The intersection of two real flag manifolds in a complex flag manifold

Transcription

1 in a complex flag manifold Takashi SAKAI (Tokyo Metropolitan University / OCAMI) joint work with Osamu Ikawa, Hiroshi Iriyeh, Takayuki Okuda and Hiroyuki Tasaki March 27, 2018 Geometry of Submanifolds and Integrable Systems The 15th OCAMI-RIRCM Joint Differential Geometry Workshop & The 3rd OCAMI-KOBE-WASEDA Joint International Workshop on Differential Geometry and Integrable Systems On Occasion of Professor Yoshihiro Ohnita for his 60th Birthday at Osaka City University

2 Introduction M : Kähler manifold L 0, L 1 M : real forms i.e. σ i : anti-holomorphic involutive isometry of M (i = 0, 1) s.t. L i = Fix(σ i, M) 0 totally geodesic Lagrangian submanifold Problems 1 Is the intersection L 0 L 1 discrete? symmetric triad 2 If so, count the intersection number #(L 0 L 1 ), and describe the geometric meaning of #(L 0 L 1 ). Floer homology 3 Study the structure of the intersection L 0 L 1. antipodal set

3 Antipodal set of a compact symmetric space M : compact Riemannian symmetric space s x : geodesic symmetry at x M Definition (Chen-Nagano 1988) 1 A M : antipodal set def s x (y) = y for all x, y A 2 # 2 M := max{#a A M : antipodal set} 2-number 3 A M : great antipodal set def #A = # 2 M Theorem (Takeuchi 1989) M : symmetic R-space = # 2 M = dim H (M, Z 2 )

4 Previous works Theorem (Tanaka-Tasaki 2012) M : Hermitian symmetric space of compact type L 0, L 1 M : real forms, L 0 L 1 = L 0 L 1 is an antipodal set of L 0 and L 1. Theorem (Iriyeh-S.-Tasaki 2013) 1 Lagrangian Floer homology of two real forms in irreducible Hermitian symmetric spacecs 2 Volume estimate of real forms under Hamiltonian deformations

5 Example RP n CP n A := {Re 1,..., Re n+1 } RP n great antipodal set For g U(n + 1), RP n grp n in CP n RP n grp n = {Ce1,..., Ce n+1 } CP n #(RP n grp n ) = n + 1 = # 2 RP n = dim H (RP n, Z 2 ) Aim of our work Generalizing the results on Hermitian symmetric spaces, study the intersection of two real forms in a complex flag manifold.

6 Complex flag manifold G : compact connected semisimple Lie group x 0 ( 0) g M := Ad(G)x 0 g : complex flag manifold = G/Gx0 = G C /P C G x0 := {g G Ad(g)x 0 = x 0 } ω : Kirillov-Kostant-Souriau symplectic form on M defined by ω(x x, Y x ) := [X, Y ], x (x M, X, Y g) J : G-invariant complex structure on M compatible with ω (, ) := ω(, J ) : G-invariant Kähler metric

7 Antipodal set of a complex flag manifold For x M G x := {g G Ad(g)x = x} Z(G x ) := {g G x gh = hg ( h G x )} Definition y M is antipodal to x M def Ad(g)y = y for all g Z(G x ) 0 A M : antipodal set def y is antipodal to x for any x, y A. Note: This definition is equivalent to the notion of an antipodal set of M defined using k-symmetric structure on M. When M is a Hermitian symmetric space, it is also equivalent to the notion of an antipodal set introduced by Chen-Nagano.

8 Antipodal set of a complex flag manifold Proposition 1 For x, y M y is antipodal to x [x, y] = 0 2 A M : maximal antipodal set = t g : maximal abelian subalgebra s.t. A = M t. Hence A is an orbit of the Weyl group of g with respect to t. In particular, any maximal antipodal sets of M are congruent with each other by G.

