The symplectic duality of Hermitian symmetric spaces Antonio J. Di - PDF document

The symplectic duality of Hermitian symmetric spaces ∗† Antonio J. Di Scala ‡ Hamburg, July 2008 ∗ -, Loi, A., Symplectic Duality of Symmetric Spaces, Advances in Mathematics 217 (2008) 2336-2352. † -; Loi, A. and Roos, G. The unicity of the symplectic duality, To appear in Transformation Groups. ‡ Politecnico di Torino 1

The unit disc ∆ ⊂ C (I). The unit disc ∆ = { z ∈ C : | z | < 1 } has two well-known symplectic forms ω 0 and ω hyp: ω 0 = i 2 d z ∧ d z, ω 0 ω hyp = (1 − | z | 2 ) 2 . The plane C has also two symplectic forms. Namely, ω 0 = i 2 d z ∧ d z, ω 0 ω FS = (1 + | z | 2 ) 2 . Actually, the Fubini-Study form ω FS on C comes from the standard embedding C ⊂ C P 1 , i.e. z ֒ → ( z : 1). Notice that ( C P 1 , ω FS ) is the compact dual of the unit disc (∆ , ω hyp). 2

The unit disc ∆ ⊂ C (II). Consider the map Φ : ∆ → C given by z Φ( z ) := � 1 − | z | 2 � Φ ∗ ω 0 = ω hyp , We claim that: Φ ∗ ω FS = ω 0 . A map Φ with the above properties is called a bisymplectomorphism of (∆ , ω hyp , ω 0 ) and ( C , ω 0 , ω FS ). 3

The unit disc ∆ ∈ C (III). What about the uniqueness of the bisymplectomorphism Φ ? Let Ψ : ∆ → C be another bisymplectomorphism, i.e. � Ψ ∗ ω 0 = ω hyp , Ψ ∗ ω FS = ω 0 . Then the composition f := Φ − 1 ◦ Ψ is a bisymplectomorphism of (∆ , ω 0 , ω hyp), i.e. f ∗ ( ω 0 ) = ω 0 f ∗ ( ω hyp) = ω hyp So we can introduce the group B (∆) of bisymplectomorphisms of the disc (∆ , ω 0 , ω hyp) . Thus, the map Φ is unique up to elements of B (∆). 4

The unit disc ∆ ⊂ C (IV). The following theorem gives a description of B (∆). Theorem 0.1. The elements f ∈ B (∆) are the maps de- fined by | z | 2 � � f ( z ) = u z ( z ∈ ∆) , where u is a smooth function u : [0 , 1) → S 1 ≃ U (1) . In other words, the restriction of a bisymplectomorphism f ∈ � r 2 � B (∆) to a circle of radius r (0 < r < 1) is the rotation u . Notice that if f ∈ B (∆) then f (0) = 0. 5

The unit disc ∆ ∈ C (Proof II). Sketch of the Proof of Theorem 0.1 : � | z | 2 � It is not difficult to show that the maps f ( z ) = u z , where u is a smooth function u : [0 , 1) → S 1 ≃ U (1) are bisymplectomorphisms. Conversely, assume now that f is a bisymplectomorphism. • Since f preserves both symplectic forms then f preserves ω 0 ω hyp = (1 − | z | 2 ) 2 . Thus, the quotient | f ( z ) | = | z | for z ∈ ∆. • A simple computation shows that f ( z ) = v ( | z | ) z for z ∈ ∆ \ { 0 } and v : (0 , 1) → U (1) smooth. • A Whitney’s Theorem can be used to show that v ( | z | ) = | z | 2 � � u for a smooth u . ✷ 6

The unit disc ∆ ∈ C (Proof I). To prove that Φ ∗ ( ω 0 ) = ω hyp notice that: d Φ − d((1 − | z | 2 ) − 1 / 2 ) z = (1 − | z | 2 ) − 1 / 2 d z. So Φ(d Φ − d((1 − | z | 2 ) − 1 / 2 ) .z ) = (1 − | z | 2 ) − 1 z d z . then − i 2 d Φ ∧ d Φ = − i 2 d(Φ(d Φ − d((1 − | z | 2 ) − 1 / 2 ) .z )) = = − i 2 d((1 − | z | 2 ) − 1 z d z ) = ω hyp , since Φ d((1 −| z | 2 ) − 1 / 2 ) z = (1 −| z | 2 ) − 1 / 2 d((1 −| z | 2 ) − 1 / 2 ) | z | 2 is exact. Thus, we get ω hyp = Φ ∗ ( ω 0 ) . The proof that Φ ∗ ( ω FS ) = ω 0 is similar. 7

