Flat Projective Connections

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1 Flat Projective Connections A. Beilinson and D. Kazhdan 1. Geometric Quantization In this section we recall basic points of Kostant s geometric quantization [K] 2. We consider a purely holomorphic version, so all the objects below will be algebraic or analytic ones. The language of complex polarizations is discussed in no Recollections from Classical Mechanics Definition. Let p : X S be a smooth morphism of smooth varieties. A connection for p, or simply, or simply, p-connection, is an O X -linear morphism S : p T S T X such that dp S = id p T S ; such S is integrable if the corresponding map T S p T X commutes with brackets. Let S be a p-connection. Then S (p T S ) T X is a subbundle transverse to fibers of p; we will call it S -horizontal subbundle. Conversely, any subbundle transverse to fibers of a smooth p defines a p-connection which is integrable iff the subbundle is integrable. For any s S an integrable Partially supported by NSF Grant. 1

2 p-connection S defines a trivialization of p over a formal neighborhood S s of s (i.e., the isomorphism X S s = X S S s, where X S = p 1 (s), etc.). Localizing on S we see that p-connections form a sheaf p-conn on S. If is a p-connection and ν Hom(T S, p T X/S ) = Ω 1 S p T X/S, then + ν is also a p-connection. This way p-conn is an Ω 1 S p T X/S -torsor. Note that an integrable p-connection S defines an action of T S on relative differential forms Ω X/S (by Lie derivatives along horizontal vector field S (T S )); we will say that a form ω Ω i X/S is S-horizontal if ω is fixed by the T S -action. Let (X, ω) be a symplectic variety, i.e., X is a smooth variety and ω is a non-degenerate closed 2-form on X. Then ω defines Poisson brackets {, } on O X in a usual manner Definition. A surjective morphism of varieties π : X Y is called polarization, or Lagrangian projection, if dim Y = 1 2 dim X and {, } vanishes on π 1 O Y O X ; such π is called smooth if π is a smooth morphism. A basic example of such π : X Y is a twisted cotangent bundle over Y (see A1.8, A1.9) Definition. Let S be a smooth variety. An S-Lagrangian triple consists of a morphism π : X Y of S-varieties (i.e., one has a commutivative diagram X p X π S Y p Y ), a relative 2-form ω Ω 2 X/S and a p-connection S such that 2

3 (i) p X, p Y and π are smooth surjective morphisms. (ii) a form ω is closed and non-degenerate, i.e., for any s S the fiber (X s, ω s ) is a symplectic variety. (iii) for any s S the morphism π s : X s Y s is a twisted cotangent bundle over Y s. (iv) S is integrable and ω is S -horizontal. Assume we have a Lagrangian triple (1.1.3). Consider the O Y -algebra A := π O X. It carries O S -linear Poisson bracket {, } and a natural filtration A i such that A 0 = O Y, A i = S i A 1 and gr.a = S T Y/S (see A1.8). Our connection S is an O S -linear morphism S : T S Der A such that for f O S A, τ T S one has S (τ)(f) = τ(f); according to (iv) S commutes with brackets and for a, b A, τ T S one has S (τ)({a, b}) = { S (τ)(a), b} + {a, S (τ)(b)}. Let n be a minimal integer such that S (τ S )(A 0 ) A n ; such n is called an order of our Lagrangian triple. For example, n = 1 means that for any f, g O Y, τ T S one has { S (τ)(f), g} O Y Lemma. (i) One has S (T S )(A i ) A i+n for any i. Hence we have an O S -linear map gr S : T S Der (n) gr.a = Der (n) S T Y/S (here Der (n) means differentiations of homogeneous degree n). (ii) Assume that n 1. There exists a unique O S -linear map σ S : T S S n+1 T Y/S such that (gr S (τ))(f) = {σ S (τ), f} for τ T S, f S T Y/S. The functions σ S (τ), τ T S, Poisson commute. 3

4 Proof: Clear. Sometimes it is convenient to describe S-Lagrangian triples in a different language. Let Y be an S-variety such that p Y : Y S is smooth and surjective Definition. An S-Hamiltonian datum on Y consists of a twisted cotangent bundle ( X, ω ex ), π : X Y over Y. Put X := X mod p Y Ω1 S : this is a T Y/S -torsor over Y ; let X projections. r X a section h : X X of r (called Hamiltonian of our datum). π Y be the Put ω X := h ω ex : this is a closed 2-form on X. The following integrability axiom should hold: for each x X the form ω x Λ 2 TX x has rank dim X-dim S. Assume we have a Hamiltonian datum Note that for each s S the map π s : X s Y s is the induced (from X) twisted cotangent bundle on Y s (see A3). The symplectic form ω s coincides with ω X Xs, so the integrability axiom asserts that ω X has minimal possible rank (in particular, in case dim S = 1 this axiom holds automatically). The kernels of ω X x, x X, form a subbundle transversal to fibers of p X := p Y π. Since ω X is closed, this subbundle is integrable, hence it defines an integrable connection S for p X. We see that (X π Y, ω, S ) is S-Lagrangian triple Proposition. This correspondence (S-Hamiltonian data on Y ) (S-Lagrangian triples with given p Y : Y S) is bijective. 4

5 Proof: Let us define the inverse correspondence. Let (X π Y, ω, S ) be an S-Lagrangian triple. The connection S extends ω Ω 2 X/S to a 2-form ω X Ω 2 X : one has ω X X s = ω s and for x X the kernel of ω X x Λ 2 T X x coincides with S -horizontal vectors at x. Since S is integrable ω X is a closed form. Let (F ωx, curv ωx ) be the corresponding Ω 1 X -torsor, so F ω X = Ω 1 X, curv ωx (ν) = dν + ω X (see A1.7). The π-vertical part of zero section of F ωx is a π-descent data for F ωx Recall that a section of F Y (see A3.1) which defines Ω 1 Y -torsor (F Y, curv Y ). is a form ν π Ω 1 X to fibers of π vanishes and curv Y (ν) := dν + ω X Ω 2 Y such that the restriction of ν π Ω 2 X. Denote by F Y/S the Ω 1 Y/S-torsor of sections of π : X Y. One has a canonical isomorphism r : F Y mod p Y Ω1 S F Y/S: here for ν F Y r(ν) is a unique section of π such that r(ν) (ν) p Y Ω1 S Ω1 Y. Let h : F Y/S F Y be the map that assigns to a section α of π a unique form h(α) F Y such that a (h(α)) Ω 1 Y vanishes (one has α = β α (β) for any β F Y ). Clearly r h = id FY /S. Let X π Y, ω ex, be a twisted cotangent bundle defined by (F Y, curv Y ), so we have the projection r : X X and the section h : X X of r. This is the desired S-Hamiltonian datum on Y. Remark: The map r : X X S T S, r( x) = (r( x), x h, r( x)) is isomorphism of symplectic manifolds: Here the symplectic form on X S T S is equal to the sum of ω X and a standard symplectic form on T S. Consider an S-Hamiltonian datum ( X, ω ex, π, h) on Y. Let x X be a point, y = π(x), s = p X (x) be the projections of x. Let {t a } be a local coordinate at s, and q i be functions at y such that {q i, t a } are local coordinates at y. Choose a function h a, p i at h(x) X such that 5

