Austere pseudo-riemannian submanifolds and s-representations Tokyo University of Science, D3 Submanifold Geometry and Lie Theoretic Methods (30 October, 2008)
The notion of an austere submanifold in a Riem. mfd. was introduced by Harvey and Lawson. We generalize the notion of an austere submfd., as follows. Definition M M : pseudo-riemannian submanifold A : the shape operator of M M : austere : for x M, ξ T def x M, SpecAC ξ is invariant (considering multiplicities) under the multiplication by 1.
The notion of an austere submanifold in a Riem. mfd. was introduced by Harvey and Lawson. We generalize the notion of an austere submfd., as follows. Definition M M : pseudo-riemannian submanifold A : the shape operator of M M : austere : for x M, ξ T def x M, SpecAC ξ is invariant (considering multiplicities) under the multiplication by 1. In general, the shape operator of a pseudo-riem. submfd. is not necessary real diagonalizable.
The notion of an austere submanifold in a Riem. mfd. was introduced by Harvey and Lawson. We generalize the notion of an austere submfd., as follows. Definition M M : pseudo-riemannian submanifold A : the shape operator of M M : austere : for x M, ξ T def x M, SpecAC ξ is invariant (considering multiplicities) under the multiplication by 1. In general, the shape operator of a pseudo-riem. submfd. is not necessary real diagonalizable. Our examples of austere pseudo-riemannian submanifolds are orbits of s-representations (the linear isotropy representations of semisimple pseudo-riem. symm. spaces).
Geometry of orbits of s-representation (G, H) : a semisimple symmetric pair g := Lie(G), h := Lie(H) σ: an involution of g such that Ker(σ id) = h g = h + q : the eigensp. decomp. of an involution σ, : the inn. product of q induced from the Killing form of g Ad : H GL(q) : the adjoint representation of H on q
Geometry of orbits of s-representation (G, H) : a semisimple symmetric pair g := Lie(G), h := Lie(H) σ: an involution of g such that Ker(σ id) = h g = h + q : the eigensp. decomp. of an involution σ, : the inn. product of q induced from the Killing form of g Ad : H GL(q) : the adjoint representation of H on q M := Ad(H)X (X q) M is called semisimple : ad(x) is a semisimple endmorphism of g def M is called hyperbolic (resp. elliptic) : M is semisimple and any eigenvalue of ad(x) is real (resp. def pure imaginary) M is called spacelike (resp. timelike) : def the quantity X, X is positive (resp. negative)
M := Ad(H)X( S) : semisimple spacelike orbit a q : a Cartan subspace of q containing X R : the restricted root system of (g, h) w.r.t. a C q g C = g C 0 + α R g C α : the restricted root sp. decomp. w.r.t. a C q h C α := (gc α + gc α ) hc, q C α := (gc α + gc α ) qc Then h C and q C are decomposed as h C = h C 0 + h C α and qc = a C q + q C α. α R + α R +
R X := {α R α(x) = 0} The complexifications of the isotropy subalgebra, the tangent space and the normal space of M in S are decomposed as h C X = hc 0 + h C α, α (R X ) + (T X M) C = q C α, α R + \(R X ) + (T X M)C = (a q RX) C + q C α. α (R X ) +
A : the shape operator of M in S Key Lemma ξ : a normal vector of M at X ξ = ξ s + ξ n : the Jordan decomp. of ξ Then ξ s and ξ n are normal vector of M at X, and the shape operator A ξ can be decomposed as A ξ = A ξs + A ξn, where A ξs and A ξn coincide with the semisimple part and the nilpotent part of A ξ respectively, and A ξs A ξn = A ξn A ξs. The spectrum of A C ξ coincides with that of AC ξ s
Proof(Sketch) By the uniqueness of the Jordan decomp. of ξ, we have ξ s, ξ n q. Since X, ξ = 0 holds, we have X, ξ s = 0, X, ξ n = 0. A ξs : semisimple, A ξn : nilpotent Since [ξ s, ξ n ] = 0 holds, A ξs and A ξn commute with each other. By the uniqueness of the Jordan decomp. of A ξ, A ξs and A ξn coincide with the semisimple part and the nilpotent part of A ξ, resp..
Austere semisimple spacelike orbits of s-representations (G, H) : semisimple symmetric pair M := Ad(H)X : semisimple spacelike orbit Lemma 1 a 1 q, a2 q : Cartan subspaces of q containing X Then φ Aut(g C ); (a 1 q )C = φ(a 2 q )C and φx = X.