9 Real flag manifolds in a complex flag manifold (G, K) : symmetric pair of compact type θ : involution of G s.t. Fix(θ, G) 0 K Fix(θ, G) x 0 ( 0) p g = k p L := Ad(K)x 0 p : real flag manifold, R-space M := Ad(G)x 0 g : complex flag mfd, C-space = G/Gx0 = G C /P C dθ : g g involutive automorphism σ := dθ : M M anti-holomorphic involutive isometry L = M p = Fix(σ, M) real form of M

10 The intersection of real flag manifolds (G, K 0 ), (G, K 1 ) : symmetric pairs of compact type θ 0, θ 1 : involutions of G x 0 ( 0) p 0 p 1 g = k 0 + p 0 = k 1 + p 1, L 0 := Ad(K 0 )x 0, L 1 := Ad(K 1 )x 0 M := Ad(G)x 0 For g G, we consider the intersection of L 0 Ad(g)L 1 in M. a : maximal abelian subspace of p 0 p 1 containing x 0 A := exp a G : toral subgroup Then G = K 0 AK 1, i.e. g = g 0 ag 1 (g 0 K 0, g 1 K 1, a A) L 0 Ad(g)L 1 = L 0 Ad(g 0 ag 1 )L 1 = Ad(g 0 )(L 0 Ad(a)L 1 )

11 Symmetric triads Hereafter we assume θ 0 θ 1 = θ 1 θ 0. g = (k 0 k 1 ) + (p 0 p 1 ) + (k 0 p 1 ) + (p 0 k 1 ) ad(a)-invariant ad(a)-invariant Then ( ) (k 0 k 1 ) + (p 0 p 1 ), (k 0 k 1 ), dθ 0 = dθ 1 is an orthogonal symmetric Lie algebra. For λ a p 0 p 1 p λ := {X p 0 p 1 [H, [H, X]] = λ, H 2 X (H a)} V λ := {X p 0 k 1 [H, [H, X]] = λ, H 2 X (H a)} Σ := {λ a \ {0} p λ {0}} W := {λ a \ {0} V λ {0}} Σ := Σ W ( Σ, Σ, W ) : symmetric triad

12 The structure of the intersection a reg := λ Σ α W { } H a λ, H πz, α, H π 2 + πz W ( Σ) : Weyl group of the root system Σ of a Theorem (Ikawa-Iriyeh-Okuda-S.-Tasaki) For a = exp H (H a), the intersection L 0 Ad(a)L 1 is discrete if and only if H a reg. Moreover, if L 0 Ad(a)L 1 is discrete, then L 0 Ad(a)L 1 = M a = W ( Σ)x 0. In particular, L 0 Ad(a)L 1 is an antipodal set of M.

13 Lagrangian Floer homology (M, ω) : closed symplectic manifold J = {J t } 0 t 1 : family of almost complex structures on M compatible with ω L 0, L 1 : closed Lagrangian submanifolds, L 0 L 1 Definition For p, q L 0 L 1, u : R [0, 1] M : J-holomorphic strip from p to q def J u := u s + J t(u) u t = 0 u(s, 0) L 0, u(s, 1) L 1 u(, t) = p, u(+, t) = q M J (L 0, L 1 : p, q) := {u : J-holomorphic strips from p to q}

14 Lagrangian Floer homology M J (L 0, L 1 : p, q) := {u : J-holomorphic strips from p to q} CF (L 0, L 1 ) := Z 2 p p L 0 L 1 : CF (L 0, L 1 ) CF (L 0, L 1 ) (p) = n(p, q) q q L 0 L 1 n(p, q) := #{isolated J-holomorphic strips from p to q} (mod 2) = 0 = HF (L 0, L 1 : Z 2 ) := ker /im HF (ϕl 0, ψl 1 : Z 2 ) = HF (L 0, L 1 : Z 2 ) for ϕ, ψ Ham(M, ω) with ϕl 0 ψl 1.

15 Lagrangian Floer homology for two real forms Theorem (Ikawa-Iriyeh-Okuda-S.-Tasaki) M : complex flag manifold with a Kähler-Einstein metric L 0, L 1 M : real flag manifolds, θ 0 θ 1 = θ 1 θ 0 minimal Maslov numbers Σ L0, Σ L1 3. = a real flag manifold L 1 = L 1 s.t. L 0 L 1 and HF (L 0, L 1 : Z 2 ) = Z 2 [p] p L 0 L 1 Corollary #(ϕl 0 ψl 1 ) #(L 0 L 1) = dim HF (L 0, L 1 : Z 2 ) for any ϕ, ψ Ham(M, ω) with ϕl 0 ψl 1.

