Toric Degeneration of Gelfand-Cetlin Systems and Potential Functions - PowerPoint PPT Presentation

Toric Degeneration of Gelfand-Cetlin Systems and Potential Functions Yuichi Nohara Mathematical Institute, Tohoku University joint work with Takeo Nishinou and Kazushi Ueda East Asian Symplectic Conference 2009 1

§ 1 Introduction Polarized toric varieties and moment polytopes : Let L → X be a polarized toric variety of dim C = N and fix a T N - invariant K¨ ahler form ω ∈ c 1 ( L ). Then the moment polytope ∆ of X appears in two different stories: • Moment map image: → R N moment map of T N -action , Φ : ( X, ω ) − ∆ = Φ( X ) . • Monomial basis of H 0 ( X, L ): � H 0 ( X, L ) = C z I (weight decomposition) . I ∈ ∆ ∩ Z N Note: Both come from the same T N -action. 2

Flag manifolds. Fl n := { 0 ⊂ V 1 ⊂ · · · ⊂ V n − 1 ⊂ C n | dim V i = i } = U ( n ) /T = GL ( n, C ) /B, where T ⊂ U ( n ) is a maximal torus and B ⊂ GL ( n, C ) is a Borel subgroup. Note that N := dim C Fl n = 1 2 n ( n − 1) . For λ = diag( λ 1 , . . . , λ n ) , λ 1 > λ 2 > · · · > λ n , we can associate ω λ Kostant-Kirillov form (a U ( n )-invariant K¨ ahler form) , L λ → Fl n U ( n )-equivariant line bundle, c 1 ( L λ ) = [ ω λ ] (if λ i ∈ Z ), ∆ λ ⊂ R N Gelfand-Cetlin polytope . 3

The Gelfand-Cetlin polytope ∆ λ ⊂ R N = { ( λ ( k ) ); 1 ≤ i ≤ k ≤ n − 1 } is i a convex polytope given by λ 1 λ 2 λ 3 · · · λ n − 1 λ n ≥ ≥ ≥ ≥ ≥ ≥ λ ( n − 1) λ ( n − 1) λ ( n − 1) 1 2 n − 1 ≥ ≥ ≥ λ ( n − 2) λ ( n − 2) 1 n − 2 ≥ ≥ · · · · · · ≥ ≥ λ (1) 1 4

Flag manifolds and Gelfand-Cetlin polytopes ∆ λ : (i) Gelfand-Cetlin basis: a basis of an irreducible representation H 0 ( Fl n , L λ ) of U ( n ) of highest weight λ , indexed by ∆ λ ∩ Z N . (ii) Gelfand-Cetlin system: a completely integrable system → R N , Φ λ : ( Fl n , ω λ ) − Φ λ ( Fl n ) = ∆ λ Remark. (i) and (ii) do not come from the same torus action: The Hamiltonian torus action of the Gelfand-Cetlin system does not pre- serve the complex structure of Fl n . The common idea is to consider U (1) ⊂ U (2) ⊂ · · · ⊂ U ( n − 1) ⊂ U ( n ) . In the case of flag manifolds, we have one more relation: (iii) Toric degeneration: Fl n degenerate into a toric variety X 0 whose moment polytope is ∆ λ . We call X 0 the Gelfand-Cetlin toric variety. 5

Question: Relation between the toric degeneration and the Gelfand- Cetlin basis/ Gelfand-Cetlin system? Kogan-Miller : The Gelfand-Cetlin basis can be deformed into the monomial basis on the Gelfand-Cetlin toric variety under the toric degeneration. This talk : The Gelfand-Cetlin system can be deformed into the mo- ment map of the torus action on the Gelfand-Cetlin toric variety. Application to symplectic geometry/ mirror symmetry : Compu- tation of the potential function for Gelfand-Cetlin torus fibers. 6

