Parametrization of PSL(n,C)-representations of surface group II - PowerPoint PPT Presentation

Parametrization of PSL(n,C)-representations of surface group II Yuichi Kabaya (Osaka University) Hakone, 31 May 2012 1

Review of part I S : a compact orientable surface (genus g , | ∂S | = b , χ ( S ) < 0) X PSL ( S ) : the PSL(2 , C )-character variety of S In part I, we have constructed a map C 6 g − 6+2 b → X PSL ( S ) essentially considering the action of PSL(2 , C ) on C P 1 . In part II, we will construct PGL( n, C )-representations using the action on the flag manifold F n based on a work of Fock and Goncharov. This is a joint work with Xin Nie. 2

Review of part I S : a compact orientable surface (genus g , | ∂S | = b , χ ( S ) < 0) X PSL ( S ) : the PSL(2 , C )-character variety of S In part I, we have constructed a map C 6 g − 6+2 b → X PSL ( S ) essentially considering the action of PSL(2 , C ) on C P 1 . In part II, we will construct PGL( n, C )-representations using the action on the flag manifold F n based on a work of Fock and Goncharov. This is a joint work with Xin Nie. 2-a

PGL( n, C ) := GL( n, C ) / C ∗ , PSL( n, C ) := SL( n, C ) / { ξ | ξ n = 1 } . These are isomorphic but PGL( n, C ) is convenient for our ar- guments. Flag A (full) flag in C n is a sequence of subspaces { 0 } = V 0 � V 1 � V 2 � · · · � V n = C n We denote the set of all flags by F n . GL( n, C ) and PGL( n, C ) act on F n from the left.     ∗ · · · ∗         F n ∼ . ... .   Fact = GL( n, C ) /B where B = .         O ∗ 3

We represent X ∈ GL( n, C ) by n column vectors: ( x n ) ( x i ∈ C n ) x 1 x 2 · · · X = . An upper triangular matrix acts as   b 11 · · · b 1 n ( b 1 n x 1 + · · · + b nn x n )   . b 11 x 1 b 12 x 1 + b 22 x 2 . . . ... .   X .  =  O b nn By setting X i = span C { x 1 , . . . , x i } , we obtain a map GL( n, C ) /B → F n . This is bijective. We call an element of AF n := GL( n, C ) /U an affine flag where     1 · · · ∗         . ... .   U = . . ( ∃ a projection AF n → F n .)         O 1 4

Generic k-tuples of flags X 1 , . . . , X k : flags Take a representative X i = ( x 1 i · · · x n i ) ∈ GL( n, C ) ( X 1 , . . . , X k ) is generic if 1 . . . x i 1 2 . . . x i 2 k . . . x i k det( x 1 1 x 1 2 . . . x 1 k ) � = 0 for any 0 ≤ i 1 , . . . , i k ≤ n satisfying i 1 + i 2 + · · · + i k = n . The genericity does not depend on the choices of the matrices X i . Moreover for X 1 , . . . , X k ∈ AF n , the determinant is a well- 2 . . . X i k defined complex number. Denote it by det( X i 1 1 X i 2 k ). 5

Generic k-tuples of flags X 1 , . . . , X k : flags Take a representative X i = ( x 1 i · · · x n i ) ∈ GL( n, C ) ( X 1 , . . . , X k ) is generic if 1 . . . x i 1 2 . . . x i 2 k . . . x i k det( x 1 1 x 1 2 . . . x 1 k ) � = 0 for any 0 ≤ i 1 , . . . , i k ≤ n satisfying i 1 + i 2 + · · · + i k = n . The genericity does not depend on the choices of the matrices X i . Moreover for X 1 , . . . , X k ∈ AF n , the determinant is a well- 2 . . . X i k defined complex number. Denote it by det( X i 1 1 X i 2 k ). 5-a

