Tài liệu Đề tài " Localization of modules for a semisimple Lie algebra in prime characteristic " pdf

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948 ROMAN BEZRUKAVNIKOV, IVAN MIRKOVI
´
C, AND DMITRIY RUMYNIN
0.0.1. Sections 1 and 2 deal with algebras of differential operators D
X
.
Equivalence D
b
(mod
fg
(U
0
))
∼
=
−→ D
b
(mod
c
(D
B
)) and its generalizations are
proved in Section 3. In Section 4 we specialize the equivalence to objects with
the χ-action of the Frobenius center Z
Fr
. In Section 5 we relate D-modules
with the χ-action of Z
Fr
to O-modules on the Springer fiber B
χ
. This leads
to a dimension formula for g-modules in terms of the corresponding coherent
sheaves in Section 6, here we also spell out compatibility of our functors with
translation functors. Finally, in Section 7 we calculate the rank of the K-group
of the Springer fiber, and thus of the corresponding category of g-modules.
0.0.2. The origin of this study was a suggestion of James Humphreys that
the representation theory of U
0
χ
should be related to geometry of the Springer
fiber B
χ
. This was later supported by the work of Lusztig [Lu] and Jantzen
[Ja1], and by [MR].
0.0.3. We would like to thank Vladimir Drinfeld, Michael Finkelberg,
James Humphreys, Jens Jantzen, Masaharu Kaneda, Dmitry Kaledin,
Victor Ostrik, Cornelius Pillen, Simon Riche and Vadim Vologodsky for various
information over the years; special thanks go to Andrea Maffei for pointing out
a mistake in example 5.3.3(2) in the previous draft of the paper. A part of the
work was accomplished while R.B. and I.M. visited the Institute for Advanced
Study (Princeton), and the Mathematical Research Institute (Berkeley); in
addition to excellent working conditions these opportunities for collaboration
were essential. R.B. is also grateful to the Independent Moscow University
where part of this work was done.
0.0.4. Notation. We consider schemes over an algebraically closed field k of
characteristic p>0. For an affine S-scheme X
q
→ S, we denote q
∗
O
X
by O
X/S
,
or simply by O
X
. For a subscheme Y of X the formal neighborhood FN
X
(Y)
is an ind-scheme (a formal scheme), the notation for the categories of modules
on X supported on Y is introduced in 3.1.7, 3.1.8 and 4.1.1. The Frobenius
neighborhood Fr N
X
(Y) is introduced in 1.1.2. The inverse image of sheaves is
denoted f
−1
and for O-modules f
∗
(both direct images are denoted f
∗
). We
denote by T
X
and T
∗
X
the sheaves of sections of the (co)tangent bundles TX
and T
∗
X.
1. Central reductions of the envelope D
X
of the tangent sheaf
We will describe the center of differential operators (without divided pow-
ers) as functions on the Frobenius twist of the cotangent bundle. Most of the
material in this section is standard.
LOCALIZATION IN CHARACTERISTIC P
949
1.1. Frobenius twist.