The Cartan’s domain D 1 [ n ] (I). D 1 [ n ] ⊂ C n 2 ∼ = M n ( C ) is given by D 1 [ n ] := { Z ∈ M n ( C ) | I n − ZZ ∗ >> 0 } . So D 1 [ n ] has two standard symplectic forms ω 0 and ω hyp given by: ω 0 = i 2 d Z ∧ d Z, ω hyp = − i 2 ∂∂ log det( I n − ZZ ∗ ) . The complex euclidean space C n 2 ∼ = M n ( C ) has two symplec- tic forms: ω 0 = i 2 d Z ∧ d Z, ω FS = i 2 ∂∂ log det( I n + ZZ ∗ ) . 8

The Cartan’s domain D 1 [ n ] (II). Notice that D 1 [ n ] ⊂ C n 2 ⊂ G n ( C 2 n ) ֒ → C P N . The last arrow is the Pl¨ ucker embedding → C P N , G n ( C 2 n ) ֒ � 2 n � − 1 and G n ( C 2 n ) is the complex Grass- where N = n mannian of complex n subspaces of C 2 n . Notice that G n ( C 2 n ) is the compact dual of D 1 [ n ]. Indeed, the form ω FS on C n 2 comes as the pullback form of ( C P N , ω FS ) via the above embedding. 9

The Cartan’s domain D 1 [ n ] (III). Now we can ask the following two questions: • Do there exist a bisymplectomorphism Φ : ( D 1 [ n ] , ω 0 , ω hyp) → ( C n 2 , ω FS , ω 0 ) , i.e. a diffeomorphism Φ : D 1 [ n ] → C n 2 such that: Φ ∗ ( ω 0 ) = ω hyp , Φ ∗ ( ω FS ) = ω 0 ? • It is possible to describe the group B ( D 1 [ n ]) of diffeomor- phisms f of D 1 [ n ] such that: f ∗ ( ω 0 ) = ω 0 f ∗ ( ω hyp) = ω hyp ? 10

The Cartan’s domain D 1 [ n ] (IV). Claim: The map Φ : D 1 [ n ] → C n 2 ∼ = M n ( C ) given by Φ( Z ) := ( I n − ZZ ∗ ) − 1 / 2 Z is a bisymplectomorphism. That is to say, Φ is a diffeomor- phism and : Φ ∗ ( ω 0 ) = ω hyp , Φ ∗ ( ω FS ) = ω 0 . 11

The Cartan’s domain D 1 [ n ] (V). First of all observe that we can write ω hyp = − i 2 ∂ ∂ log det( I n − ZZ ∗ ) = i 2 d ∂ log det( I n − ZZ ∗ ) = = i 2 d ∂ tr log( I n − ZZ ∗ ) = i 2 d tr ∂ log( I n − ZZ ∗ ) = = − i 2 d tr[ Z ∗ ( I n − ZZ ∗ ) − 1 d Z ] , where we use the decomposition d = ∂ + ¯ ∂ and the identity log det A = tr log A . By substituting X = ( I n − ZZ ∗ ) − 1 2 Z in the last expression one gets: − i 2 d tr[ Z ∗ ( I n − ZZ ∗ ) − 1 d Z ] = = − i 2 d tr( X ∗ dX ) + i 2 d tr { X ∗ d [( I n − ZZ ∗ ) − 1 2 ] Z . Finally, notice that the 1-form tr[ X ∗ d( I n − ZZ ∗ ) − 1 2 Z ] is exact being equal to d tr( C 2 2 − log C ), where C = ( I n − ZZ ∗ ) − 1 2 . So Φ ∗ ( ω 0 ) = ω hyp . The proof that Φ ∗ ( ω FS ) = ω 0 is similar. 12