6 ω ex = dp i dq i + dh a dt a. Then {q i, p i, t a } are coordinates at x on X, and the Hamiltonian h is given by the functions h a (p, q, t) Lemma. One has S ( ta ) = ta + i q i (h a ) pi pi (h a ) qi. Proof: Follows from ω X ( pi S ( ta )) = ω X ( qi S ( ta )) = 0. Note that the integrability axiom asserts that ω X ( tb S ( ta )) = tb (h a ) ta (h b ) + i p i (h a ) qi (h b ) qi (h a ) pi (h b ) = Corollary. Let m be a minimal order (with respect to y Y ) of polynomial maps h y : X y X y (note that X y, Xy are affine spaces). Then m 1 is equal to the order of corresponding Lagrangian triple (see 1.1.4) Remark: (i) We see that a Hamiltonian datum is just a system of commuting Hamiltonians in a classical sense. (ii) Let F X map curve ex h : F X be the Ω 1 Y/S-torsor of sections of π : X Y ; one has the Ω 2 Y, curv e X h(γ) = (h γ) (ω ex ). The equation curv ex h(?) = 0 is a classical Hamilton-Jacobi equation. 1.2 D-Connections Let p : Y S be any smooth morphism of smooth varieties, and let D Y be a tdo on Y. Denote by D Y/S the centralizer of π 1 O S in D Y. This is a flat π 1 O S -algebra. One may consider D Y/S as a family of tdo parameterized by S. Namely, for s S denote by m s O S the maximal ideal of functions equal to zero at s. Then the quotient D Y/S /m s D Y/S is tdo on Y s = p 1 (s) that coincides with the inverse image of D Y on Y s (see A3). If D Y = D L for 6

7 some line bundle L, then D Y/S consists of differential operators on L acting along fibers of p Definition. (i) A D Y -connection on p is an O S -linear mapping DY : T S p Der(D Y/S ) such that for τ T S, f O S one has DY (τ)(π 1 f) = π 1 τ(f) π 1 O S D Y/S. Such DY is integrable if it commutes with brackets. (ii) A D Y -connection DY is admissible if for any τ T S there exists (locally on S) an element τ p D Y such that for any D Y/S one has DY (τ)( ) = [ τ, ] Remark: One may easily define an obstruction for DY to be admissible; it lies in H 0 (S, Ω 1 S H1 DR (Y/S)). In particular, if the first de Rham cohomology of fibers vanish, any D Y -connection is admissible. We define the order of a D Y -connection as a smallest n such that DY (τ)(o Y ) D Y/S n = (D Y/S ) n for each τ T S Lemma. (i) One has DY (τ)(d Y/S i) D Y/S i+n for any i. Hence we have an O S -linear map gr DY : T S Der (n) gr D Y/S = Der (n) S T Y/S. (ii) If n 1 then there is a unique O S -linear map σ DY : T S p S n+1 T Y/S such that (gr DY (τ))(f) = {σ DY (τ), f} for f S T Y/S. If DY is integrable then the functions σ DY (τ), τ T S, Poisson commute. Proof: (i): Induction by i using D Y/S i = { D Y/S : [, O Y ] D Y/S i 1}. (ii) follows since gr DY (τ) is a differentiation for Poisson brackets Now let D S be a tdo on S. A p-morphism α : D S D Y is a morphism of C-algebras α : D S p D Y that coincides on O S D S with 7

8 O S p 1 p O X p D Y. Clearly α is injective, so α identifies D S with a subalgebra in p D Y containing O S Remark: Consider a filtration L. on D Y by degree along S : so L o = D Y/S, L i = { D Y : ad (π 1 O S ) L i 1 }. One has gr L. D Y = S T S OS D Y/S. Then for a p-morphism α one has α(d S i) L i, and gr α coincides with an obvious embedding S T S S T S OS D Y/S Let α : D S p D Y be a p-morphism. One associates with α an admissible integrable D Y -connection on p as follows. For τ T S choose τ T DS such that σ( τ) = τ. Then α( τ) L 1, hence ad α ( τ) maps D Y/S to itself. Put α (τ) := ad α(τ) DY /S Der D Y/S. It is easy to see that α (τ) does not depend on choice of τ. This morphism α : T S Der D Y/S is our D Y -connection. It is admissible and integrable Lemma. If the fibers of p are connected, then (D S, α) α is a bijection between the set of pairs (D S, α) and admissible integrable D Y - connections on p. Proof: Here is a construction of an inverse map. For an admissible integrable connection = DY put T = {(τ, τ) T S p D Y : for any D Y/S one has (τ)( ) = [ τ, ]}. One has short exact sequence 0 O S i T σ T S 0, where i(f) = (0, p 1 (f)), σ(τ, τ) = τ, and an obvious O S -module and Lie algebra structure on T make T an O S -extension of T S (see A1.3, A1.4). Let D S be the corresponding tdo. The embedding T p D Y, (τ, τ) τ, extends uniquely to a morphism of rings α : D S p D Y which is a p- morphism. This (D S, α ) is a desired pair. 8

9 1.2.8 Remark: For a p-morphism α : D S D Y consider the smallest integer m such that α( T DS ) D Ym. Then m 1 is equal to the order of α, and σ α (τ) = α( τ) mod D Ym 1 S m T Y/S for τ T DS, τ = σ τ T S. 1.3 Quantization Let (X π Y p Y S; ω; S ) be an S-Lagrangian triple (see 1.1.3), so we have a filtered commutative O Y -algebra A = π O X with Poisson bracket {, }, and the Ω 1 Y -torsor (F Y, curv Y ) (see 1.1.6: this torsor corresponds to the twisted cotangent bundle of the Hamiltonian datum). Let Ω = det Ω 1 Y/S be the sheaf of volume forms along the fibers of p Y, and (F Ω, curv Ω ) = d log Ω be the corresponding Ω 1 Y -torsor (see A1.12). Put (F Y, curv Y ) = (F Y, curv Y ) + 1 (F 2 Ω, curv Ω ); let D Y = D (F Y,curv Y ) be the corresponding tdo. From now on we will assume that our Lagrangian triple has order 1, i.e., for τ T S one has S (τ)(o Y ) A 1. According to A2.5 one has a canonical isomorphism σ : T DY /S = D Y/S A Definition. A quantization of our Lagrangian triple is an order 1 integrable D Y -connection DY on p Y such that for τ T S, f O Y = D Y/S 0 one has σ[ DY (τ)(f)] = S (τ)(f). Let DY be any order 1 integrable connection. The following lemma explains how to verify whether DY is a quantization, and also why we took the Ω 1/2 -twist in the definition of D Y. Consider the following sheaves on Y : F A := {(τ, l), τ p 1 Y T S, l : O Y A 1 l(fg) = fl(g) + gl(f), {l(f), g} = {l(g), f} for f, g O Y, l(t) = τ(t) for t p 1 Y O S O Y }. 9