1.1.1. Frobenius twist of a k-scheme. Let X be a scheme over an
algebraically closed field k of characteristic p>0. The Frobenius map of
schemes X→X is defined as the identity on topological spaces, but the pull-
back of functions is the p
th
power: Fr
∗
X
(f)=f
p
for f ∈O
X
(1)
= O
X
. The
Frobenius twist X
(1)
of X is the k-scheme that coincides with X as a scheme
(i.e. X
(1)
= X as a topological space and O
X
(1)
= O
X
as a sheaf of rings), but
with a different k-structure: a ·
(1)
f
def
= a
1/p
· f, a ∈ k,f∈O
X
(1)
. This makes
the Frobenius map into a map of k-schemes X
Fr
X
−→ X
(1)
. We will use the twists
to keep track of using Frobenius maps. Since Fr
X
is a bijection on k-points,
we will often identify k-points of X and X
(1)
. Also, since Fr
X
is affine, we may
identify sheaves on X with their (Fr
X
)
∗
-images. For instance, if X is reduced
the p
th
power map O
X
(1)
→(Fr
X
)
∗
O
X
is injective, and we think of O
X
(1)
as a
subsheaf O
p
X
def
= {f
p
,f∈O
X
} of O
X
.
1.1.2. Frobenius neighborhoods. The Frobenius neighborhood of a sub-
scheme Y of X is the subscheme (Fr
X
)
−1
Y
(1)
⊆ X; we denote it Fr N
X
(Y )or
simply X
Y
. It contains Y and O
X
Y
= O
X
⊗
O
X
(1)
O
Y
(1)
= O
X
⊗
O
p
X
O
p
X
/I
p
Y
=
O
X
/I
p
Y
·O
X
for the ideal of definition I
Y
⊆O
X
of Y .
1.1.3. Vector spaces. For a k-vector space V the k-scheme V
(1)
has a
natural structure of a vector space over k; the k-linear structure is again given
by a ·
(1)
v
def
= a
1/p
v, a ∈ k,v∈ V . We say that a map β : V →W between
k-vector spaces is p-linear if it is additive and β(a · v)=a
p
· β(v); this is the
same as a linear map V
(1)
→W . The canonical isomorphism of vector spaces
(V
∗
)
(1)
∼
=
−→ (V
(1)
)
∗
is given by α→α
p
for α
p
(v)
def
= α(v)
p
(here, V
∗(1)
= V
∗
as a
set and (V
(1)
)
∗
consists of all p-linear β : V →k). For a smooth X, canonical
k-isomorphisms T
∗
(X
(1)
)=(T
∗
X)
(1)
and (T (X))
(1)
∼
=
−→ T(X
(1)
) are obtained
from definitions.
1.2. The ring of “crystalline” differential operators D
X
. Assume that X
is a smooth variety. Below we will occasionally compute in local coordinates:
since X is smooth, any point a has a Zariski neighborhood U with ´etale coor-
dinates x
1
, ,x
n
; i.e., (x
i
) define an ´etale map from U to A
n
sending a to 0.
Then the dx
i
form a frame of T
∗
X at a; the dual frame ∂
1
, ,∂
n
of T
X
is
characterized by ∂
i
(x
j
)=δ
ij
.
Let D
X
= U
O
X
(T
X
) denote the enveloping algebra of the tangent Lie al-
gebroid T
X
; we call D
X
the sheaf of crystalline differential operators. Thus
D
X
is generated by the algebra of functions O
X
and the O
X
-module of vec-
tor fields T
X
, subject to the module and commutator relations f ·∂ = f∂,
950 ROMAN BEZRUKAVNIKOV, IVAN MIRKOVI
´
C, AND DMITRIY RUMYNIN
∂·f − f·∂ = ∂(f ),∂∈T
X
,f∈O
X
, and the Lie algebroid relations ∂