From Key Lemma and Lemma 1, M = Ad(H)X : austere : for x M, ξ T def x M, SpecAC ξ is invariant (considering multiplicities) under the multiplication by 1 for a Cartan subspace a q cont. X, ξ a q RX, SpecA C ξ is invariant (considering multiplicities) under the multiplication by 1
From Key Lemma and Lemma 1, M = Ad(H)X : austere : for x M, ξ T def x M, SpecAC ξ is invariant (considering multiplicities) under the multiplication by 1 for a Cartan subspace a q cont. X, ξ a q RX, SpecA C ξ is invariant (considering multiplicities) under the multiplication by 1 R : the restricted root system w.r.t. a C q SpecA C ξ = { for all ξ a q RX α(ξ) α(x) α R + \ (R X ) + }
θ : a Cartan involution of g which commutes with σ and preserves a q invariant g = k + p : the Cartan decomp. of g corr. to θ B : the Killing form of g C A α a C q (α R) : B(A α, A) = α(a) ( A a C def q ) Then A α 1(a q k) + a q p( q).
θ : a Cartan involution of g which commutes with σ and preserves a q invariant g = k + p : the Cartan decomp. of g corr. to θ B : the Killing form of g C A α a C q (α R) : B(A α, A) = α(a) ( A a C def q ) Then A α 1(a q k) + a q p( q). Proposition For each α R which is real-valued on a q, the orbit through the restricted root vector A α is austere
Assume that R is irreducible Theorem A If a semisimple spacelike orbit of the s-representation of G/H is an austere pseudo-riemannian submanifold in S, then the orbit is hyperbolic.
Proof(Sketch) M := Ad(H)X : semisimple spacelike orbit Without loss of generalities, we assume that X a q. If M is an austere pseudo-riem. submfd. in S, then there exists f : R + \ (R X ) + R + \ (R X ) + which satisfies the following three conditions: f has no fixed point, for any α R + \ (R X ) +, n α R \ {0}, ε α = ±1 s.t. ( Aα X = n α A α + ε A ) f α α, A f α for any α R + \ (R X ) +, m(γ) = γ R + \(R X ) + ;γ//α γ R + \(R X ) + ;γ//f α m(δ), or there exists α R s.t. X = A α and α is real-valued on a q. Therefore X a q Span R {A α α R} = a q p.
To classify all austere semisimple spacelike orbits, (1) classify X a q admitting f : R + \ (R X ) + R + \ (R X ) + which satisfies the following three conditions: f has no fixed point, for any α R + \ (R X ) +, n α R \ {0}, ε α = ±1 s.t. ( Aα X = n α A α + ε A ) f α α, A f α for any α R + \ (R X ) +, m(γ) = γ R + \(R X ) + ;γ//α γ R + \(R X ) + ;γ//f α (2) classify all α R such that α is real valued on a q. m(δ),
Remark In the classification of (1), we make use of the classification of all austere orbits of the linear isotropy representations of semisimple Riem. symm. sp., which is given by Ikawa, Sakai and Tasaki.
The restricted root systems with respect to Cartan subspaces with maximal vector parts (G, H) : semisimple symmetric pair (g, h) : the infinitesimal pair of (G, H) g = h + q : the eigen. decomp. of g corr. to an involution σ θ : a Cartan involution of g commuting with σ g = k + p : the Cartan decomp. of g corr. to θ K : the maximal cpt. subgroup of G whose Lie algebra is k a q : a θ-invariant Cartan subspace of q a q is said to have maximal vector part : def p a q is a maximal abelian subsp. of p q a q is said to have maximal toroidal part k a q is a maximal abelian subsp. of k q : def
Remark There exists, up to conjugate under the action of K H, the only Cartan subspace with maximal vector part and the only Cartan subspace with maximal troidal part.
Remark There exists, up to conjugate under the action of K H, the only Cartan subspace with maximal vector part and the only Cartan subspace with maximal troidal part. a q : a Cartan subspace of q with maximal vector part R : the restricted root system w.r.t. a C q Assume that R is irreducible
Remark There exists, up to conjugate under the action of K H, the only Cartan subspace with maximal vector part and the only Cartan subspace with maximal troidal part. a q : a Cartan subspace of q with maximal vector part R : the restricted root system w.r.t. a C q Assume that R is irreducible θ : a q a q : θ(x) = θ(x) for all X aq def α θ (a C q ) (α R) : def α θ(x) := α( θx) for all X a C q R 0 := {α R α p aq 0} = {α R α = α θ } α is real-valued on a q α = α θ
Ψ = Ψ(R) : a θ-fundamental system of R Ψ 0 := Ψ R 0 Proposition There exists a permutation p of Ψ \ Ψ 0 of order 2 such that for each α Ψ \ Ψ 0, α θ pα (mod Span Z {β β Ψ 0 }). From the Dynkin diagram of Ψ and the permutation p as in above proposition we define the Satake diagram of (R, R 0, θ) as follows.