The general picture (I). Let Ω be a symmetric bounded domain and let Ω ∗ be its compact dual. Assume dim C (Ω) = n . The following inclusions are well-known: Ω ⊂ C n ⊂ Ω ∗ ֒ → C P N , where the last arrow is the Borel-Weil embedding. So the compact dual Ω ∗ and C n can be endowed with the pullback form of the Fubini-Study form ω FS of C P N . Thus, we can regard C n as a complex euclidean space equipped with two symplectic forms ω 0 and ω FS . 13

The general picture (II). We can ask about the existence and uniqueness of a symplectic duality map Φ. Namely, • Do there exist a bisymplectomorphism Φ : (Ω , ω 0 , ω hyp) → ( C n , ω FS , ω 0 ) , i.e. a diffeomorphism Φ : Ω → C n such that: Φ ∗ ( ω 0 ) = ω hyp , Φ ∗ ( ω FS ) = ω 0 ? • It is possible to describe the group B (Ω) of diffeomor- phisms f of Ω such that: f ∗ ( ω 0 ) = ω 0 f ∗ ( ω hyp) = ω hyp ? 14

Related results (I). The existence of a symplectomorphism: ψ : (Ω , ω hyp) → ( C n , ω 0 ) was proved by D. McDuff in The symplectic structure of K¨ ahler manifolds of non-positive curvature , J. Diff. Ge- ometry 28 (1988), pp. 467-475. As a conclusion it follows that the symplectic struture ω hyp on R 2 n is not exotic. 15

Related results (II). • Notice that our question is stronger. Namely, we ask about the existence of a BISYMPLECTOMORPHISM , i.e. : Φ ∗ ( ω 0 ) = ω hyp , Φ ∗ ( ω FS ) = ω 0 ? • Observe that McDuff’s theorem is existencial ,i.e. there is not given an explicit symplectomorphism. Actually, we are going to give an explicit formula for our bisymplectomorphism Φ. Moreover, we are going to give an explicit description of all bysimplectomorphism Φ’s. 16

Bounded Symmetric Domains and Hermitian Jordan Triple systems (I). We use the approach ”via” Jordan Algebras, due to Max Koecher, to construct all the symmetric bounded domains Ω ⊂ C n by starting with a Hermitian Positive Jordan Triple Sys- tem ( V, { , , } ) : • V = C n and { , , } : V 3 → V , • { x, y, z } is C -bilinear in ( x, z ) and C -anti-linear in y . • satisfying the Jordan identity : { x, y, { u, v, w }} − { u, v, { x, y, w }} = = {{ x, y, u } , v, w } − { u, { v, x, y } , w } . • the sesquilinear form ( x | y ) := traceD ( x, y ) is positive, where D ( x, y )( · ) := { x, y, ·} . 17

Bounded Symmetric Domains and Hermitian Jordan Triple systems (II). Each element x ∈ V has a spectral decomposition : x = λ 1 c 1 + λ 2 c 2 + · · · + λ r c r , where λ 1 ≥ λ 2 ≥ · · · ≥ 0 and ( c 1 , c 2 , · · · , c r ) is a frame a maximal system of mutually orthogonal tripotents, i.e. { c i , c i , c i } = 0 and D ( c i , c j ) = 0 for i � = j . Unique just for elements x ∈ V of maximal rank r . There exist polynomials m 1 , . . . , m r on M × M , homoge- neous of respective bidegrees (1 , 1) , . . . , ( r, r ), such that for x ∈ M , the polynomial m ( T, x, y ) = T r − m 1 ( x, y ) T r − 1 + · · · + ( − 1) r m r ( x, y ) satisfies r � ( T − λ 2 m ( T, x, x ) = i ) , i =1 where x is the spectral decomposition of x = � λ j c j . The inohomogeneous polynomial N ( x, y ) = m (1 , x, y ) is called the generic norm . 18

Bounded Symmetric Domains and Hermitian Jordan Triple systems (III). Construction of the bounded domain Ω. • The Spectral Norm | z | of z ∈ V is defined as | z | 2 := � D ( z, z ) � 2 where �·� is the operator norm in V endowed with ( ·| · ). • The bounded domain attached to the HPJTS ( V, { , , } ) is given by: Ω := { z ∈ V : | z | < 1 } . That is to say, Ω is the unit sphere w.r.t. the Spectral Norm . 19