Generic k-tuples of flags X 1 , . . . , X k : flags Take a representative X i = ( x 1 i · · · x n i ) ∈ GL( n, C ) ( X 1 , . . . , X k ) is generic if 1 . . . x i 1 2 . . . x i 2 k . . . x i k det( x 1 1 x 1 2 . . . x 1 k ) � = 0 for any 0 ≤ i 1 , . . . , i k ≤ n satisfying i 1 + i 2 + · · · + i k = n . The genericity does not depend on the choices of the matrices X i . Moreover for X 1 , . . . , X k ∈ AF n , the determinant is a well- 2 . . . X i k defined complex number. Denote it by det( X i 1 1 X i 2 k ). 5-b

n-triangulation A triple ( i, j, k ) of integers satisfying 0 ≤ i, j, k ≤ n and i + j + k = n corresponds to an integral point of a triangle. (4 , 0 , 0) (3 , 0 , 1) (2 , 0 , 2) (1 , 0 , 3) (0 , 4 , 0) (0 , 0 , 4) We give a ‘counter-clockwise’ orientation to each interior edges of the n -triangulation. 6

n-triangulation A triple ( i, j, k ) of integers satisfying 0 ≤ i, j, k ≤ n and i + j + k = n corresponds to an integral point of a triangle. (4 , 0 , 0) (4 , 0 , 0) (3 , 0 , 1) (2 , 1 , 1) (2 , 0 , 2) (1 , 0 , 3) (0 , 4 , 0) (0 , 0 , 4) (0 , 4 , 0) (0 , 0 , 4) We give a ‘counter-clockwise’ orientation to each interior edges of the n -triangulation. 6-a

n-triangulation A triple ( i, j, k ) of integers satisfying 0 ≤ i, j, k ≤ n and i + j + k = n corresponds to an integral point of a triangle. (4 , 0 , 0) (4 , 0 , 0) (3 , 0 , 1) (2 , 1 , 1) (2 , 0 , 2) (1 , 0 , 3) (0 , 4 , 0) (0 , 0 , 4) (0 , 4 , 0) (0 , 0 , 4) We give a ‘counter-clockwise’ orientation to each interior edges of the n -triangulation. 6-b

n-triangulation A triple ( i, j, k ) of integers satisfying 0 ≤ i, j, k ≤ n and i + j + k = n corresponds to an integral point of a triangle. (4 , 0 , 0) (4 , 0 , 0) (3 , 0 , 1) (2 , 1 , 1) (2 , 0 , 2) (1 , 0 , 3) (0 , 4 , 0) (0 , 0 , 4) (0 , 4 , 0) (0 , 0 , 4) We give a ‘counter-clockwise’ orientation to each interior edges of the n -triangulation. 6-c

Definition of the triple ratio X, Y, Z ∈ F n : a generic triple of flags We fix lifts of X, Y, Z to AF n and denote ∆ i,j,k := det( X i Y j Z k ). X ( i + 1 , j, k − 1) ( i + 1 , j − 1 , k ) ( i, j + 1 , k − 1) ( i, j − 1 , k + 1) ( i, j, k ) Y Z ( i − 1 , j + 1 , k ) ( i − 1 , j, k + 1) The triple ratio is defined (for 1 ≤ i, j, k ≤ n − 1) by T i,j,k ( X, Y, Z ) := ∆ i +1 ,j,k − 1 ∆ i − 1 ,j +1 ,k ∆ i,j − 1 ,k +1 ∆ i +1 ,j − 1 ,k ∆ i,j +1 ,k − 1 ∆ i − 1 ,j,k +1 . This does not depend on the choice of the representatives. 7

Definition of the triple ratio X, Y, Z ∈ F n : a generic triple of flags We fix lifts of X, Y, Z to AF n and denote ∆ i,j,k := det( X i Y j Z k ). X ( i + 1 , j, k − 1) ( i + 1 , j − 1 , k ) ( i, j + 1 , k − 1) ( i, j − 1 , k + 1) ( i, j, k ) Y Z ( i − 1 , j + 1 , k ) ( i − 1 , j, k + 1) The triple ratio is defined (for 1 ≤ i, j, k ≤ n − 1) by T i,j,k ( X, Y, Z ) := ∆ i +1 ,j,k − 1 ∆ i − 1 ,j +1 ,k ∆ i,j − 1 ,k +1 ∆ i +1 ,j − 1 ,k ∆ i,j +1 ,k − 1 ∆ i − 1 ,j,k +1 . This does not depend on the choice of the representatives. 7-a