Ψ = Ψ(R) : a θ-fundamental system of R Ψ 0 := Ψ R 0 Proposition There exists a permutation p of Ψ \ Ψ 0 of order 2 such that for each α Ψ \ Ψ 0, α θ pα (mod Span Z {β β Ψ 0 }). From the Dynkin diagram of Ψ and the permutation p as in above proposition we define the Satake diagram of (R, R 0, θ) as follows. (step 1) Replace a white circle of the Dynkin diagram, which belongs to Ψ 0, with a black circle. (step 2) If restricted roots α, β Ψ \ Ψ 0 satisfy α β and pα = β, join α and β with an arrowed segment.
Example (g, h) = (e 6( 14), sp(2, 2)), k = so(10) + so(2) {α 1,..., α 6 } : a θ-fundamental system «¾ «««««½ Satake diagram of (R, R 0, θ)
Example (g, h) = (e 6( 14), sp(2, 2)), k = so(10) + so(2) {α 1,..., α 6 } : a θ-fundamental system «¾ «««««½ Satake diagram of (R, R 0, θ) From the Satake diagram we have α θ 3 = α 3, α θ 4 = α 4, α θ 5 = α 5, pα 1 = α 6, pα 2 = α 2.
Since θ leaves R invariant, we have α θ 1 = α 3 + α 4 + α 5 + α 6, α θ 2 = α 2 + α 3 + 2α 4 + α 5, α θ 6 = α 1 + α 3 + α 4 + α 5.
Since θ leaves R invariant, we have α θ 1 = α 3 + α 4 + α 5 + α 6, α θ 2 = α 2 + α 3 + 2α 4 + α 5, α θ 6 = α 1 + α 3 + α 4 + α 5. {α α θ = α} = {α = i l iα i l 3 = l 5 = (2l 1+l 2 ), l 2 4 = l 1 +l 2, l 6 = l 1 } = {±(α 1 + 2α 2 + 2α 3 + 3α 4 + 2α 5 + α 6 )} {±(α 1 + α 3 + α 4 + α 5 + α 6 )}
Classification Theorem B Let (G, H) be a semisimple symmetric pair whose restricted root system R is irreducible. A semisimple spacelike orbit of the s-representation of (G, H), which is an austere submanifold in the pseudo-hypersphere is one of the following list. R = a n Base point Remark AI all restricted roots 2α 1 + α 2, α 1 + 2α 2 rank = 2 α 1 + 2α 2 + α 3 rank = 3 AII α 1 + 2α 2 + α 3 rank = 3 α i + + α j 1 1 i l, i + j = r + 2 AIII α 1 + 2α 2 + α 3 rank = 3, s-rank = 2
R = b n Base point Remark α i + + α j 1 1 i < j l α i + + α r + α j + + α r 1 i < j l BI α i + + α r 1 i l α 1 + α 2 + (1/ 2)(α 1 + 2α 2 ) rank = s-rank = 2 : the multiplicities of the restricted roots are constant R = (bc) n Base point Remark α i + + α j 1 1 i < j l BCI α i + + α r + α j + + α r 1 i < j l α i + + α r 1 i l 2(α i + + α r) 1 i l BCIII α 2i 1 + + α 2r + α 2i + + α 2r 1 i r
R = c n Base point Remark α i + + α j 1 1 i < j l α i + + α j 1 + 2α j + + 2α r 1 + α r 1 i < l CI 2α i + + 2α r 1 + α r 1 i l (1/2)(α 1 + α 2 ) + (1/ 2)(α 1 + 2α 2 ) rank = s-rank = 2 CII α 2i 1 + 2α 2i + + 2α r 1 + α r 1 i l CIII α 2i 1 + 2α 2i + + 2α 2r + α 2r+1 1 i r : the multiplicities of the restricted roots are constant R = d n Base point Remark α i + + α j 1 1 i < j l α i + + α r 2 + α j + + α r 1 i < j l DI (1/2)(2α 1 + + α r 2 + α 2 + + α r ) α 1 + 2α 2 + α 3 + 2α 4 rank = s-rank = 4 α 1 + 2α 2 + 2α 3 + α 4 rank = s-rank = 4 α 2i 1 + + α 2r 2 + α 2i + + α 2r 1 i r, rank:even DIII α 2i 1 + + α 2r 1 + α 2i + + α 2r+1 1 i r, rank:odd α 1 + 2α 2 + α 3 + 2α 4 rank = 4