10 F D := {(τ, l ), τ p 1 Y T S, l : O Y A 1 l (fg) = fl (g)+l (f)g, [l (f), g]+ [fl (g)] = 0 for f, g O Y, l (t) = τ(t) for t p 1 Y O S O Y }. One has a short exact sequence of p 1 Y O S-modules: 0 A 2 /A 0 i A F A j A p 1 Y T S 0, 0 D Y 2/D Y 0 i D F D j D p 1 Y T S 0, defined by formulas i A (a) = (0, l(a)), l(a)(f) = {a, f}, j A (τ, l) = τ, i D ( ) = (0, l ( )), l ( )(f) = [, f], j D (τ, l ) = τ. Our connections S, DY define the splittings 0 S, 0 D of j A, j D, respectively, by formulas 0 S (τ) = (τ, S (τ) OY ), 0 D (τ) = (τ, D Y (τ) OY ) Lemma. (i) One has a canonical commutative diagram 0 D Y2 /D Y0 F D p 1 Y T S 0 σ σ F 0 A 2 /A 0 F A p 1 Y T S 0 where σ F is defined by formula σ F (τ, l ) = (τ, l), l(f) = σl (f), and σ : D Yi /D Yi 2 A i /A i 2 was defined in A2.5. (ii) DY is a quantization iff σ F 0 D 0 S Hom(p 1 Y T S, A 2 /A 0 ) is 0. In particular, DY is always a quantization if p Y (A 2 /A 0 ) = 0. Proof: (i) It suffices to verify that σ F (τ, l ) actually lies in F A by a direct computation. (ii) Clear. Let DY be a quantization Lemma. (i) One has σ( S ) = σ( DY ) Ω 1 S p Y S 2 T Y/S. (ii) DY is admissible D Y -connection. 10

11 Proof: (i) Clear, (ii) follows since π has affine fibers, see According to a quantization DY defines a tdo D S on S together with embedding α : D S p Y D Y, which is our primary object of interest. 1.4 Symmetries Assume that we are in a situation 1.2, i.e., we have a smooth map p Y : Y S and a tdo D Y on Y. Let ν Y : g T Y, ν S : g T S be actions of a Lie algebra g on Y and S that commute with p. Let ν DY Der D Y g on a tdo D Y the subalgebra D Y/S D Y. be a weak ν Y -action of (see A4.1). Clearly the derivations ν DY (γ), γ g, preserve Definition. (i) The action ν DY preserves a D Y -connection DY for p if for any γ g, τ T S one has [ν DY (γ), DY (τ)] = DY ([ν S (γ), τ]) Der D Y/S. (ii) The action ν DY preserves a p-morphism α : D S D Y if the derivations ν DY (γ), γ g, preserve a subalgebra D S α p D Y. It is easy to see that if ν DY preserves α, then it preserves α (see 1.2.4); conversely if the fibers of p are connected, then ν DY preserves α if it preserves DY (see 1.2.5). Assume that ν DY preserves a p-morphism α. Then the restriction of operators ν DY (γ), γ g, to D S on D S. α p D Y define a weak ν S -action ν DS of g Let (X π Y ; ω; S ) be an S-Lagrangian triple, and our Lie algebra g acts on it. This means that we have compatible g-actions ν X, ν Y, ν S on 11

12 X, Y and S that fix ω and S (note that, since π and p Y are surjective, ν Y and ν S are uniquely determined by ν X ). We get a canonical weak ν X -action on D ωx and weak ν Y -action of g on D Y (since D ωx, D Y were defined in a canonical way). We will say that g preserves a quantization if ν DY preserves DY. In this case we get a weak ν S -action of g on corresponding D S. Sometimes one needs strong actions on D ωx, D Y rather than just weak ones. One has Lemma. The strong liftings ν ωx : g T DωX for ν ωx are in 1 1 correspondence with ones ν DY : g T DY for ν DY. Proof: Let N T DωX be a normalizer of ωx (T X/Y ). One has T DY = π (N/ ωx (T X/Y )). Now assume we have ν ωx. Clearly ν ωx (g) N, hence ν DY := ν ωx mod ωx (T X/Y ) is a strong lifting of ν DY. Conversely, assume we have ν DY. For γ g an element ν ωx (γ) N such that ad νωx (γ) = ν ωx (γ) defines it up to a constant. The condition that ν ωx (γ) mod ωx T X/Y ) = ν DY defines it uniquely. Note that ν ωx is just an ω X -Hamiltonian lifting of ν ωx (see A4.3(ii)). 1.5 Kostant D-modules Assume we are in a situation 1.2, so we have p : Y S, a tdo D Y on Y, D S on S and a p-morphism α : D S D Y. Let M be a D Y -module. The algebra p D Y acts on sheaf-theoretic direct images R i p M in an obvious manner, hence α defines the functors R i p : D Y -modules D S -modules. If R i p transforms O Y -coherent modules to O S -coherent ones, then it transforms lisse D Y -modules to lisse D S -ones (see A1.14). 12

13 Assume we have an action of a Lie algebra g on our data such that ν DY preserves α (see 1.4.1). Let ν DY : g T DY, ν DS : g T DS be strong liftings of ν DY, ν DS. For a D Y -module M consider a canonical ν DY -action νm 0 of g on M (see A4.4). The induced action of g on Ri p M is obviously a ν DS -action. Hence ν DS defines a canonical action [νm 0 ] : g End D S R i p M of g on R i p M (see A4.5). We get a canonical action of g on the functor R i p, i.e., R i p transforms D Y -modules to D S C U(g)-ones. Now let (X π Y ; ω; S ) be an S-Lagrangian triple Definition. (i) Kostant line bundle is a line bundle L Y on Y equipped with a D Y -module structure (which is an isomorphism D Y D LY ). (ii) An ω X -line bundle is a line bundle L X on X equipped with a D ωx - module structure (which is the same as a connection X on L X with curv X = ω X ). (iii) An ω X -line bundle (L X, X ) is admissible if for any y Y its restriction of (L Xy, Xy ) to the fiber X y is a trivial bundle with connection Remark: Since the fibers X y are affine spaces, in analytic situation any ω X -line bundle is admissible. In algebraic situation admissibility just means that Xy has regular singularities at infinity (see [Bo], [D]). Assume that there exists a line bundle Ω 1/2 on Y together with an isomorphism (Ω 1/2 ) 2 Ω (for notations see 1.3); choose one. Let M be a D Y -module. Then M Ω 1/2 := Ω 1/2 OY M is a D (FY,curv Y )-module. Since D (FY,curv Y ) coincides with π-descent of D ωx, we see that π M Ω 1/2 = O X OY M Ω 1/2 is a D ωx -module. If M is a line bundle, then π M Ω 1/2 is 13

14 an admissible ω X -bundle, so we obtained the functor π Ω 1/2 : (Kostant line bundles) (admissible ω X -bundles), π Ω 1/2 (L Y ) = π (Ω 1/2 L Y ) Lemma. This functor is equivalence of categories. Proof: The inverse functor assigns to (L X, X ) a line bundle Ω 1/2 π L X/Y X, where X/Y is vertical part of X Let DY be a quantization of our symplectic triple, and D S be the corresponding tdo on S. Let L Y be a Kostant line bundle. Then R i p Y L Y are D S -modules, we will call them Kostant D S -modules. If a Kostant D S -module E is lisse (which happens, e.g., when p Y is proper), then E is a vector bundle on S with a canonical integrable projective connection (see A1.14 A1.17) Assume we are in a situation 1.4.2, so we have a Lie algebra g that acts on our Lagrangian triple and preserves a quantization DY. Choose strong liftings ν DY, ν DS are g-modules. By these define an action of g on Kostant D-module Remark: ν DY is the same as ν DY -action of g on a Kostant line bundle. 1.6 Example: Metaplectic Representation Let W be a symplectic C-vector space with symplectic form ω. Let S be a Grassmannian of Lagrangian planes in W, and L W S be a canonical Lagrangian subbundle of a constant vector bundle W S on S. Denote by S the space of the line bundle det L on S with zero section removed. 14

