Compactifications of reductive groups, non-abelian symplectic - PowerPoint PPT Presentation

Main Question Modular compactifications Symplectic cutting Compactifications of reductive groups, non-abelian symplectic cutting and geometric quantisation of non-compact spaces Johan Martens QGM, Aarhus University E T R M Y O Q E O G C E N T R S E E M F F C U T O A G M M P O R S D N U Q I L A U joint work with Michael Thaddeus (Columbia University) arXiv:1105.4830, arXiv:1210.8161, Transform. Groups Dec 2012 EMS-DMF meeting, Århus, April 2013 Johan Martens Group compactifications & symplectic cutting 1 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups Let G be (connected) split reductive group over a field (i.e. over C , G = K C , with K compact Lie group) e.g. G = semi-simple, GL ( n , C ) , ( C ∗ ) n , Spin c C , . . . Johan Martens Group compactifications & symplectic cutting 2 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups Let G be (connected) split reductive group over a field (i.e. over C , G = K C , with K compact Lie group) e.g. G = semi-simple, GL ( n , C ) , ( C ∗ ) n , Spin c C , . . . Question What are ‘good’ compactifications G of G ? Here ‘good’ should mean G × G -equivariant smooth, with all orbit closures smooth boundary G \ G is a smooth normal crossing divisor nice enumeration of orbits Johan Martens Group compactifications & symplectic cutting 2 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups Ideally want some conceptual understanding of boundary G \ G modular compactification? Johan Martens Group compactifications & symplectic cutting 3 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups Toric varieties & fans Toric varieties T are normal T -equivariant varieties with open dense orbit Determined by fans: collection of strongly convex, rational cones in Λ T ⊗ Z Q every cone simplicial ⇒ at worst finite quotient singularities non-minimal element on ray ⇒ extra orbifold-structure fan complete ⇒ T compact Johan Martens Group compactifications & symplectic cutting 4 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups Wonderful compactfication of adjoint groups G adjoint, i.e. ZG = { 1 } e.g. PGL ( n ) , SO ( 2 n + 1 , C ) , E 8 , F 4 , G 2 Johan Martens Group compactifications & symplectic cutting 5 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups Wonderful compactfication of adjoint groups G adjoint, i.e. ZG = { 1 } e.g. PGL ( n ) , SO ( 2 n + 1 , C ) , E 8 , F 4 , G 2 λ regular dominant weight highest weight representation V λ have G End ( V λ ) P ( End ( V λ )) Johan Martens Group compactifications & symplectic cutting 5 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups Wonderful compactification Definition (De Concini - Procesi) w of an adjoint group is the The wonderful compactification G closure in P ( End ( V λ )) Independent of choice of λ Johan Martens Group compactifications & symplectic cutting 6 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups Wonderful compactification Definition (De Concini - Procesi) w of an adjoint group is the The wonderful compactification G closure in P ( End ( V λ )) Independent of choice of λ w T G maximal torus in G , take closure in G ⇒ get toric variety T G Fan of T G = Weyl chambers + Λ G ( Λ G = co-weight lattice since G is adjoint) Johan Martens Group compactifications & symplectic cutting 6 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups e.g. PGL ( 3 ) : ̟ ∨ 2 ̟ ∨ 1 Smooth since ̟ ∨ i generate co-weight lattice! Johan Martens Group compactifications & symplectic cutting 7 of 22

Main Question Reductive groups Modular compactifications Review: Toric varieties Symplectic cutting Review: Wonderful compactifications of adjoint groups e.g. SL ( 3 , C ) Corresponding toric variety no longer smooth! Johan Martens Group compactifications & symplectic cutting 8 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Moduli Problem Moduli problem: Johan Martens Group compactifications & symplectic cutting 9 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Moduli Problem Moduli problem: G m -equivariant G -principal bundles on chains of projective lines Johan Martens Group compactifications & symplectic cutting 9 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Moduli Problem Moduli problem: G m -equivariant G -principal bundles on chains of projective lines Johan Martens Group compactifications & symplectic cutting 9 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Moduli Problem Moduli problem: n G m -equivariant G -principal bundles on chains of projective lines Framed at north- and south-poles s Johan Martens Group compactifications & symplectic cutting 9 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Moduli Problem Moduli problem: n G m -equivariant G -principal bundles on chains of projective lines Framed at north- and south-poles Lenght of chain is arbitrary finite, can vary in families s Johan Martens Group compactifications & symplectic cutting 9 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Moduli Problem Moduli problem: n G m -equivariant G -principal bundles on chains of projective lines Framed at north- and south-poles Lenght of chain is arbitrary finite, can vary in families Problem: Too many objects: stack is not separated nor of finite type s Johan Martens Group compactifications & symplectic cutting 9 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Moduli Problem Moduli problem: n G m -equivariant G -principal bundles on chains of projective lines Framed at north- and south-poles Lenght of chain is arbitrary finite, can vary in families Problem: Too many objects: stack is not separated nor of finite type s Cure this by imposing a stability condition Johan Martens Group compactifications & symplectic cutting 9 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Which bundles are stable? Theorem (Birkhoff-Grothendieck-Harder) Every G -principal bundle on P 1 reduces to the maximal torus and up to isomorphims is entirely determined by a co-character Λ ∋ ρ : G m → G unique up to W G -action Johan Martens Group compactifications & symplectic cutting 10 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Which bundles are stable? Theorem (Birkhoff-Grothendieck-Harder) Every G -principal bundle on P 1 reduces to the maximal torus and up to isomorphims is entirely determined by a co-character Λ ∋ ρ : G m → G unique up to W G -action Take two charts given by stereographic projection from s and n , use ρ as transition function Johan Martens Group compactifications & symplectic cutting 10 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Every G m - equivariant G -principal bundle on P 1 is determined by action of G m on fibers over n and s , n ρ n s ρ s given by co-characters ρ n and ρ s , unique up to W G . Johan Martens Group compactifications & symplectic cutting 11 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Every G m - equivariant G -principal bundle on P 1 is determined by action of G m on fibers over n and s , n ρ n s ρ s given by co-characters ρ n and ρ s , unique up to W G . Underlying non-equivariant bundle determined by ρ n − ρ s Johan Martens Group compactifications & symplectic cutting 11 of 22

Main Question Moduli problem Modular compactifications Stability Symplectic cutting Every G m - equivariant G -principal bundle on P 1 is determined by action of G m on fibers over n and s , n ρ n s ρ s given by co-characters ρ n and ρ s , unique up to W G . Underlying non-equivariant bundle determined by ρ n − ρ s Theorem (M.-Thaddeus) Every G m -equivariant G -principal bundle on a chain-of-lines of length n reduces to the maximal torus T G and is given up to isomorphism by an element of Λ n + 1 / W G Johan Martens Group compactifications & symplectic cutting 11 of 22