R = e 6 EI EII EIII EIV Base point all restricted roots α 3, α 4, α 3 + α 4, α 2 + α 4 + α 5, α 2 + α 3 + α 4 + α 5, α 2 + α 3 + 2α 4 + α 5, α 1 + α 2 + α 3 + α 4 + α 5 + α 6, α 1 + α 2 + α 3 + 2α 4 + α 5 + α 6, α 1 + α 2 + 2α 3 + 2α 4 + α 5 + α 6, α 1 + 2α 2 + 3α 3 + 2α 4 + 2α 5 + α 6 α 1 + α 3 + α 4 + α 5 + α 6, α 1 + 2α 2 + 2α 3 + 3α 4 + 2α 5 + α 6 no austere orbit
R = e 7 EV EVI EVII Base point all restricted roots α 1, α 3, α 1 + α 3, α 2 + α 3 + 2α 4 + α 5, α 1 + α 2 + α 3 + 2α 4 + α 5, α 1 + α 2 + 2α 3 + 2α 4 + α 5, α 1 + α 2 + 2α 3 + 2α 4 + 2α 5 + 2α 6 + α 7, α 1 + α 2 + α 3 + 2α 4 + 2α 5 + 2α 6 + α 7, 2α 1 + 2α 2 + 3α 3 + 4α 4 + 3α 5 + 2α 6 + α 7, α 1 + 2α 2 + 2α 3 + 4α 4 + 3α 5 + 2α 6 + α 7, α 1 + 2α 2 + 3α 3 + 4α 4 + 3α 5 + 2α 6 + α 7 α 7, α 2 + α 3 + 2α 4 + 2α 5 + 2α 6 + α 7, 2α 1 + 2α 2 + 3α 3 + 4α 4 + 3α 5 + 2α 6 + α 7
R = e 8 EVIII EIX Base point all restricted roots α 7, α 8, α 7 + α 8, α 2 + α 3 + 2α 4 + 2α 5 + 2α 6 + α 7, α 2 + α 3 + 2α 4 + 2α 5 + 2α 6 + α 7 + α 8, α 2 + α 3 + 2α 4 + 2α 5 + 2α 6 + 2α 7 + α 8, 2α 1 + 2α 2 + 3α 3 + 4α 4 + 3α 5 + 2α 6 + α 7, 2α 1 + 2α 2 + 3α 3 + 4α 4 + 3α 5 + 2α 6 + 2α 7 + α 8, 2α 1 + 2α 2 + 3α 3 + 4α 4 + 3α 5 + 2α 6 + α 7 + α 8, 2α 1 + 3α 2 + 4α 3 + 6α 4 + 5α 5 + 4α 6 + 2α 7 + α 8, 2α 1 + 3α 2 + 4α 3 + 6α 4 + 5α 5 + 4α 6 + 3α 7 + α 8, 2α 1 + 3α 2 + 4α 3 + 6α 4 + 5α 5 + 4α 6 + 3α 7 + 2α 8
R = f 4 FI Base point all restricted roots FII α 1 + 2α 2 + 3α 3 + 2α 4 FIII α 1 + α 2 + α 3, α 1 + 2α 2 + 2α 3, α 1 + 2α 2 + 3α 3 + 2α 4, 2α 1 + 3α 2 + 4α 3 + 3α 4 R = g 2 G Base point all restricted roots α 1 + (1/ 3)α 2
Remarks (g, h) : semisimple symmetric pair g := h + 1q (g, h) : the c-dual pair of (g, h) Any semisimple timelike orbit of an s-representation, which is austere submfd. in the pseudo-hyperbolic space is elliptic.
Remarks (g, h) : semisimple symmetric pair g := h + 1q (g, h) : the c-dual pair of (g, h) Any semisimple timelike orbit of an s-representation, which is austere submfd. in the pseudo-hyperbolic space is elliptic. The classification of such orbits reduces to the classification of semisimple spacelike orbits which are austere submfd. in the pseudo-hypersphere. semisimple spacelike orbits of the s-representation of (g, h) which are austere submanifolds in the pseudohypershpere semisimple timelike orbits of the s-representation of (g, h) which are austere submanifolds in the pseudohyperbolic space