15 Define an S-Lagrangian triple (X π Y ; ω; S ) as follows. Put X = W S = W S, Y = W S /L, π = canonical projection, ω X is a lifting to X of a constant 2-form ω on W, S = constant connection. We may also consider an S -Lagrangian triple (X π Y ; ω ; S ) defined in the same way (this is just a base change by S S of the previous triple). Let g = W Sp(W ) be a Lie algebra of affine symplectic symmetries of W (so W acts by translations). It acts on our Lagrangian triples in an obvious manner (so W acts trivially on S, S ). Let g be a central extension of g by C such that for w 1, w 2 W g one has [ w 1, w 2 ] = ω(w 1 w 2 ). Such g exists and unique up to a unique isomorphism. In fact H 1 ( g, C) = H 2 ( g, C) = 0. By A4.2 one has a canonical strong lifting ν DY : g T DY. It restricts to the Lie algebra map W T DY /S which defines the isomorphism of associative O S -algebra U 1 ( W ) C O S D Y/S ; here U 1 ( W ) is a quotient of a universal envelopping algebra U( W ) modulo relation 1 = 1 C W. Let DY be a D Y -connection for p Y with U 1 ( W ) being the horizontal sections. This is a quantization of our Lagrangian triple; let D S be the corresponding tdo on S. The g-action preserves the quantization and, as above, we get a canonical strong lifting ν DS : g T DS. Certainly, in all these things we may replace the S-Lagrangian triple by the S -one. It is easy to see that Ω 1/2 does not exist globally on Y. On Y the sheaf Ω is canonically trivialized, hence we get a canonical Ω 1/2. Take an admissible ω X -bundle (it comes from a line bundle L W on W equipped with a connection W with curvatuer ω). By we get a Kostant line bundle L Y on Y, hence a Kostant D S -module E = p Y (L Y ) on S. By 15

16 1.5.6 it carries a canonical metaplectic g-action: for s S a fiber E s is a metaplectic representation of g on vectors algebraic with respect to a polarization L s. 1.7 Complex Polarizations In this section we will relate the above purely holomorphic construction with a complex polarization approach. We will start with a general lemma on a C -description of twisted cotangent bundles. Everywhere below variety means complex analytic variety. Let Y be a smooth variety and φ = (π φ : X φ Y ; ω φ ) be a twisted cotangent bundle over Y. Let (F φ, curv) be the corresponding Ω 1 γ -torsor of holomorphic sections of π φ, and let C F φ be the Ω 10 C Y -torsor of C sections of π φ (so C F φ is the pushout of F φ by Ω 1 Y Ω10 C Y ). For γ C F φ put curv(γ) := γ (ω φ ): this is a closed C -class 2-form on Y with zero (0,2)- component Lemma. The map (φ, γ) curv(γ) is a 1-1 correspondence between the set of pairs (twisted cotangent bundle φ on Y, a C -class section of π φ ) and the set of closed C -class 2-forms with zero (0,2)-component. Proof: Here is a construction of inverse map. Let ν = ν 11 + ν 20 be a closed C -form. We need to construct an Ω 10 C Y ν 11 = 0 the sheaf F ν := 1 ( ν 11 ) Ω 10 C Y -trivialized Ω 1-torsor. Since Y is an Ω1 Y -torsor; it carries an obvious Ω 10 C Y -trivialization. Define curv ν : F ν Ω 2 Y formula curv ν(γ) = dγ + ν. This (F ν, curv ν ) is our Ω 1 Y -torsor.. 16

17 1.7.2 Remarks: (i) Consider the sheaf A = π φ O Xφ ; it carries a canonical filtration A i (see A1.3). A C -section γ defines the map γ : A O C Y. If curv(γ) = ω γ is a nondegenerate 2-form then γ is injective and one may determine A i γ O C Y by induction: one has A = O Y, A i = {f O C } : {f, O Y } A i 1 ; here { } is Poisson bracket on O C Y defined by ω γ. (ii) Certainly is a particular case of a general nonsense that claims, in the notations of A1.5, that a quasi-isomorphism A B of length 2 complexes defines an equivalence between categories of A - and B -torsors. Consider an S-Lagrangian triple (X π Y ; ω; S ) Definition. A C -class section γ : Y X is called admissible if it satisfies the properties (i) (iii) below: (i) S is tangent to γ(y ), i.e., for y Y the R-subspace dγ(t Y,y ) T Xγ(y) contains the S -horizontal subspace S (T SpY (y)) γ(y). Clearly the S - horizontal planes tangent to γ(y ) form an integrable C -class connection γ S for p Y. (ii) This γ S is globally trivial, i.e., it comes from a global C -class trivialization Y Y 0 S. Consider a C -class 2-form ω γ := γ (ω) along the fibers of p Y. (iii) For s S the form ωs γ on Y s is nondegenerate and real-valued Lemma. (i) The form ω γ s is a closed form of type (1,1) on Y s. (ii) ω γ is γ S-horizontal, i.e., by 1.7.3(ii), it comes from a single symplectic form ω 0 on Y 0. (iii) For each y 0 Y 0 the section S Y 0 S = Y, s (y 0, s), is holomorphic. 17

18 Proof: Clear. Let us describe the above structure from a Y 0 viewpoint. Let (Y 0, ω 0 ) be any C -class (real) symplectic manifold Definition. A complex polarization of (Y 0, ω 0 ) is a complex structure on Y 0 such that ω 0 has type (1,1). According to integrability theorem of Newlander-Nirenberg, a complex structure s on Y 0 is the same as an integrable C-subbundle T 01 s T Y0 C such that T 01 s T 01 T Y0 C (here integrable means [T 01 s, T 01 s ] T 01 s ). Such s is a complex polarization iff T 01 s is an ω 0 -Lagrangian subbundle Note that 1-jet of a deformation of a C-subbundle T 01 s T Y0 C is an element ϕ Hom(T 01 s, T 01 s ) = Ω 01 s T 10 s, wheret 10 s := T Y0 C/T 01 s, Ω 01 S 01 := (Ts ). If Ts 01 is a complex structure, then ϕ is a 1-jet of a deformation of complex structure iff ϕ Ω 02 s Ts 10 is equal to zero (here is taken with respect to the holomorphic structure on T 10 s ). If T 01 s is Lagrangian, then ω 0 identifies Ω 01 s with Ts 10, and ϕ is a 1-jet of a deformation of a Lagrangian subbundle iff ϕ S 2 Ts 10 Ts 10 Ts 10. If Ts 01 is a complex polarization and both above-mentioned conditions hold, then ϕ is a 1-jet of a deformation of a polarization. Let S be a C manifold, and Ts 01, s C, be a C -class family of complex polarizations of (Y 0, ω 0 ). Put Y := Y 0 S. Our T 01 s form a subbundle T 01 Y/S of T Y/S C, same for TY/S 01, etc. The 1-jets of deformation form a section C Ω 1 C S S2 T 10 Y/S. Assume now that S is a C-analytic manifold. 18

