Hermitian symmetric spaces of compact type II, to appear in J. Math. Soc. Japan. [4] M. S. Tanaka and H. Tasaki, Correction to: The intersection of two real forms in Hermitian symmetric spaces of compact type, to appear in J. Math. Soc. Japan. 17

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