·∂

−
∂

·∂

=[∂

,∂

],∂

,∂

∈T
X
. In terms of a local frame ∂
i
of vector fields we
have D
X
= ⊕
I
O
X
·∂
I
. One readily checks that D
X
coincides with the ob-
ject defined (in a more general situation) in [BO, §4], and called there “PD
differential operators”.
By the definition of an enveloping algebra, a sheaf of D
X
modules is just
an O
X
module equipped with a flat connection. In particular, the standard
flat connection on the structure sheaf O
X
extends to a D
X
-action. This action
is not faithful: it provides a map from D
X
to the “true” differential operators
D
X
⊆End
k
(O
X
) which contain divided powers of vector fields; the image of
this map is an O
X
-module of finite rank p
dim X
; see [BO] or 2.2.5 below.
For f ∈O
X
the p
th
power f
p
is killed by the action of T
X
, hence for any
closed subscheme Y ⊆ X we get an action of D
X
on the structure sheaf O
X
Y
of the Frobenius neighborhood.
Being defined as an enveloping algebra of a Lie algebroid, the sheaf of
rings D
X
carries a natural “Poincar´e-Birkhoff-Witt” filtration D
X
= ∪D
X,≤n
,
where D
X,n+1
= D
X,≤n
+ T
X
·D
X,≤n
, D
X,≤0
= O
X
. In the following Lemma
parts (a,b) are proved similarly to the familiar statements in characteristic
zero, while (c) can be proved by a straightforward use of local coordinates.
1.2.1. Lemma. a) There is a canonical isomorphism of the sheaves of
algebras:gr(D
X
)
∼
=
O
T
∗
X
.
b) O
T
∗
X
carries a Poisson algebra structure, given by {f
1
,f
2
} =[
˜
f
1
,
˜
f
2
]
mod D
X,≤n
1
+n
2
−2
,
˜
f
i
∈D
X,≤n
i
, f
i
=
˜
f
i
mod D
X,≤n
i
−1
∈O
T
∗
X
, i =1, 2.
This Poisson structure coincides with the one arising from the standard
symplectic form on T
∗
X.
c) The action of D
X
on O
X
induces an injective morphism D
X,≤p−1
→
End(O
X
).
We will use the familiar terminology, referring to the image of d ∈D
X,≤i
in D
X,≤i
/D
X,≤i−1
⊂O
T
∗
X
as its symbol.
1.3. The difference ι of p
th
power maps on vector fields. For any vector
field ∂ ∈T
X
, ∂
p
∈D
X
acts on functions as another vector field which one
denotes ∂
[p]
∈T
X
.For∂ ∈T
X
set ι(∂)
def
= ∂
p
− ∂
[p]
∈D
X
. The map ι lands in
the kernel of the action on O
X
; it is injective, since it is injective on symbols.
1.3.1. Lemma. a) The map ι : T
X
(1)
→D
X
is O
X
(1)
-linear, i.e., ι(∂)+
ι(∂

)=ι(∂ + ∂

) and ι(f∂)= f
p
·ι(∂),∂,∂

∈T
X
(1)
,f∈O
X
(1)
.
b) The image of ι is contained in the center of D
X
.
LOCALIZATION IN CHARACTERISTIC P
951
Proof.
1
For each of the two identities in (a), both sides act by zero on
O
X
. Also, they lie in D
X,≤p
, and clearly coincide modulo D
X,≤p−1
. Thus the
identities follow from Lemma 1.2.1(c).
b) amounts to: [f,ι(∂)] = 0, [∂

