Universal centralizers and Poisson transversals Ana B alibanu - PowerPoint PPT Presentation

Universal centralizers and Poisson transversals Ana B˘ alibanu Harvard University Friday Fish Online – September 11, 2020 1

The universal centralizer G semisimple algebraic group of adjoint type over C rank p G q “ l g “ Lie G The regular locus of g is g r “ t x P g | dim G x “ l u . ‚ this is the regular locus of the KKS Poisson structure ‚ x regular semisimple � G x is a maximal torus ‚ x regular nilpotent � G x is a abelian group – C l 2

The universal centralizer Let t e , h , f u Ă g be a regular sl 2 -triple. Theorem (Kostant) The principal slice S “ f ` g e Ă g r meets each regular G -orbit on g exactly once, transversally. Remark S is a Poisson transversal for the KKS Poisson structure. Definition The universal centralizer of g is Z “ tp a , x q P G ˆ g | x P S , a P G x u S . 3

The universal centralizer Z is a smooth, symplectic variety: T ˚ G ˆ G G – G ˆ g ý µ g ˆ g 4

The universal centralizer Z is a smooth, symplectic variety: T ˚ G ˆ G G – G ˆ g p a , x q ý µ µ g ˆ g p a ¨ x , x q ‚ µ ´ 1 p x , x q “ G x Z “ µ ´ 1 p S ∆ q . ñ ‚ the image of µ is tp x , y q P g ˆ g | x P G ¨ y u ñ Z “ µ ´ 1 p S ∆ q “ µ ´ 1 p S ˆ S q is a Poisson transversal in T ˚ G . 4

The universal centralizer G has a canonical smooth compactification G , called the wonderful compactification. Plan Compactify the centralizer fibers of Z in G . � G G T ˚ � T ˚ G G , D Extend the symplectic structure on Z to a log-symplectic structure on its partial compactification. 5

The partial compactification of Z Let ˜ G be the simply-connected cover of G , V a regular irreducible ˜ G -representation. Definition (DeConcini–Procesi) ˜ p End V qzt 0 u G G P p End V q . 6

The partial compactification of Z Let ˜ G be the simply-connected cover of G , V a regular irreducible ˜ G -representation. Definition (DeConcini–Procesi) ˜ p End V qzt 0 u G χ G P p End V q . The wonderful compactification of G is G : “ χ p G q . ‚ independent of V ‚ smooth projective G ˆ G -variety ‚ D : “ G z G is a simple normal crossing divisor 6

The partial compactification of Z Example ˜ V “ C 2 . Then Let G “ PGL 2 G “ SL 2 , � " „ *  a b χ p G q “ P P p M 2 ˆ 2 q | ad ´ bc ‰ 0 , c d and G “ P p M 2 ˆ 2 q – P 3 . " „ *  – P 1 ˆ P 1 . a b D “ P P p M 2 ˆ 2 q | ad ´ bc “ 0 c d Non-example Let G “ PGL n for n ě 3. Then V “ C n is not a regular rep of ˜ G “ SL n , and G fl P n 2 ´ 1 . 7

The partial compactification of Z D “ D 1 Y . . . Y D l . G ˆ G -orbits on G Ð Ñ J Ă t 1 , . . . , l u , in the sense that č O J “ D j . j P J For each J Ă t 1 , . . . , l u : parabolic subgroups P J and P ´ J common Levi L J : “ P J X P ´ J , corresponding Lie algebras p J , p ´ J , l J . L J { Z p L J q O J 8 G { P J ˆ G { P ´ J

The partial compactification of Z The log-cotangent bundle T ˚ G , D of G fits into a short exact sequence T ˚ Ý Ñ G ˆ g ˆ g Ý Ñ T G , D . G , D ã � T ˚ G , D is a Lie algebroid over G with trivial anchor map. The fibers of T ˚ G , D are subalgebras of g ˆ g : for each J Ă t 1 , . . . , l u , there is a basepoint z J P O J such that T ˚ G , D , z J “ p J ˆ l J p ´ J . 9

The partial compactification of Z Definition � p a , x q P G ˆ g | x P S , a P G x ( Z “ S . ‚ generic fiber is a smooth toric variety ‚ special fibers are singular 10

The partial compactification of Z T ˚ G , D has a natural log-symplectic Poisson structure, and T ˚ µ G ˆ G Ý Ý Ñ g ˆ g . ý G , D ‚ µ is projection onto the fibers of G ˆ g ˆ g . ‚ the image of µ is g ˆ g {{ G g . ñ µ ´ 1 p S ∆ q “ µ ´ 1 p S ˆ S q Ă T ˚ G , D is a Poisson transversal. Theorem (B.) Z – µ ´ 1 p S ∆ q Ă T ˚ G , D is a smooth, log-symplectic partial compactification of Z . 11

The multiplicative analogue Plan Integrate this to a multiplicative picture: g � ˜ G . ˜ G G by conjugation ý � corresponding regular locus G r “ t g P ˜ G | dim G g “ l u . ˜ Remark This is the regular locus of the AKM quasi-Poisson structure on ˜ G , whose nondegenerate leaves are the conjugacy classes. 12

The multiplicative analogue Theorem (Steinberg) There is an l -dimensional affine subspace Σ Ă ˜ G r which meets each regular conjugacy class in ˜ G exactly once, transversally. Definition The (multiplicative) universal centralizer of ˜ G is ! G | g P Σ , a P G g ) p a , g q P G ˆ ˜ Z “ Σ . 13

The multiplicative analogue The double D G : “ G ˆ ˜ G has a natural q-Poisson structure with group-valued moment map G ˆ ˜ ˜ G D G ý µ G ˆ ˜ ˜ G 14

The multiplicative analogue The double D G : “ G ˆ ˜ G has a natural q-Poisson structure with group-valued moment map G ˆ ˜ ˜ G D G p a , g q ý µ µ G ˆ ˜ ˜ G p aga ´ 1 , g ´ 1 q Proposition (Finkelberg-Tsymbaliuk) Z “ µ ´ 1 p Σ ∆ q “ µ ´ 1 p Σ ˆ ι p Σ qq Ă D G is a smooth, symplectic algebraic variety. 14

The multiplicative analogue Recall the inclusions T ˚ T ˚ G ˆ g ˆ g G G , D G G . Proposition (B.) T ˚ G , D integrates to a smooth subgroupoid G ˆ ˜ G ˆ ˜ G D G G whose source/target fiber at z I P G is P I ˆ L I P ´ I . 15

The multiplicative analogue Recall the inclusions T ˚ T ˚ G ˆ g ˆ g G G , D G G . Proposition (B.) T ˚ G , D integrates to a smooth subgroupoid G ˆ ˜ G ˆ ˜ G D G D G G G whose source/target fiber at z I P G is P I ˆ L I P ´ I . 15

The multiplicative analogue Definition ! ) p a , g q P G ˆ ˜ G | g P Σ , a P G g Z : “ . D G has a Hamiltonian q-Poisson structure with moment map Ñ ˜ G ˆ ˜ µ G ˆ G Ý Ý D G G . ý Theorem (B. in progress) Z – µ ´ 1 p Σ ∆ q “ µ ´ 1 p Σ ˆ ι p Σ qq Ă D G is a smooth, log-symplectic partial compactification of Z . 16