Geometry of Soergel Bimodules Ben Webster (joint with Geordie Williamson) IAS/MIT June 17th, 2007 Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 1 / 20
Outline 1 Soergel bimodules Algebra Geometry 2 Hochschild homology 3 Equivariant cohomology Equivariant formality Bott-Samelsons Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 2 / 20
References: This slide show can be downloaded from http://math.berkeley.edu/˜bwebste/HH-SB.pdf Some references: B. W. and G. W., A geometric model for the Hochschild homology of Soergel bimodules . ( http://math.berkeley.edu/ bwebste/hochschild-soergel.pdf ) W. Soergel, Kategorie O, Perverse Garben und Moduln über den Koinvarianten zur Weylgruppe . W. Soergel, The combinatorics of Harish-Chandra bimodules . M. Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules . J. Bernstein and V. Lunts, Equivariant sheaves and functors . Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 3 / 20
Soergel bimodules Algebra Soergel bimodules Let R = C [ x 1 , . . . , x n ] / ( x 1 + · · · + x n ) , and s i be the map permuting x i and x i + 1 and let G = SL ( n , C ) . Like so many objects in mathematics, Soergel bimodules have a number of definitions: 1 One which explains why anyone ever cared: Definition A Soergel bimodule is the image of a projective object in category ˜ O under Soergel’s “combinatoric” functor V . 2 One which is hands-on but totally unilluminating: 3 One which involves disgusting levels of machinery, but which ultimately is the best for working with: Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 4 / 20
Soergel bimodules Algebra Soergel bimodules Let R = C [ x 1 , . . . , x n ] / ( x 1 + · · · + x n ) , and s i be the map permuting x i and x i + 1 and let G = SL ( n , C ) . Like so many objects in mathematics, Soergel bimodules have a number of definitions: 1 One which explains why anyone ever cared: Definition A Soergel bimodule is the image of a projective object in category ˜ O under Soergel’s “combinatoric” functor V . 2 One which is hands-on but totally unilluminating: 3 One which involves disgusting levels of machinery, but which ultimately is the best for working with: Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 4 / 20
Soergel bimodules Algebra Soergel bimodules Let R = C [ x 1 , . . . , x n ] / ( x 1 + · · · + x n ) , and s i be the map permuting x i and x i + 1 and let G = SL ( n , C ) . Like so many objects in mathematics, Soergel bimodules have a number of definitions: 1 One which explains why anyone ever cared: projectives in ˜ O 2 One which is hands-on but totally unilluminating: Definition A Soergel bimodule is a direct sum of summands of tensor products R ⊗ R si 1 R ⊗ R si 2 · · · ⊗ R sim R 3 One which involves disgusting levels of machinery, but which ultimately is the best for working with: Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 4 / 20
Soergel bimodules Algebra Soergel bimodules Let R = C [ x 1 , . . . , x n ] / ( x 1 + · · · + x n ) , and s i be the map permuting x i and x i + 1 and let G = SL ( n , C ) . Like so many objects in mathematics, Soergel bimodules have a number of definitions: 1 One which explains why anyone ever cared: projectives in ˜ O 2 One which is hands-on but totally unilluminating: tensor products 3 One which involves disgusting levels of machinery, but which ultimately is the best for working with: Definition A Soergel bimodule is the hypercohomology of a semi-simple B × B-equivariant perverse sheaf on G. Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 4 / 20
Soergel bimodules Algebra Soergel bimodules Let R = C [ x 1 , . . . , x n ] / ( x 1 + · · · + x n ) , and s i be the map permuting x i and x i + 1 and let G = SL ( n , C ) . Like so many objects in mathematics, Soergel bimodules have a number of definitions: 1 One which explains why anyone ever cared: projectives in ˜ O 2 One which is hands-on but totally unilluminating: tensor products 3 One which involves disgusting levels of machinery, but which ultimately is the best for working with: perverse sheaves While intimidating at first, a multiplicity of definitions is, in fact, a strength rather than a weakness, allowing us to our problems translate back and forth at will. Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 4 / 20