§ 2 Gelfand-Cetlin systems We identify Fl n with the adjoint orbit O λ of λ = diag( λ 1 , . . . , λ n ): � � � � � x ∗ = x, eigenvalues = λ 1 , . . . , λ n U ( n ) /T ∼ = O λ = x ∈ M n ( C ) gT ↔ gλg ∗ For each k = 1 , . . . , n − 1 and x ∈ O λ , set x ( k ) = upper-left k × k submatrix of x, λ ( k ) 1 ( x ) ≥ · · · ≥ λ ( k ) ( x ) : eigenvalues of x ( k ) . k Theorem (Guillemin-Sternberg) . � � λ ( k ) → R N , Φ λ : O λ − x �− → ( x ) i k =1 ,...,n − 1 , i =1 ,...,k is a completely integrable system on ( Fl n , ω λ ) and Φ λ ( O λ ) = ∆ λ . Φ λ is called the Gelfand-Cetlin system. 7

Remark. (i) For k = 1 , . . . , n − 1, we embed U ( k ) in U ( n ) by � � U ( k ) 0 U ( k ) ∼ ⊂ U ( n ) . = 0 1 n − k x �→ x ( k ) ∈ √− 1 u ( k ) ∼ = u ( k ) ∗ is a moment map of the U ( k )-action. (ii) The moment map of the action of maximal torus T is given by x ∈ O λ �− → diag( x 11 , x 22 , . . . , x nn ) . Since � � x kk = tr x ( k ) − tr x ( k − 1) = λ ( k ) λ ( k − 1) − , i i i i the T -action is contained in the Hamiltonian torus action of the Gelfand-Cetlin system. (iii) The Hamiltonian torus action of G-C system is not holomorphic. Hence inverse image of a face of ∆ λ is not a subvariety in general. 8

Example (the case of Fl 3 ) . (1) � 1 Φ λ = ( λ (2) , λ (2) , λ (1) → R 3 . ) : Fl 3 − 1 2 1 (2) � 1 Gelfand-Cetlin polytope ∆ λ : (2) � 2 For every u ∈ Int ∆ λ , L ( u ) := Φ − 1 λ ( u ) is a Lagrangian T 3 . The fiber of the vertex emanating four edges is a Lagrangian S 3 . 9

§ 3 Gelfand-Cetlin basis (Gelfand-Cetlin) Borel-Weil: H 0 ( Fl n , L λ ) is an irred. rep. of U ( n ) of h.w. λ . � H 0 ( Fl n , L λ ) = V λ ( n − 1) irred. decomp. as a U ( n − 1)-rep. λ ( n − 1) Fact: • Each irreducible component has multiplicity at most 1. • multiplicity = 1 iff λ 1 ≥ λ ( n − 1) ≥ λ 2 ≥ λ ( n − 1) ≥ λ 3 ≥ · · · ≥ λ n − 1 ≥ λ ( n − 1) ≥ λ n . 1 2 n − 1 Repeating this process we obtain Gelfand-Cetlin decomposition: � H 0 ( X, L λ ) = dim V Λ = 1 . V Λ , Λ ∈ ∆ λ ∩ Z N Taking v Λ ( ̸ = 0) ∈ V Λ for each Λ, we have Gelfand-Cetlin basis. 10

§ 4 Toric degeneration of flag manifolds Toric degeneration is given by deforming the Pl¨ ucker embedding n − 1 � i C n � � � ( V 1 ⊂ · · · ⊂ V n − 1 ) �→ ( � 1 V 1 , . . . , � n − 1 V n − 1 ) . Fl n ֒ → P , i =1 Theorem (Gonciulea-Lakshmibai, ...) . There exists a flat family � i P ( � i C n ) × C X t ⊂ X ⊂ ↓ ↓ t ∈ C of projective varieties such that X 1 = Fl n , X 0 = Gelfand-Cetlin toric variety . 11

Example. Pl¨ ucker embedding of Fl 3 is given by �� � � � � ∈ P 2 × P 2 � Fl 3 = [ z 0 : z 1 : z 2 ] , [ w 0 : w 1 : w 2 ] � z 0 w 0 = z 1 w 1 + z 2 w 2 . Its toric degeneration: � � �� � � � X = [ z 0 : z 1 : z 2 ] , [ w 0 : w 1 : w 2 ] , t � tz 0 w 0 = z 1 w 1 + z 2 w 2 ⊂ P 2 × P 2 × C � � X 1 = z 1 w 1 + z 2 w 2 = z 0 w 0 Flag manifold, � � X 0 = z 1 w 1 + z 2 w 2 = 0 Gelfand-Cetlin toric variety. Remark. General X t does not have U ( k )-actions. 12

