Equivariant K -theory and tangent spaces to Schubert varieties - PowerPoint PPT Presentation

Equivariant K -theory and tangent spaces to Schubert varieties William Graham and Victor Kreiman

Flag varieties Notation ◮ G = simple algebraic group ◮ B = Borel subgroup, B − = opposite Borel subgroup ◮ T = maximal torus contained in B ◮ B = TU , B − = TU − ◮ If V is a representation of T , the set of weights of V is denoted Φ( V ) ◮ X = G / B , the flag variety ◮ g , b , t , u , u − denote Lie algebras of the corresponding groups. ◮ W = Weyl group, equipped with Bruhat order ◮ The T -fixed points of X are xB for x ∈ W .

Tangent spaces to Schubert varieties There is an open cell in X containing xB : ◮ Let U − ( x ) = xU − x − 1 with Lie algebra u − ( x ) ◮ U − ( x ) xB is an open cell C x containing xB . Schubert varieties ◮ X = G / B , X w = B − · wB , Schubert variety, codim ℓ ( w ) . ◮ The T -fixed point xB is in X w if and only if x ≥ w in the Bruhat order. ◮ One would like to understand the singularities of X w at xB . ◮ Write T x X w for T xB X w . ◮ More modest goal: Understand the Zariski tangent space T x X w , or equivalently, the set of weights Φ( T x X w ) . ◮ Φ( T x X w ) ⊆ Φ( T x C x ) = x Φ − .

Equivariant K -theory ◮ For classical groups, Φ( T x X w ) has been described. ◮ The description is complicated except in type A . ◮ Goal: obtain some information about Φ( T x X w ) from equivariant K -theory. Motivation ◮ There are ways to do calculations in equivariant K -theory which are uniform across types. ◮ One can obtain information about multiplicities from these calculations but some cancellations are required. ◮ The set of weights Φ( T x X w ) is related to these cancellations.

Generalized flag varieties ◮ Suppose P = LU P ⊃ B is a parabolic subgroup. ◮ X P = G / P generalized flag variety. P = B − · wP , Schubert variety in G / P . ◮ X w ◮ W P = minimal coset representatives of W with respect to W P = Weyl group of L . ◮ Let π : X → X P . If w ∈ W P , then π − 1 ( X w P ) = X w . ◮ Because π is a fiber bundle map, if we understand Φ( T x X w P ) then we can understand Φ( T x X w ) .

Generalized flag varieties Remark Sometimes it is useful to take P to be the largest parabolic subgroup such that w is in W P , and then study X w P . ◮ The simple roots of the Levi factor L are the α such that ws α > w . Tangent and normal spaces ◮ Let x , w ∈ W P with x ≥ w . P x − 1 → X P , y �→ y · xP , gives an isomorphism ◮ The map xU − P x − 1 with an open cell C x , P in X P containing xP . of xU − ◮ Let Φ amb = Φ( T x X P ) = x Φ( u − P ) . (“Amb” for “ambient”.) ◮ Let Φ tan = Φ( T x X w P ) . ◮ Let Φ nor = Φ amb \ Φ tan .

Equivariant K -theory ◮ If T acts on a smooth scheme M , K T ( M ) denotes the Grothendieck group of T -equivariant coherent sheaves (or vector bundles) on M . ◮ K T ( M ) is a module for K T ( point ) , which equals the representation ring R ( T ) of T (spanned by e λ for λ ∈ ˆ T ). ◮ A T -invariant closed subscheme Y of M has structure sheaf O Y , which defines a class [ O Y ] ∈ K T ( M ) ◮ If i m : { m } ֒ → M is the inclusion of a T -fixed point, there is a pullback i ∗ m : K T ( M ) → K T ( { m } ) = R ( T ) .

Pullbacks of Schubert classes If Y is a Schubert variety in a flag variety M , the pullback i ∗ m [ O Y ] can be computed. Notation ◮ Let i x : { xP } → X P denote the inclusion. ◮ i ∗ x [ O X w P ] denotes the pullback of the Schubert class to xP . ◮ This is the same as the pullback of [ O X w ] to xB .

The 0-Hecke algebra The 0-Hecke algebra arises in the formulas for the K -theory pullbacks. Definition The 0-Hecke algebra is a free R ( T ) -algebra with basis H w , for w ∈ W . Multiplication: Let s be a simple reflection. ◮ H s H w = H sw if l ( sw ) > l ( w ) ◮ H s H w = H w if l ( sw ) < l ( w ) ◮ H 2 s = H s ◮ H 1 is the identity element.

Sequences of reflections Let s = ( s 1 , s 2 , . . . , s l ) be a sequence of simple reflections. Define the Demazure product δ ( s ) ∈ W by the formula H s 1 · · · H s l = H δ ( s ) . ◮ δ ( s ) ≥ w iff s contains a subexpression multiplying to w (Knutson-Miller). ◮ In particular, δ ( s ) ≥ s 1 s 2 · · · s l , with equality if s is reduced. Subsequences ◮ Let w ∈ W . Define T w , s to be the set of sequences t = ( i 1 , . . . , i m ) , where 1 ≤ i 1 < · · · < i m ≤ l , such that H s i 1 · · · H s im = H w . ◮ Define the length ℓ ( t ) = m and the excess e ( t ) = ℓ ( t ) − ℓ ( w ) .