Soergel bimodules Algebra Soergel bimodules for n = 2 When n = 2, then R = C [ x 1 , x 2 ] / ( x 1 + x 2 ) ∼ = C [ y ] with the action of s 1 sending y �→ − y . Thus, R s 1 = C [ y 2 ] and = C [ y ⊗ 1 , 1 ⊗ y ] · r / ( y 2 ⊗ 1 − 1 ⊗ y 2 ) R 1 ∼ = R ⊗ R s 1 R ∼ Proposition The elements r and ( 1 ⊗ y − y ⊗ 1 ) · r generate R 1 ⊗ R R 1 as an R-bimodule, and generate two summands, so R 1 ⊗ R R 1 ∼ = R 1 ⊕ R 1 { 2 } . Corollary Every indecomposable Soergel bimodule for n = 2 is isomorphic to R or R 1 . Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 5 / 20
Soergel bimodules Algebra Soergel bimodules for n = 2 When n = 2, then R = C [ x 1 , x 2 ] / ( x 1 + x 2 ) ∼ = C [ y ] with the action of s 1 sending y �→ − y . Thus, R s 1 = C [ y 2 ] and = C [ y ⊗ 1 , 1 ⊗ y ] · r / ( y 2 ⊗ 1 − 1 ⊗ y 2 ) R 1 ∼ = R ⊗ R s 1 R ∼ Proposition The elements r and ( 1 ⊗ y − y ⊗ 1 ) · r generate R 1 ⊗ R R 1 as an R-bimodule, and generate two summands, so R 1 ⊗ R R 1 ∼ = R 1 ⊕ R 1 { 2 } . Corollary Every indecomposable Soergel bimodule for n = 2 is isomorphic to R or R 1 . Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 5 / 20
Soergel bimodules Algebra Soergel bimodules for n = 2 When n = 2, then R = C [ x 1 , x 2 ] / ( x 1 + x 2 ) ∼ = C [ y ] with the action of s 1 sending y �→ − y . Thus, R s 1 = C [ y 2 ] and = C [ y ⊗ 1 , 1 ⊗ y ] · r / ( y 2 ⊗ 1 − 1 ⊗ y 2 ) R 1 ∼ = R ⊗ R s 1 R ∼ Proposition The elements r and ( 1 ⊗ y − y ⊗ 1 ) · r generate R 1 ⊗ R R 1 as an R-bimodule, and generate two summands, so R 1 ⊗ R R 1 ∼ = R 1 ⊕ R 1 { 2 } . Corollary Every indecomposable Soergel bimodule for n = 2 is isomorphic to R or R 1 . Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 5 / 20
Soergel bimodules Algebra Soergel bimodules for n = 2 When n = 2, then R = C [ x 1 , x 2 ] / ( x 1 + x 2 ) ∼ = C [ y ] with the action of s 1 sending y �→ − y . Thus, R s 1 = C [ y 2 ] and = C [ y ⊗ 1 , 1 ⊗ y ] · r / ( y 2 ⊗ 1 − 1 ⊗ y 2 ) R 1 ∼ = R ⊗ R s 1 R ∼ Proposition The elements r and ( 1 ⊗ y − y ⊗ 1 ) · r generate R 1 ⊗ R R 1 as an R-bimodule, and generate two summands, so R 1 ⊗ R R 1 ∼ = R 1 ⊕ R 1 { 2 } . Corollary Every indecomposable Soergel bimodule for n = 2 is isomorphic to R or R 1 . Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 5 / 20
Soergel bimodules Algebra Soergel bimodules for n = 3 When n = 3, similar calculations show Proposition Every indecomposable Soergel bimodule for n = 2 is isomorphic to one of R , R 1 , R 2 , R 1 ⊗ R R 2 , R 2 ⊗ R R 1 or R ⊗ R S 3 R. Anyone used to playing with SL ( 3 ) will probably note that we have an obvious bijection from S 3 to the set of indecomposable Soergel bimodules: 1 ↔ R ( 12 ) ↔ R 1 ( 23 ) ↔ R 2 ( 123 ) ↔ R 2 ⊗ R R 1 ( 132 ) ↔ R 1 ⊗ R R 2 ( 13 ) ↔ R ⊗ R S 3 R Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 6 / 20
Soergel bimodules Algebra Soergel bimodules for n = 3 When n = 3, similar calculations show Proposition Every indecomposable Soergel bimodule for n = 2 is isomorphic to one of R , R 1 , R 2 , R 1 ⊗ R R 2 , R 2 ⊗ R R 1 or R ⊗ R S 3 R. Anyone used to playing with SL ( 3 ) will probably note that we have an obvious bijection from S 3 to the set of indecomposable Soergel bimodules: 1 ↔ R ( 12 ) ↔ R 1 ( 23 ) ↔ R 2 ( 123 ) ↔ R 2 ⊗ R R 1 ( 132 ) ↔ R 1 ⊗ R R 2 ( 13 ) ↔ R ⊗ R S 3 R Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 6 / 20
Soergel bimodules Algebra Soergel bimodules for n = 3 When n = 3, similar calculations show Proposition Every indecomposable Soergel bimodule for n = 2 is isomorphic to one of R , R 1 , R 2 , R 1 ⊗ R R 2 , R 2 ⊗ R R 1 or R ⊗ R S 3 R. Anyone used to playing with SL ( 3 ) will probably note that we have an obvious bijection from S 3 to the set of indecomposable Soergel bimodules: 1 ↔ R ( 12 ) ↔ R 1 ( 23 ) ↔ R 2 ( 123 ) ↔ R 2 ⊗ R R 1 ( 132 ) ↔ R 1 ⊗ R R 2 ( 13 ) ↔ R ⊗ R S 3 R Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 6 / 20
Soergel bimodules Geometry Indecomposable Soergel bimodules Question In general, is the set of indecomposable Soergel bimodules in bijection with S n ? Definition 2 is perfectly useless at answering this sort of question. But from the perspectives of Definitions 1 or 3, it borders on obvious: Proposition (Soergel) Every indecomposable Soergel bimodule is of the form R w = IH ∗ B × B ( BwB ) = H ∗ B × B ( IC ( BwB )) , What? for w ∈ S n (and these are pairwise not isomorphic). Let G w = BwB . Ben Webster (IAS/MIT) Geometry of Soergel Bimodules June 17th, 2007 7 / 20
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