19 1.7.7 Definition. A S family of polarizations is holomorphic if C Ω 1 S O S S 2 T 10 Y/S = Ω10 C S S2 T 10 Y/S Proposition. One has a canonical 1 1 correspondence between S- Lagrangian triples (X π Y ; ω; S ) equipped with an admissible C section γ : Y X, and a C -class (real) symplectic manifolds (Y 0, ω 0 ) equipped with a holomorphic S family of polarizations. Proof: As was explained in 1.7.3, an admissible section defines (Y 0, ω 0 ) and a holomorphic S-family of polarizations. Conversely, consider a holomorphic family of polarizations of (Y 0, ω 0 ). Put Y = Y 0 S. The subbundle TY 01 T Y C with fiber at (y, s) Y equal to TY 01 s (y) TS 01 (s) defines the complex structure on Y such that the projection p Y : Y S is holomorphic. Let ω Y be the inverse image of ω 0 via the projection Y Y 0. This is a closed (1,1)-form on Y. Let ( X, ω ex ), π : X Y be the twisted cotangent bundle over Y with the C -section γ : Y X defined by ω Y Put X = X mod p Y Ω1 S according to π Y : this is a twisted cotangent bundle along the fibers of p Y. By a holomorphic section of X is a C -class 1-form ν along the fibers of π (which is the same as a family ν s of 1-forms on Y 0 parameterized by s S) such that ν s is a 10-form on Y S (i.e., ν s T 01 = 0), s ν s = ω 0 Ω 11 Y s and ν s depends on s S in a holomorphic way. Denote by H(ν) the 1,0-form on Y which coincides with ν in fiberwise directions and vanishes on horizontal ones (i.e., H(ν) yo S = 0 for each y 0 Y 0 ). One has ν = ω Y, hence we have defined a holomorphic section H : X X. This ( X, H) is an S-Hamiltonian datum on Y, so, by 1.1.6, we have S-Lagrangian triple (X π Y, ω; S ). It is easy to see that γ = γ mod p Y Ω1 S is an admissible section. This construction is clearly inverse to one of 1.7.3,

20 1.7.9 Lemma. Consider a holomorphic S-family of polarizations of (Y 0, ω 0 ). The corresponding S-Lagrangian triple has order n (see 1.1) iff for any s S and a tangent vector s at s the tensor C(s) S 2 T 10 s in A n 1 S 2 T Ys. Here A n 1 = A sn 1 O C Y 0 (see 1.7.7) lies is the sheaf of functions on Y 0 defined in 1.7.2(i) for the complex structure Y s and the form ω 0. For example, our triple has order 1 iff C(s) is a holomorphic tensor on Y s. Proof: Clear. 20

21 2. D-Rational Varieties and Canonical Quantization In some situations a quantization is uniquely defined by a Lagrangian triple. In this section we desribe some sufficient conditions for this. 2.1 D-Rationality Let Y be a smooth variety and D be a tdo on Y Definition. Y is D-rational if H 0 (Y, D) = C and H i (Y, D) = 0 for i > 0. For arbitrary D consider the class c 1(D) := c 1 (D) 1 2 c 1(det Ω 1 Y ) H1 (Y, Ω 1 Y ). For c H 1 (Y, Ω 1 Y ) let c denote the image of c in H1 (Y, Ω 1 Y ). Let δ D : H j (Y, S i T Y ) H j+1 (Y, S i 1 T Y ) be the convolution with c 1(D) Lemma. Assume that H 0 (Y, O Y ) = C and for each i > 0 the sequence 0 H 0 (Y, S i T Y ) δ D H 1 (Y, S i 1 T Y ) δ D δd H i (Y, O Y ) 0 is exact. Then Y is D-rational. Proof: By A2.6 δ D is the boundary map for the short exact sequence 0 S i 1 T Y D i /D i 2 S i T Y 0, i.e., δ D is the first differential in the spectral sequence E p,q that computes H (Y, D) using filtration D i. Our conditions mean that E 0,0 2 = C, E p,q 2 = 0 for p, q (0, 0) Remark: One may interpret δ D microlocally as follows. Let π : T Y Y be cotangent bundle to Y. The symplectic form on T Y defines 21

22 the isomorphism Ω 1 T Y T T Y (which coincides with translation action along the fibers on π Ω 1 Y Ω1 T Y ). Hence we get the class c 1(D) = π c 1(D) H 1 (T Y, T T Y ). One has H j (T Y, O T Y ) = H j (Y, S i T Y ), so we have δ : i H j (T Y, O T Y ) H j+1 (T Y, O T Y ). Clearly δ coincides with the product with c 1(D) via the map T C O O, f (f) Example: Let Y be a compact complex torus, or an abelian variety. Then det Ω 1 Y O Y, hence c 1(D) = c 1 (D). One has H 1 (Y, Ω 1 Y ) = H 0 (Y, Ω 1 Y ) H1 (Y, O) = Hom(H 0 (Y, T Y ), H 1 (Y, O Y )) We will say that a class c H 1 (Y, Ω 1 Y ) is non-degenerate if the map c : H0 (Y, T Y ) H 1 (Y, O Y ) is isomorphism; a class c F 1 HDR 1 (Y ) is non-degenerate if such is an element c = c mod F 2 H 2 DR of H1 (Y, Ω 1 Y ) Lemma. Y is D-rational iff c 1 (D) is non-degenerate. Proof: Note that H j (Y, S i T Y ) = S i H 0 (Y, T Y ) Λ j H 1 (Y, O Y ). This isomorphism identifies the complex from with i-th symmetric power of the 2-term complex H 0 (Y, T Y ) c 1(D) H 1 (Y, O Y ). Hence if c 1 (D) is non-degenerate then the conditions of hold (the Koszul complex is acrylic, and Y is D- rational. If c(d) is degenerate, the exact sequence 0 C H 0 (Y, T D ) H 0 (Y, T Y ) c 1(D) H 1 (Y, O Y ) shows that C H 0 (Y, T D ) H 0 (Y, D), so Y is not D-rational Remark. Let L be a line bundle on a compact complex torus Y. If c 1 (L) is non-degenerate, then all the cohomologies H j (Y, L) vanish but a single one. We do not know whether one has a similar statement for a line bundle L on arbitrary D L -rational variety. 22

23 2.2 Canonical D-Connections Assume we are in a situation 1.2, so we have a smooth morphism p : Y S of smooth varieties, and D Y is a tdo on Y Definition. We will say that p is D Y -rigid if one has p D Y/S = O S and for any τ T S there exists (locally along S) an element τ p D Y that for f O S one has p τ(f) = [ τ, p f]. such Lemma. Consider the short exact sequence 0 T DY /S T DY p T S 0. It defines the morphism KS : T S R 1 p TDY /S (the Kodaira- Spencer class). Then p is D-rigid iff p D Y/S = O S and the composition T S KS R 1 π TDY /S R 1 p D Y/S equal to 0. Proof. Clear. Assume that p is D-rigid. Let T S be the sheaf of all pairs (τ, τ) from We have a short exact sequence 0 O S i T S σ T S 0, i(f) = (o, f), σ(τ, τ = τ). Also T S carries an obvious Lie algebra and O S -module structure, so T S is an O-extension of T S. Let D S = D ets be the corresponnding tdo (see A1.4). The map T S p D Y, (τ, τ) τ, extends (uniquely) to p-morphism α : D S D Y. We will call α a canonical p-morphism, and the corresponding D Y -connection DY a canonical D Y -connection Lemma. (i) A canonical D Y -connection is actually a unique D Y - connection for p. (ii) A degree of α is equal to minimal degree of τ for (τ, τ) T S minus 1. 23