,ι(∂)] = 0, for f ∈O
X
, ∂,∂

∈T
X
. In both
cases the left-hand sides lie in D
X,≤p−1
: this is obvious in the first case, and
in the second one it follows from the fact that the p
th
power of an element in
a Poisson algebra in characteristic p lies in the Poisson center. The identities
follow, since the left-hand sides kill O
X
.
Since ι is p-linear, we consider it as a linear map ι : T
X
(1)
→D
X
.
1.3.2. Lemma. The map ι : T
X
(1)
→D
X
extends to an isomorphism of
Z
X
def
= O
T
∗
X
(1)
/X
(1)
and the center Z(D
X
). In particular, Z(D
X
) contains
O
X
(1)
.
Proof.Forf ∈O
X
we have f
p
∈ Z(D
X
), because the identity ad(a)
p
=
ad(a
p
) holds in an associative ring in characteristic p, which shows that [f
p
,∂]
= 0 for ∂ ∈T
X
. This, together with Lemma 1.3.1, yields a homomorphism
Z
X
→ Z(D
X
). This homomorphism is injective, because the induced map on
symbols is the Frobenius map ϕ → ϕ
p
, Z = O
T
∗
X
(1)
→O
T
∗
X
. To prove that it
is surjective it suffices to show that the Poisson center of the sheaf of Poisson
algebras O
T
∗
X
is spanned by the p
th
powers. Since the Poisson structure arises
from a nondegenerate two-form, a function ϕ ∈O
T
∗
X
lies in the Poisson center
if and only if dϕ = 0. It is a standard fact that a function ϕ on a smooth variety
over a perfect field of characteristic p satisfies dϕ = 0 if and only if ϕ = η
p
for
some η.
Example.IfX = A
n
, so that D
X
= kx
i
,∂
i
 is the Weyl algebra, then
Z(D
X
)=k[x
p
i
,∂
p
i
].
1.3.3. The Frobenius center of enveloping algebras. Let G be an algebraic
group over k, g its Lie algebra. Then g is the algebra of left invariant vector
fields on G, and the p
th
power map on vector fields induces the structure
of a restricted Lie algebra on g. Considering left invariant sections of the
sheaves in Lemma 1.3.2 we get an embedding O(g
∗(1)
)
ι
g
→ Z(U(g)); we have
ι
g
(x)=x
p
− x
[p]
for x ∈ g. Its image is denoted Z
Fr
(the “Frobenius part” of
the center).
From the construction of Z
Fr
we see that if G acts on a smooth variety
X then g→ Γ(X, T
X
) extends to U(g)→ Γ(X, D
X
) and the constant sheaf
(Z
Fr
)
X
= O(g
∗(1)
)
X
is mapped into the center Z
X
= O
T
∗
X
(1)
. The last map
comes from the moment map T
∗
X→ g
∗
.
1
Another proof of the lemma follows directly from Hochschild’s identity (see [Ho, Lemma
1]).
952 ROMAN BEZRUKAVNIKOV, IVAN MIRKOVI
´
C, AND DMITRIY RUMYNIN
Ug is a vector bundle of rank p
dim(
g
)
over g
∗(1)
.Anyχ ∈ g
∗
defines a
point χ of g
∗(1)
and a central reduction U
χ
(g)
def
= U(g)⊗
Z
Fr
k
χ
.
1.4. Central reductions. For any closed subscheme Y⊆T
∗
X one can
restrict D
X
to Y
(1)
⊆ T
∗
X
(1)
; we denote the restriction
D
X,Y
def
= D
X
⊗
O
T
∗
X
(1)
/X
(1)
O
Y
(1)
/X
(1)
.
1.4.1. Restriction to the Frobenius neighborhood of a subscheme of X.
A closed subscheme Y→X gives a subscheme T
∗
X|Y ⊆ T
∗
X, and the corre-
sponding central reduction
D
X
⊗
O
T
∗
X
(1)
O
(T
∗
X|Y )
(1)
= D
X
⊗
O
X
(1)
O
Y
(1)
= D
X
⊗
O
X
O
X
Y
,
is just the restriction of D
X
to the Frobenius neighborhood of Y . Alternatively,
this is the enveloping algebra of the restriction T
X
|X
Y
of the Lie algebroid T
X
.
Locally, it is of the form ⊕
I
O
X
Y
∂
I
. As a quotient of D
X
it is obtained by
imposing f
p
= 0 for f ∈I
Y
. One can say that the reason we can restrict Lie
algebroid T
X
to the Frobenius neighborhood X
Y
is that for vector fields (hence
also for D
X
), the subscheme X
Y
behaves as an open subvariety of X.
Any section ω of T
∗
X over Y ⊆ X gives ω(Y )⊆ T
∗
X|Y , and a further
reduction D
X,ω(Y )
. The restriction to ω(Y )⊆ T
∗
X|Y imposes ι(∂)=ω, ∂
p
,
i.e., ∂
p
= ∂
[p]
+ ω, ∂
p
,∂∈T
X
. So, locally, D
X,ω(Y )
= ⊕
I∈{0,1, ,p−1}
n
O
X
Y
∂
I
and ∂
p
i
= ∂
[p]
i
+ ω, ∂
i