Definition of the triple ratio X, Y, Z ∈ F n : a generic triple of flags We fix lifts of X, Y, Z to AF n and denote ∆ i,j,k := det( X i Y j Z k ). X ( i + 1 , j, k − 1) ( i + 1 , j − 1 , k ) ( i, j + 1 , k − 1) ( i, j − 1 , k + 1) ( i, j, k ) Y Z ( i − 1 , j + 1 , k ) ( i − 1 , j, k + 1) The triple ratio is defined (for 1 ≤ i, j, k ≤ n − 1) by T i,j,k ( X, Y, Z ) := ∆ i +1 ,j,k − 1 ∆ i − 1 ,j +1 ,k ∆ i,j − 1 ,k +1 ∆ i +1 ,j − 1 ,k ∆ i,j +1 ,k − 1 ∆ i − 1 ,j,k +1 . This does not depend on the choice of the representatives. 7-b

Definition of the triple ratio X, Y, Z ∈ F n : a generic triple of flags We fix lifts of X, Y, Z to AF n and denote ∆ i,j,k := det( X i Y j Z k ). X ( i + 1 , j, k − 1) ( i + 1 , j − 1 , k ) ( i, j + 1 , k − 1) ( i, j − 1 , k + 1) ( i, j, k ) Y Z ( i − 1 , j + 1 , k ) ( i − 1 , j, k + 1) The triple ratio is defined (for 1 ≤ i, j, k ≤ n − 1) by T i,j,k ( X, Y, Z ) := ∆ i +1 ,j,k − 1 ∆ i − 1 ,j +1 ,k ∆ i,j − 1 ,k +1 ∆ i +1 ,j − 1 ,k ∆ i,j +1 ,k − 1 ∆ i − 1 ,j,k +1 . This does not depend on the choice of the representatives. 7-c

Definition of the triple ratio X, Y, Z ∈ F n : a generic triple of flags We fix lifts of X, Y, Z to AF n and denote ∆ i,j,k := det( X i Y j Z k ). X ( i + 1 , j, k − 1) ( i + 1 , j − 1 , k ) ( i, j + 1 , k − 1) ( i, j − 1 , k + 1) ( i, j, k ) Y Z ( i − 1 , j + 1 , k ) ( i − 1 , j, k + 1) The triple ratio is defined (for 1 ≤ i, j, k ≤ n − 1) by T i,j,k ( X, Y, Z ) := ∆ i +1 ,j,k − 1 ∆ i − 1 ,j +1 ,k ∆ i,j − 1 ,k +1 ∆ i +1 ,j − 1 ,k ∆ i,j +1 ,k − 1 ∆ i − 1 ,j,k +1 . This does not depend on the choice of the representatives. 7-d

Definition of the triple ratio X, Y, Z ∈ F n : a generic triple of flags We fix lifts of X, Y, Z to AF n and denote ∆ i,j,k := det( X i Y j Z k ). X ( i + 1 , j, k − 1) ( i + 1 , j − 1 , k ) ( i, j + 1 , k − 1) ( i, j − 1 , k + 1) ( i, j, k ) Y Z ( i − 1 , j + 1 , k ) ( i − 1 , j, k + 1) The triple ratio is defined (for 1 ≤ i, j, k ≤ n − 1) by T i,j,k ( X, Y, Z ) := ∆ i +1 ,j,k − 1 ∆ i − 1 ,j +1 ,k ∆ i,j − 1 ,k +1 ∆ i +1 ,j − 1 ,k ∆ i,j +1 ,k − 1 ∆ i − 1 ,j,k +1 . This does not depend on the choice of the representatives. 7-e

Definition of the triple ratio X, Y, Z ∈ F n : a generic triple of flags We fix lifts of X, Y, Z to AF n and denote ∆ i,j,k := det( X i Y j Z k ). X ( i + 1 , j, k − 1) ( i + 1 , j − 1 , k ) ( i, j + 1 , k − 1) ( i, j − 1 , k + 1) ( i, j, k ) Y Z ( i − 1 , j + 1 , k ) ( i − 1 , j, k + 1) The triple ratio is defined (for 1 ≤ i, j, k ≤ n − 1) by T i,j,k ( X, Y, Z ) := ∆ i +1 ,j,k − 1 ∆ i − 1 ,j +1 ,k ∆ i,j − 1 ,k +1 ∆ i +1 ,j − 1 ,k ∆ i,j +1 ,k − 1 ∆ i − 1 ,j,k +1 . This does not depend on the choice of the representatives. 7-f