24 (iii) Let L. be the filtration by degree along S on D Y (see 1.2.5). One has T S = p L 1. (iv) Any (compatible) Lie algebra action on Y, S, D Y preserves D S and α (see 1.4.1). Proof: Clear Proposition. Let p : Y S be any smooth surjective morphism and D Y be a tdo on Y such that for each s S the fiber Y s is D Ys -rational. Then p is D Y -rigid, and one has p D Y = D S, R i p D Y = 0 for i > 0. If, moreover, D Y satisfies conditions 2.1.2, then a canonical D Y -connection has order 1. Proof. One has O S p D Y/S, R i p D Y/S = 0 since D Y/S is flat O S -module and we have fiberwise rationality. Consider the filtration L. on D Y. Since L i /L i 1 = D Y/S S i T S one has Rp L i /L i 1 = S i T S. This implies that R i p D Y = 0 for Ki > 0 and p D Y is a tdo with a canonical filtration equal to p L i. By we see that p is D Y -rigid and p D Y = D S K. 2.3 Canonical Quantization Let (X π Y ; ω; S ) be an S-Lagrangian triple of order 1, D Y be a corresponding tdo on Y Definition. We will say that our Lagrangian triple is canonically quantizable if p Y : Y S is D Y -rigid and a canonical D Y -connection DY is a quantization (see 1.3). In this case DY (which is a unique D Y -connection for p Y ) is called a canonical quantization of our triple. By 2.2.3(iv) DY is preserved by any symme- 24

25 tries of the triple. In some cases the compatibility 1.3 holds automatically, e.g., one has Lemma. Assume that for each s S one has H 0 (Y s, O Ys = C, H 0 (Y s, T Ys ) = 0 and the maps δ DY /S : H 0 (Y s, S i T Ys ) H 1 (Y s, S i 1 T Ys ) are injective for i > 1. Then our Lagrangian triple is canonically quantizable iff the composition T S KS R 1 π Y TDY /S = R 1 p Y D D/S 1 R 1 p Y D Y/S 2 vanishes. Proof: Our conditions obviously imply that p Y D Y/S = O S, p Y (D Y/S /O Y ) = 0. By 2.2.2, the above map T S R 1 p Y D Y/S 2 vanishes iff p Y is D Y -rigid and a canonical D Y -connection DY has order 1. By 1.3.2(ii) DY is a quantization Remark. Let π : X Y be a morphism of S-varieties and ω Ω 2 X/S. Assume that these data satisfy conditions 1.1.3(i) (ii). Note that the sheaf p -conn ω of those p X -connections S that ω is S -horizontal is an Ω 1 S p X T ω X/S -torsor (where T ω X/S T X/S is a subsheaf of vector fields that preserve ω). Therefore in case p X T ω X/S = 0 there exists at most one such S which is automatically integrable (since the curvature lies in Ω 2 S p X T ω X/S = 0). Hence (π : X Y, ω, S ) is an S-Lagrangian triple. We will call such triples canonical Lagrangian triples. 2.4 Example: Heat Equation for θ-functions Let Y be a complex torus or an abelian variety. Denote by ( 1) the involution y y of Y. Let D Y be a tdo on Y. 25

26 2.4.1 Definition. A symmetric structure on D Y is an isomorphism D α Y ( 1) D Y. A symmetric tdo is a tdo equipped with a symmetric structure. A symmetric tdo forms a category T DOS(Y ) in an obvious manner. Certainly, we may repeat the above definition of symmetric structure for Ω 1 Y -torsors or twisted cotangent bundles Lemma. Any tdo admits a symmetric structure. A symmetric tdo (D Y, α) has no automorphisms. One has ( 1) (α) α = id DY, so D Y is a Z/2-equivariant tdo. Two symmetric tdo s are isomorphic iff they are isomorphic as usual tdo s. Proof: Follows from A1.6, A1.13 since ( 1) acts on H 1 (Y, Ω 1 Y ) = F 1 HDR 2 ) HDR 2 (Y ) as identity map, and on H0 (Y, Ω 1 Y ) = F 1 HDR 1 (Y ) H1 DR (Y ) as minus identity We see that c 1 defines equivalence between T DOS(Y ) and a discrete category with the set of objects F 1 HDR 2 (Y ). For c F 1 HDR 2 (Y ) we will denote by D c the corresponding symmetric tdo, and by (π c : X c Y ; ω c ), the symmetric twisted cotangent bundle. Note that if c lies in F 2 H 2 DR (Y ) = H 0 (Y, Ω 2 Y ) then the tdo D c carries a unique symmetric (in an obvious sense) connection c with curvature c (cf. A1.7) Here is an explicit construction of the twisted cotangent bundle X c for a non-degenerate c F 1 HDR 2 (Y ). Let 0 H1 (Y, O Y ) X π Y 0 be a universal extension of Y (see, e.g. [MM]); we consider here the vector space H 1 (Y, O Y ) as an algebraic group). So X is a commutative algebraic group with Lie algebra Lie X canonically identified with H 1 DR (Y ). One 26

27 may describe points of X as line bundles with connection on a dual abelian variety Y 0 ; in the analytic case one identifies X with H 1 (Y, C/Z). Our class c F 1 H 2 DR (Y ) H2 DR (Y ) = Λ2 H 1 DR (Y ) defines an invariant 2-form ω c on X; this form is closed, non-degenerate (since such was c), and π is a polarization for ω c (since c F 1 H 2 DR ), so X c = {(π : X Y ; ω c )} is a twisted cotangent bundle on Y. The involution ( 1) x : x x is a symmetric structure on X c. Since π : H 2 DR (Y ) H2 DR (X) is isomorphism, and π X c carries a section with curvature ω c, we see that c 1 (X c ) = c Now let p Y : Y S be an abelien scheme over S, i.e., a family Y s, s S, of abelian varieties (so we are in an algebraic situation). Let c be a horizontal section of HDR 2 (Y/S) (with respect to Gauss-Manin connection) that lies in F 1 H 2 DR (Y/S). For any s S the element c s F 1 H 2 DR (Y s) defines a symmetric twisted cotangent bundle (π cs : X cs Y s, ω cs ), in a canonical way. These spaces form a relative symmetric twisted cotangent bundle π c : X Y, ω c H 0 (X, Ω 2cl X/S ), ( 1) X : X c X c. We will say that c is non-degenerate if for some (or any) s S the class c s F 1 H 2 DR (Y s) is non-degenerate Proposition. Assume that c is non-degenerate. Then (i) p X = p Y π c : X c S (i.e., the one such that ( 1) X S = S ). admits a unique symmetric connection S (ii)(π c : X c Y ; ω c ; S ) is an S-Lagrangian triple which is canonically quantizable. Proof: (i) One has H 0 (X s, O Xs ) = C, H i (X s, O Xs ) = 0 for i > 0 (to see this note that H (X s, O Xs ) = H (H s, π s O Xs ) since π s is affine; the standard 27