p
= ω, ∂
i

p
.
1.4.2. The “small” differential operators D
X,0
. When Y is the zero section
of T
∗
X (i.e., X = Y and ω = 0), we get the algebra D
X,0
by imposing in D
X
the relation ι∂ = 0, i.e., ∂
p
= ∂
[p]
,∂∈T
X
(in local coordinates ∂
i
p
= 0). The
action of D
X
on O
X
factors through D
X,0
since ∂
p
and ∂
[p]
act the same on
O
X
. Actually, D
X,0
is the image of the canonical map D
X
→D
X
from 1.2 (see
2.2.5).
2. The Azumaya property of D
X
2.1. Commutative subalgebra A
X
⊆D
X
. We will denote the centralizer
of O
X
in D
X
by A
X
def
= Z
D
X
(O
X
), and the pull-back of T
∗
X
(1)
to X by
T
∗,1
X
def
= X×
X
(1)
T
∗
X
(1)
.
2.1.1. Lemma. A
X
= O
X
·Z
X
= O
T
∗,1
X/X
.
Proof. The problem is local so assume that X has coordinates x
i
. Then
D
X
= ⊕O
X
∂
I
and Z
X
= ⊕O
X
(1)
∂
pI
(recall that ι(∂
i
)=∂
i
p
). So, O
X
·Z
X
=
LOCALIZATION IN CHARACTERISTIC P
953
⊕O
X
∂
pI
∼
=
←− O
X
⊗
O
X
(1)
Z
X
, and this is the algebra O
X
⊗
O
X
(1)
O
T
∗
X
(1)
of
functions on T
∗,1
X. Clearly, Z
D
X
(O
X
) contains O
X
·Z
X
, and the converse
Z
D
X
(O
X
)⊆⊕O
X
∂
pI
was already observed in the proof of Lemma 1.3.2.
2.1.2. Remark. In view of the lemma, any D
X
-module E carries an action
of O
T
∗,1
X
; such an action is the same as a section ω of Fr
∗
(Ω
1
X
) ⊗End
O
X
(E).
As noted above E can be thought of as an O
X
module with a flat connection;
the section ω is known as the p-curvature of this connection. The section ω is
parallel for the induced flat connection on Fr
∗
(Ω
1
X
) ⊗End
O
X
(E).
2.2. Point modules δ
ζ
. A cotangent vector ζ =(b, ω) ∈ T
∗
X
(1)
(i.e., b ∈
X
(1)
and ω ∈ T
∗
a
X
(1)
) defines a central reduction D
X,ζ
= D
X
⊗
Z
X
O
ζ
(1)
. Given
a lifting a ∈ T
∗
X of b under the Frobenius map (such a lifting exists since k is
perfect and it is always unique), we get a D
X
-module δ
ξ
def
= D
X
⊗
A
X
O
ξ
, where
we have set ξ =(a, ω) ∈ T
∗,(1)
X. It is a central reduction of the D
X
-module
δ
a
def
= D
X
⊗
O
X
O
a
of distributions at a, namely δ
ξ
= δ
a
⊗
Z
X
O
ζ
. In local coor-
dinates at a, 1.4.1 says that D
X,ζ
has a k-basis x
J
∂
I
,I,J∈{0, 1, ,p− 1}
n
with x
p
i
= 0 and ∂
p
i
= ω, ∂
i

p
.
2.2.1. Lemma. Central reductions of D
X
to points of T
∗
X
(1)
are matrix
algebras. More precisely, in the above notations,
Γ(X, D
X,ζ
)
∼
=
−→ End
k
(Γ(X, δ
ξ
)).
Proof. Let x
1
, ,x
n
be local coordinates at a. Near a,
D
X
= ⊕
I∈{0, ,p−1}
n
∂
I
·A
X
;
hence δ
ξ
∼
=
⊕
I∈{0, ,p−1}
n
k∂
I
. Since x
i
(a)=0,
x
k
·∂
I
= I
k
·∂
I−e
k
and ∂
k
·∂
I
=

∂
I+e
k
if I
k
+1<p,
ω(∂
i
)
p
·∂
I−(p−1)e
k
if I
k
= p − 1.

.
Irreducibility of δ
ξ
is now standard and x
i
’s act on polynomials in ∂
i
’s by
derivations; so for 0 = P =