A pullback formula Reduced expressions and inversion sets ◮ Let s = ( s 1 , s 2 , . . . , s l ) be a reduced expression for x . ◮ Let γ i = s 1 · · · s i − 1 ( α i ) . ◮ The inversion set I ( x − 1 ) = Φ + ∩ x Φ − = { γ 1 , . . . , γ l } . The pullback formula Theorem (G.-Willems) Let x , w ∈ W P , x ≥ w. Then � ( − 1 ) e ( t ) � ( 1 − e − γ i ) . i ∗ x [ O X w P ] = t ∈ T w , s i ∈ t Let P s denote the right hand side of this expression.

The expression P s ◮ The expression P s is a sum of monomials in 1 − e − γ 1 , . . . , 1 − e − γ l . ◮ There is one monomial for each t ∈ T w , s , that is, for each subexpression t = ( i 1 , . . . , i m ) such that H s i 1 · · · H s im = H w . ◮ That monomial is � i ∈ t ( 1 − e − γ i ) (up to sign). ◮ We will be interested in the weights γ i such that 1 − e − γ i occurs as a factor in each of these monomials. ◮ This is equivalent to saying that i lies in every subexpression t ∈ T w , s .

Indecomposable elements Recall that for x ≥ w in W P , we defined ◮ Φ amb = Φ( T x X P ) = x Φ( u − P ) . (“Amb” for “ambient”.) ◮ Φ tan = Φ( T x X w P ) . ◮ Φ nor = Φ amb \ Φ tan . An element α ∈ Φ amb is called indecomposable if α cannot be written as a positive linear combination of other elements of Φ amb .

Weights of the normal space The main result of the talk is: Theorem Let γ i be indecomposable in Φ amb . Then γ i is in Φ nor if and only if i lies in every subexpression t ∈ T ( w , s ) . Remark ◮ If i lies in every subexpression t ∈ T ( w , s ) , then 1 − e − γ i is a factor of i ∗ x [ O X w P ] . ◮ To motivate why the theorem might be true, we look at the connection between normal spaces and factors of i ∗ x [ O X w P ] .

Equivariant K -theory and tangent spaces By replacing X P by the cell C x , P , which is isomorphic to a vector space V , and X w P by its intersection with the cell, we can assume we are in the following model situation: ◮ V = representation of T such that all weights Φ( V ) lie in an open half-space and all weight spaces are 1-dimensional ◮ Y = closed T -stable subvariety of V ◮ The T -fixed point is the origin, and i x corresponds to i : { 0 } ֒ → V . ◮ In our model situation, i ∗ is an isomorphism in equivariant K -theory, so we can simply omit the pullbacks to the origin. ◮ Let � ( 1 − e − α ) . λ − 1 ( V ∗ ) = α ∈ Φ( V )

Equivariant K -theory and tangent spaces More definitions ◮ Let C = tangent cone to Y at 0; then C ⊂ V ′ = T 0 Y . ◮ The normal space is V / V ′ . ◮ Write Φ amb = Φ( V ) , Φ tan = Φ( V ′ ) , Φ nor = Φ amb \ Φ tan .

Equivariant K -theory and tangent spaces ◮ Since C ⊂ V ′ , we have classes [ O C ] V ′ ∈ K T ( V ′ ) and [ O C ] V ∈ K T ( V ) . ◮ We also have [ O Y ] V ∈ K T ( V ) . ◮ In our Schubert situation, [ O Y ] V corresponds to i ∗ x [ O X w P ] = P w , s . ◮ [ O C ] V = [ O Y ] V , and [ O C ] V = λ − 1 (( V / V ′ ) ∗ )[ O C ] V ′ . ◮ Conclude: If α ∈ Φ nor , then 1 − e − α is a factor of [ O Y ] V . ◮ One can show that if α is indecomposable in Φ amb , then the converse holds: If 1 − e − α is a factor of [ O Y ] V then α ∈ Φ nor . ◮ This implies one implication of our main theorem. Suppose γ i is indecomposable in Φ amb . If i is in each subxpression t in T w , s , then 1 − e − γ i is a factor of i ∗ x [ O X w P ] = P w , s , so γ i ∈ Φ nor .

Sketch of the proof of the converse For the other implication, again suppose γ i is indecomposable in Φ amb . ◮ Suppose that there exists some subexpression t in T w , s such that i is not in t . We want to show that γ i is in Φ tan . ◮ One can describe the set of weights of the coordinate ring C [ C ] of the tangent cone in terms of the pullback i ∗ x [ O X w P ] . ◮ The hypothesis that i is not in some t , combined with the formula for P w , s , can be used to show that − γ i is a weight of C [ C ] . ◮ Since γ i is indecomposable, the weight − γ i must occur in the degree 1 component of the graded ring C [ C ] . ◮ The weights of this degree 1 component are exactly − Φ tan , so γ i ∈ Φ tan .