28 filtration A i on A = π S O X gives a spectral sequence with first term equal to Koszul complex, cf ). The connections for p X form a p X Ω1 S T X/Storsor on X. Since T Xs = H 1 DR (Y s) O Xs (see 2.4.3) we see that connections for p X (global along the fibers of p X ) exist and form an H 1 DR (Y/S) Ω1 S - torsor. Since ( 1) X acts on HDR 1 (Y/S) as multiplication by 1, we see that there exists a unique symmetric connection S. (ii) Note that X is naturally a group scheme over S. It follows easily by unicity that S is actually a unique connection for p X compatible with group structure on X, and the induced connection on Lie X/S = H 1 DR (Y/S) is (dual to) Gauss-Manin connection (see [MM]). Also S is flat. Since ω c is invariant 2-form on X-horizontal with respect to Gauss-Manin connection (see 2.4.3) we see that it is S -horizontal, so (π : X Y ; ω c ; S ) is an S- Lagrangian triple. By 2.1.5, 2.2.4, p Y is D Y -rigid and a canonical connection DY has order 1. Since DY is symmetric (being unique) the section 0 S σ 0 D Y Ω 1 S p Y (A 2 /A 0 ) = Ω 1 S p Y (A 1 /A) is symmetric (see 1.3.2), hence vanishes. By 1.3.2(ii) this means that DY is a quantization Remark: (i) If we are in an analytic situation, i.e., p Y : Y S is a family of compact complex tori, then remains valid with the only correction: in (i) one should also demand that S has finite order (i.e., for f O Y, t T S the function S (τ)(f) O X should be polynomial along the fibers of π). The connection S defines an obvious topological local trivialization of the fibration X = H 1 (Y/S, C/Z) Y. (ii) In the language of the above Proposition 2.4.6(i) says that (X/ ± 1 π Y/ ± 1, ω, S ) is a canonical S-Lagrangian triple. Here / ± 1 means quotient modulo the involution ( 1) which is an S-family of smooth 28

29 orbifolds or stack Assume that our class c is integral, i.e., c s H 2 (Y s, Z(1)). Localizing S, if necessary, one finds a symmetric line bundle L c on Y together with a trivialization e L c O S of its restriction to zero section e of Y. Put λ = e Ω = p Y Ω, and choose a square-root of λ, i.e., a line bundle λ 1/2 on S together with isomorphism λ 1/2 2 = λ. Then L c p Y λ1/2 is a Kostant line bundle, and a corresponding integrable projective connection on R i p Y classicial heat equation for θ-functions. is a 29

30 5. Centralizers of Regular Elements Let G be a connected reductive group, g its Lie algebra, Ḡ := G/ center G be the adjoint group, and B the variety of Borel subalgebras of g. We can also interpret B as the variety of Borel subgroups of G. Recall the definition of the Cartan group of G, the Cartan Lie algebra and the Weyl group. The action of Ḡ on B is transitive, so for each pair B 1, B 2 of Borel subgroups we may choose g Ḡ such that Ad(g)B 1 = B 2 which induces the isomorphism Ad(g) : B 1 /[B 2, B 1 ] B 2 /[B 2, B 2 ]. In fact, this isomorphism does not depend on a choice of g, hence we may identify canonically all the toruses B/[B, B]. This torus H is called the Cartan group of G. Its Lie algebra h is called Cartan Lie algebra of g; one has a canonical isomorphism h = b/[b, b] for b B. Put Γ := Hom(G m, H). One defines similarly the Weyl group W = W (G); it acts on H and h in a canonical way. We also have the root data; denote by the set of roots, and by S the subset of simple roots. Denote by p : h h/w := Y the projection, and by R Y the ramification locus of p. We have a canonical AdG-invariant projection f : g Y such that f/b coincides with the composition b b/[b, b] = h any b B. p Y for 5.1. Let g reg g be the open subset of regular (not necessarily semi-simple) elements of g. Put g reg := {(a, b) : a b} g reg B. Then g reg is a smooth variety, and the projection p : g reg g reg is finite. The group Ḡ acts on 30

31 these objects in an obvious manner. Consider the commutative diagram g reg p freg h p g reg f reg where f reg := f greg, and f reg (a, b) = a mod [b, b] = h. One knows (see [K2]) that (i) this diagram is Cartesian, hence W acts along the fibers of p, and g reg = W \ g reg. (ii) f reg is a smooth projection. The adjoint action of Ḡ is transitive along the fibers of f reg. Hence Y = Ḡ\g reg, h = Ỹ =: Ḡ\ g reg. (iii) f reg admits a global section s : Y g reg. Y 5.2 Let a g reg be a regular element. Denote by H a the centralizer of a, and by i a : H a G the embedding. One knows that H a is a commutative group of dimension dim h. For any Borel subgroup B G such that a b = Lie B one has H a B, hence the projection B B/[B, B] = H defines the morphism ϕ B : H a H. For ã = (a, b) g reg we put ϕã := ϕ B : H a H. If a is a regular semi-simple, then ϕã is an isomorphism. One may describe H a as follows. Let M a G be the centralizer of a ss (:= semi-simple part of a). This is a Levi subgroup of G. Since a Lie M a, one has Center M a H a. In fact, Center M a coincides with the reductive part of H a : if H aun denotes the unipoint radical of H a then one has H a = Center M a H aun. Note that H aun = ker ϕã, hence Center M a ϕã(h a ). We will need a bit of information on the structure of the group H a /Ha 0 = Center M a /Center 0 M a of connected components of H a. For a root γ let γ Γ 31

32 be the corresponding co-root, and σ γ W be the reflection a a γ(a)γ ; let χ γ : H G m, i γ : G m H be the corresponding character and 1- parameter subgroup. We will say that γ is a type 1 root if Γ = Γ σγ Zγ, that γ is of type 2 if Γ = Γ σγ 1 2 Zγ, and that γ is of type 3 in other cases (in other words, γ is of type 3 if the projection Γ σγ Γ σγ from σ γ -invariants to σ γ -coinvariants is isomorphism). Therefore γ is of type 2 if the rank 1 subgroup that corresponds to γ equal to P GL 2. Denote by S i S, i = 1, 2, 3, the subset of simple roots of type i. If M a is our Levi subgroup and B is a Borel subgroup such that a ss b = Lie B, then B a := B M a is a Borel subgroup of M a and B a /[B a, B a ] = B/[B, B]. Hence a choice of B defines the isomorphism between the Cartan groups of M a and G, and identifies the root system of M a with the subsystem of the one of G. In particular, S a (:= simple roots of M a ) S, and W Sa W is the Weyl group of M a Lemma. (i) One has Center M a = γ S a ker χ γ, H W Sa = γ S a ker(i γ χ γ ), (ii) One has H W Sa /Center Ma Z/2 S a2, where A a2 := S a S 2. root. (iii) In each orbit of S Sa in the roots of M a there is at most one simple (iv) If a Lie B = b, then W Sa equals the stabilizer of ã = (a, b) g reg with respect to W -action (see 5.1(i)). (v) If S a = {γ}, then the group H q /Hq 0 equals Z/2 if γ is of type 1 and is trivial otherwise. Proof: Easy, e.g., morphism (χ γ ) : H γ S a G m is surjective and ker i γ is {±1} if γ is of type 2 and trivial otherwise. Therefore the map (χ γ ) γ Sa2 32