There exists an ( n − 1)-parameter family � i P ( � i C n ) × C n − 1 � X ( t 2 ,...,t n ) ⊂ X ⊂ ↓ ↓ C n − 1 ( t 2 , . . . , t n ) ∈ such that • � X | t 2 = ··· = t n = X , • X (1 ,..., 1) = Flag manifold, • X (0 ,..., 0) = Gelfand-Cetlin toric variety, • U ( k − 1) acts on X (1 ,..., 1 ,t k ,...,t n ) , • T n − 1 × · · · × T k acts on X ( t 2 ,...,t k , 0 ,..., 0) , where T k is a torus ( S 1 ) k corresponding to ( λ ( k ) ) i =1 ,...,k . i ( T n − 1 × · · · × T 1 is the torus acting on X 0 .) 13

Remark. The fact that Pl¨ ucker coordinates are Gelfand-Cetlin basis f : � X → C n − 1 . is important for the construction of � Degeneration is stages (Kogan-Miller): X → C n − 1 to the following piecewise linear path Restrict � f : � ( t 2 , . . . , t n ) = (1 , . . . , 1) ❀ (1 , . . . , 1 , 0) ❀ · · · ❀ (1 , 0 , . . . , 0) ❀ (0 , . . . , 0) The ( n − k + 1)-th stage is given by X k = � f k : X | t 2 = ··· = t k − 1 =1 − → C t k +1 = ··· = t n =0 ∪ ∈ X k,t = X (1 ,..., 1 ,t, 0 ,..., 0) − → t. Then T n − 1 × · · · × T k and U ( k − 1) acts on X k,t for each t . 14

� § 5 Toric degeneration of Gelfand-Cetlin systems Theorem. The Gelfand-Cetlin system can be deformed into the mo- ment map on X 0 in the following sense: (i) For each stage f k : X k → C , there exists Φ k : X k → R N s.t. • Φ k | X k,t : X k,t → R N is a completely integrable system, • Φ n | X n, 1 is the Gelfand-Cetlin system on X n, 1 = Fl n , • Φ 2 | X 2 , 0 is the moment map on X 2 , 0 = X 0 , • Φ k | X k, 0 = Φ k − 1 | X k − 1 , 1 on X k, 0 = X k − 1 , 1 . (ii) There exists a vector field ξ k on X k such that exp(1 − t ) ξ k � X k,t X k, 1 � � � �������� � � � � Φ k � Φ k � � ∆ λ 15

Constructions: Using U ( k − 1) and T n − 1 × · · · × T k -actions, we have λ ( l ) � : X k − → R , eigenvalues for moment maps of U ( l )-action , i � � ν ( j ) → R N : X k − moment map of T j -action . � i i =1 ,...,j Then Φ k is given by � � ν ( n − 1) ν ( k ) λ ( k − 1) λ (1) → R N . , � , . . . , � Φ k = , . . . , � : X k − � 1 i j l ξ k = gradient-Hamiltonian vector field introduced by W.-D. Ruan. Remark. Theorem is true for • partial flag manifolds of type A, • orthogonal flag manifolds (?) 16

(2) � 1 (3) � 1 Example. The Gelfand-Cetlin system on F = SO (4) /T = P 1 × P 1 is � � not the standard moment map. In fact the G-C polytope is given by λ 1 | λ 2 | ≥ ≥ λ (3) 1 ≥ | λ (2) | , 1 which is moment polytope of X 0 = P ( O P 1 (2) ⊕ O P 1 ). Gelfand-Cetlin system on F is the pull-back of the moment map on X 0 under a diffeomorphism F ∼ = X 0 . Remark. In the case of SO ( n ) /T , T -action is not contained in the Hamiltonian torus action of the G-C system, since we need to consider both of even and odd orthogonal groups. 17