Criteria for rational smoothness of some symmetric orbit closures - PowerPoint PPT Presentation

Criteria for rational smoothness of some symmetric orbit closures Axel Hultman KTH, Stockholm Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 1

Schubert varieties Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

Schubert varieties • G – reductive algebraic group over C Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

Schubert varieties Example: • G – reductive algebraic group over C • GL n ( C ) Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

Schubert varieties Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

Schubert varieties Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices • complete flags in C n • B = G / B – the flag variety Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

Schubert varieties Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices • complete flags in C n • B = G / B – the flag variety • B -orbits in B ↔ Weyl group W • W = S n (symmetric group) Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

Schubert varieties Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices • complete flags in C n • B = G / B – the flag variety • B -orbits in B ↔ Weyl group W • W = S n (symmetric group) The closure of the orbit indexed by w ∈ W is the Schubert variety X ( w ). Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

Schubert varieties Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices • complete flags in C n • B = G / B – the flag variety • B -orbits in B ↔ Weyl group W • W = S n (symmetric group) The closure of the orbit indexed by w ∈ W is the Schubert variety X ( w ). Well-studied problem: Describe geometric properties, such as the (rationally) singular locus, of X ( w ) in terms of the combinatorics of W . Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 2

Rational smoothness Definition: A complex variety X is rationally smooth at x if � Q if m = 2 · topdim, H m ( X , X \ { y } ; Q ) = 0 otherwise, for all y in some neighbourhood of x . Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 3

Rational smoothness Definition: A complex variety X is rationally smooth at x if � Q if m = 2 · topdim, H m ( X , X \ { y } ; Q ) = 0 otherwise, for all y in some neighbourhood of x . Theorem (Carrell-Kuttler) : For Schubert varieties in simply laced types, rational smoothness is equivalent to smoothness. Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 3

Symmetric orbits Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices • complete flags in C n • B = G / B – the flag variety Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 4

Symmetric orbits Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices • complete flags in C n • B = G / B – the flag variety • g �→ ( g − 1 ) T • θ – an involution on G Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 4

Symmetric orbits Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices • complete flags in C n • B = G / B – the flag variety • g �→ ( g − 1 ) T • θ – an involution on G • K – the fixed point subgroup • O n ( C ) Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 4

Symmetric orbits Example: • G – reductive algebraic group over C • GL n ( C ) • B – Borel subgroup • upper triangular matrices • complete flags in C n • B = G / B – the flag variety • g �→ ( g − 1 ) T • θ – an involution on G • K – the fixed point subgroup • O n ( C ) Fact: K acts on B with finitely many orbits. Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 4

Twisted involutions Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

Twisted involutions The map θ restricts to W as a Coxeter diagram involution. Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

Twisted involutions The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

Twisted involutions The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : I ( θ ) = { w ∈ W | θ ( w ) = w − 1 } ( twisted involutions ) Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

Twisted involutions The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : I ( θ ) = { w ∈ W | θ ( w ) = w − 1 } ( twisted involutions ) ι ( θ ) = { θ ( x − 1 ) x | x ∈ W } ⊆ I ( θ ) ( twisted identities ) Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

Twisted involutions The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : I ( θ ) = { w ∈ W | θ ( w ) = w − 1 } ( twisted involutions ) ι ( θ ) = { θ ( x − 1 ) x | x ∈ W } ⊆ I ( θ ) ( twisted identities ) Richardson-Springer constructed an order preserving map from K -orbit closures to twisted involutions ordered by Bruhat order. Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

Twisted involutions The map θ restricts to W as a Coxeter diagram involution. Two important subsets of W : I ( θ ) = { w ∈ W | θ ( w ) = w − 1 } ( twisted involutions ) ι ( θ ) = { θ ( x − 1 ) x | x ∈ W } ⊆ I ( θ ) ( twisted identities ) Richardson-Springer constructed an order preserving map from K -orbit closures to twisted involutions ordered by Bruhat order. Problem: Describe geometric properties, such as the (rationally) singular locus, of K -orbit closures in terms of the combinatorics of I ( θ ). Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 5

Restricting the setup Being outsmarted by the general problem, we put restrictions on G and θ : Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

Restricting the setup Being outsmarted by the general problem, we put restrictions on G and θ : Hypotheses: Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

Restricting the setup Being outsmarted by the general problem, we put restrictions on G and θ : Hypotheses: The image of the dense K -orbit is in ι ( θ ). ( essential ) Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

Restricting the setup Being outsmarted by the general problem, we put restrictions on G and θ : Hypotheses: The image of the dense K -orbit is in ι ( θ ). ( essential ) K is connected. ( convenient ) Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

Restricting the setup Being outsmarted by the general problem, we put restrictions on G and θ : Hypotheses: The image of the dense K -orbit is in ι ( θ ). ( essential ) K is connected. ( convenient ) Consequence: The RS map is injective with image ι ( θ ). Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 6

Examples Four classes of examples satisfy the hypotheses: Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7

Examples Four classes of examples satisfy the hypotheses: Class 1. ( ∼ Schubert varieties) With G = G ′ × G ′ and θ (( x , y )) = ( y , x ), ι ( θ ) is in bijection with the Weyl group W ′ . Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7

Examples Four classes of examples satisfy the hypotheses: Class 1. ( ∼ Schubert varieties) With G = G ′ × G ′ and θ (( x , y )) = ( y , x ), ι ( θ ) is in bijection with the Weyl group W ′ . Class 2. With G = SL 2 n ( C ), K = Sp 2 n ( C ), ι ( θ ) is in bijection with fixed point free involutions in W = S 2 n (with dual Bruhat order). Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7

Examples Four classes of examples satisfy the hypotheses: Class 1. ( ∼ Schubert varieties) With G = G ′ × G ′ and θ (( x , y )) = ( y , x ), ι ( θ ) is in bijection with the Weyl group W ′ . Class 2. With G = SL 2 n ( C ), K = Sp 2 n ( C ), ι ( θ ) is in bijection with fixed point free involutions in W = S 2 n (with dual Bruhat order). Class 3. G of type D n +1 , K of type B n . Axel Hultman Criteria for rational smoothness of some symmetric orbit closures 7