33 defines the isomorphism γ S a ker(i γ χ γ )/ γ S a ker(χ γ ) (±1) S a2, hence (ii) follows from (i). The morphism χ γ : H W {±1} depends only on the W -orbit of γ, hence (iii) follows from (ii) When a g reg varies the groups H a form a flat commutative group scheme H greg on g reg equipped with the embedding i : H greg G greg to the constant group scheme G on g reg. The morphisms ϕã form a canonical morphism ϕ greg : H greg := p H greg H greg. The W -action on g reg lifts to our group schemes: namely, W acts on H greg in an obvious manner, and on H greg = H g reg in a diagonal one. The morphism ϕ commutes with W -action. All the picture is equivariant with respect to (adjoint) action of G on all our schemes. Note that the stabilizer of a point a g reg, equal to the image of H a in Ḡ, acts on the fiber H a trivially (since H a is commutative). Therefore, according to 5.1(ii), the scheme H g descents to Y : we have a canonical group scheme H Y on Y such that H greg = f H Y. For any section s of f reg one has a canonical isomorphism s H greg = H Y, hence the embedding i s := s (i) : H Y = s H greg s G greg = G Y. The morphism ϕ greg descents to a canonical morphism ϕ ey : H ey := p H Y H ey equivariant with respect to W -action. By adjointness we have the morphism ϕ Y : H Y (p H ey ) W. This is an embedding which is isomorphism off R. As follows from 5.2.1, the cokernel of ϕ Y is a constructible sheaf with a stalk at y R equal to Z/2 S y 2, where S y 2 S 2 is the set of type 2 simple roots vanishing at y. In particular, Γ(Y, Coker ϕ Y ) = (Coker ϕ Y ) 0 = Z/2 S 2. Clearly H 1 (Y, Coker ϕ Y ) = 0. Note that Center G H a for any a g reg, hence Center G Γ(Y, H Y ). 33

34 Precisely, one has Lemma. Γ(Y, H Y ) = Center G, H 1 (Y, H Y ) = 0. Proof: Note that all the global (algebraic) H-valued functions on Ỹ = h are constant. Hence Γ(Y, H Y ) = ker(γ(y, (p H ey ) W ) Z/2 S 2 ) = ker(h W Z/2 S 2 ) = Center G by 6.3. Now let F be any H Y -torsor, and F := ϕ ey (p F) be the corresponding W -equivariant H-torsor on Ỹ. Since any H-torsor on Ỹ is trivial, the value at 0 map defines the isomorphism Γ(Ỹ, F) F (0). Therefore for the (p H ey ) W -torsor ϕ Y (F) = (p F) W F one has Γ(Y, ϕ Y (F)) = Γ(Ỹ, F) W = F W (0), and Γ(Y, F) = Im(ϕ e Y0 : F 0 F W (0), q.e.d Consider the canonical embedding i : H greg G greg. We would like to descent it down to Y. We assume that Ḡ acts on G g reg = G g reg by a diagonal adjoint action. Then i is Ḡ-equivariant. Note that the stabilizer of a point a g reg acts on a fiber G a in a nontrivial way; hence we need for G greg a bit more clever descent then the obvious one used for H greg in 5.3. Namely, Π denotes the set of global sections s : Y g reg of f greg ; according to 5.1(iii) Π is a nonempty Ḡ(Y )-set Lemma. Π is a Ḡ(Y )-torsor. Proof: For s 1, s 2 Π consider the sheaf φ s2 s 1 on Y, defined by formula φ s2 s 1 (U) := {g G(U) : Ad(g)s 1 U = s 2 U }. This is an H Y -torsor with respect to right multiplication by i s1 : H Y G Y. By the global sections Γ(Y, φ s2 s 1 ) form a torsor with respect to the action of Center G = Γ(y, H Y ). 34

35 Hence for any s 1, s 2 Π there exists a unique element g s2 s 1 Ḡ(Y ) = G(Y )/Center G such that Ad(g)s 1 = s 2.We are done. Denote by G Y the group scheme on Y obtained from G Y by Π-twist (with respect to adjoint action of Ḡ(Y )). Hence for any s Π we have a canonical isomorphism j s : G Y G Y such that k s2 j 1 s 1 = Ad(g s2 s 1 ). There is a canonical embedding i : H Y G Y such that j si = i s. Note that we have no canonical isomorphism between f regg Y and G g reg The variation considered in 5.1 also carry a natural Gm-action that commutes with Ḡ- and W -actions. Namely, Gm acts on G reg and h by homotheties, and this determines the Gm-actions on Y = W \h Ḡ\g reg and g reg = g reg Y h. Explicitly, if p i are homogeneous generators of S(h ) W of degree d i, so (p i ) : Y formula λ(p i ) = (λ d i p i ). C dim h, then Gm acts on Y in coordinates p i by This Gm-action lifts to our group schemes H Y, G Y the Gm-action on H ey and H e Y. Namely, = H Ỹ is the trivial one. For a g reg and λ C one has H a = H λa, which defines the Gm-action on H greg which descents down to H Y. The group Gm acts on the set Π of global sections of f reg by formula (λs)(y) = λs(λ 1 y); for g = g(y) Ḡ(Y ) we have λ(gs) = (λg)(λs), where (λg)(y) = g(λ 1 y). This defines the Gm-action on G Y j s λ = j λs : G Y G Y for s Π, λ C. The morphisms ϕ ey : H ey G ey, i : H Y H Y such that commute with Gm-action. We will need to know whether there exists a Gm-equivariant G-torsor T such that the group G Y with Gm-action is isomorphic to Aut T. Or, eqeuivalently, whether the Gm-equivariant Ḡ(Y )-torsor Π lifts to a Gm-equivariant 35

36 G(Y )-torsor. Since G(Y ) is a central extension of Ḡ(Y ) by Center G, the obstruction α for lifting an element cl Π H 1 (Gm, Ḡ(Y )) to H1 (Gm, G(Y )) lies in a finite group H 2 (Gm, Center G) = H 2 (Gm, A) = A( 1), where A denotes the group of connected components of Center G (and (-1) is Tate twist). The obstruction α could be easily computed. Namely, let α be a regular nilpotent element, and ν : SL 2 G be a morphism such that Lie ν = (( )) = a (so ν is a Kostant principal T DS). Let ν : Gm G, ν(λ) = ν (( λ 0 0 λ 1 )) be the corresponding one-parameter subgroup, so one has Ad ν(λ)(a) = λ 2 a and ν( 1) Center G Lemma. The obstruction α equals the image of ν( 1) in A. In particular 2α = 0. Proof: Put ψ := {λa, λ 0} g reg, Π a := {s : s(0) ψ} Π, G a := ν(gm) Center G, G(Y ) a := {g G(Y ) : g(0) G a } G(Y ), Ḡ a = G a /Center G Ḡ, Ḡ(Y ) a := {ḡ Ḡ(Y ) : ḡ(0) Ḡa} Clearly, G(Y ) a, G a Ḡ(Y ). are central extensions of the corresponding -groups by Center G. We have canonical morphisms G(Y ) µ G(Y ) a π G a, π(g) = g(0), identical on Center G, and the corresponding morphisms of -groups. Now Π a, ψ are Ḡ(Y ) a- and Ḡ-torsors, respectively, and the obvious maps Π µ Π a π ψ, π (s) = s(0), are µ- and π-compatible. Note that all our groups and torsors carry an obvious Gm-action. Hence, by functoriality, the obstructions for lifting Π, Π a and ψ to, respectively Gm-equivariant G(Y )-, G(Y ) a - and G a -torsors. The obstruction for ψ coincides with the image of ν( 1) in A, and we are done. 36