16 Example (G, K 0, K 1 ) = (SU(2n), SO(2n), Sp(n)) θ 0 (g) = ḡ, θ 1 (g) = J n ḡj 1 n (g G) where J n := [ 1X ] 1Y p 0 p 1 = 1Y 1X [ X, Y M n (R) tracex = 0 t X = X, t Y = Y O I n I n O ] Fix a maximal abelian subspace a in p 0 p 1 as {[ 1X ] } O X = diag(t a = 1,..., t n ), O 1X t 1,..., t n R, t t n = 0 Σ = Σ = W = {±(e i e j ) 1 i < j n} where e i e j a (i j) is defined by e i e j, H = t i t j.

17 [ 1X ] O x 0 = a O 1X where X = diag(x 1 I n1,..., x r+1 I nr+1 ) and x i are distinct real numbers satisfying n 1 x n r+1 x r+1 = 0. L 0 = Ad(K 0 )x 0 = F R 2n1,...,2n r (R 2n ) L 1 = Ad(K 1 )x 0 = F H n1,...,n r (H n ) M = Ad(G)x 0 = F C 2n1,...,2n r (C 2n ) K = R, C or H n, n 1,..., n r satisfying n r+1 := n (n n r ) > 0 V j is a K-subspace of K n, Fn K 1,...,n r (K n ) = (V 1,..., V r ) dim K V j = n n j, V 1 V 2 V r K n

18 [ 1Y ] O a = exp H, H = a O 1Y where Y = diag(t 1,..., t n ) and t 1,..., t n R which satisfy t t n = 0. Then L 0 Ad(a)L 1 is discrete { H a reg = H a e i e j, H π } 2 Z (1 i < j n) L 0 Ad(a)L 1 = M a = W ( Σ)x 0. In this case, a maximal abelian subspace a in p 0 p 1 is also a maximal abelian subspace in p 1.

19 We express the intersection in the flag model F C 2n 1,...,2n r (C 2n ). v 1,..., v 2n : standard basis of C 2n W i := v i, v n+i C = v i H (1 i n) Proposition For a = exp H (H a reg ), F R 2n 1,...,2n r (R 2n ) af H n 1,...,n r (H n ) = {(W i1 W in1, W i1 W in1 +n 2,..., W i1 W in1 + +nr ) 1 i 1 < < i n1 n, 1 i n1 +1 < < i n1 +n 2 n,..., 1 i n1 + +n r 1 +1 < < i n1 + +n r n, #{i 1,..., i n1 + +n r } = n n r }, which is an antipodal set of F C 2n 1,...,2n r (C 2n ).

20 Example dim HF (F R 2n 1,...,2n r (R 2n ), F H n 1,...,n r (H n ) : Z 2 ) = # ( F R 2n 1,...,2n r (R 2n ) af H n 1,...,n r (H n ) ) = # I (Fn H 1,...,n r (H n )) = dim H (Fn H 1,...,n r (H n ) : Z 2 ) n! = n 1!n 2! n r+1! < # I (F2n R 1,...,2n r (R 2n )) = dim H (F2n R 1,...,2n r (R 2n ) : Z 2 ) = # k (F C 2n 1,...,2n r (C 2n )) = dim H (F C 2n 1,...,2n r (C 2n ) : Z 2 ) = (2n)! (2n 1 )!(2n 2 )! (2n r+1 )! for a = exp H (H a reg )

21 Further problems 1 Study the intersection of two real flag manifolds in the case where θ 0 θ 1 θ 1 θ 0. 2 Determine Hamiltonian volume minimizing properties of all real forms in irreducible Hermitian symmetric spaces, more generally, in complex flag manifolds. Thank you very much for your attention

22 Further problems 1 Study the intersection of two real flag manifolds in the case where θ 0 θ 1 θ 1 θ 0. 2 Determine Hamiltonian volume minimizing properties of all real forms in irreducible Hermitian symmetric spaces, more generally, in complex flag manifolds. Thank you very much for your attention

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