I∈{0, ,p−1}
n
c
I
∂
I
∈ δ
ξ
and a maximal K with
c
K
=0,x
K
·P is a nonzero scalar. Now multiply with ∂
I
’s to get all of δ
ξ
.
Thus δ
ξ
is an irreducible D
X,ζ
-module. Since dim D
X,ζ
= p
2 dim(X)
= (dim δ
ξ
)
2
we are done.
Since the lifting ξ ∈ T
∗,(1)
X of a point ζ ∈ T
∗
X
(1)
exists and is unique,
we will occasionally talk about point modules associated to a point in T
∗
X
(1)
,
and denote it by δ
ζ
, ζ ∈ T
∗
X
(1)
.
2.2.2. Proposition (Splitting of D
X
on T
∗,1
X). Consider D
X
as an
A
X
-module (D
X
)
A
X
via the right multiplication. Left multiplication by D
X
954 ROMAN BEZRUKAVNIKOV, IVAN MIRKOVI
´
C, AND DMITRIY RUMYNIN
and right multiplication by A
X
give an isomorphism
D
X
⊗
Z
X
A
X
∼
=
−→ E nd
A
X
((D
X
)
A
X
).
Proof. Both sides are vector bundles over T
∗,1
X = Spec(A
X
); the
A
X
-module (D
X
)
A
X
has a local frame ∂
I
,I∈{0, ,p − 1}
dim X
; while
x
J
∂
I
,J,I∈{0, ,p − 1}
dim X
is a local frame for both the Z
X
-module
D
X
and the A
X
-module D
X
⊗
Z
X
A
X
. So, it suffices to check that the map
is an isomorphism on fibers. However, this is the claim of Lemma 2.2.1,
since the restriction of the map to a k-point ζ of T
∗,1
X is the action of
(D
X
⊗
Z
X
A
X
)⊗
A
X
O
ζ
= D
X
⊗
Z
X
O
ζ
= D
X,ζ
on (D
X
)
A
X
⊗
A
X
O
ζ
= δ
ζ
.
2.2.3. Theorem. D
X
is an Azumaya algebra over T
∗
X
(1)
(nontrivial if
dim(X) > 0).
Proof. One of the characterizations of Azumaya algebras is that they
are coherent as O-modules and become matrix algebras on a flat cover [MI].
The map T
∗,1
X→T
∗
X
(1)
is faithfully flat; i.e., it is a flat cover, since the
Frobenius map X→X
(1)
is flat for smooth X (it is surjective and on the formal
neighborhood of a point given by k[[x
p
i
]]→k[[x
i
]]). If dim(X) > 0, then D
X
is nontrivial, i.e. it is not isomorphic to an algebra of the form End(V ) for
a vector bundle V , because locally in the Zariski topology of X, D
X
has no
zero-divisors, since gr(D
X
)=O
T
∗
X
; while the algebra of endomorphisms of a
vector bundle of rank higher than one on an affine algebraic variety has zero
divisors.
2.2.4. Remarks.(1) A related Azumaya algebra was considered in [Hur].
(2) One can give a different, somewhat shorter proof of Theorem 2.2.3
based on the fact that a function on a smooth k-variety has zero differential
if and only if it is a p
th
power, which implies that any Poisson ideal in O
T
∗
X
is induced from O
T
∗
X
(1)
. This proof applies to a more general situation of the
so called Frobenius constant quantizations of symplectic varieties in positive
characteristic, see [BeKa, Prop. 3.8].
(3) The statement of the theorem can be compared to the well-known fact
that the algebra of differential operators in characteristic zero is simple: in
characteristic p it becomes simple after a central reduction. Another analogy
is with the classical Stone – von Neumann Theorem, which asserts that L
2
(R
n
)
is the only irreducible unitary representation of the Weyl algebra: Theorem
2.2.3 implies, in particular, that the standard quantization of functions on the
Frobenius neighborhood of zero in A
2n
k
has unique irreducible representation
realized in the space of functions on the Frobenius neighborhood of zero in A
n
k
.
LOCALIZATION IN CHARACTERISTIC P
955
(4) The class of the Azumaya algebra in the Brauer group can be described
as follows. In [MI, II.4.14] one finds the following exact sequence of sheaves
in ´etale topology available for any smooth variety M over a perfect field of
characteristic p:
0 →O
∗
M
Fr
−→ O
∗
M
dlog
−→ Ω
1
M,cl
C−1
−→ Ω
1
M
→ 0,
where Fr : f → f
p
, C is the Cartier operator and Ω
1
M,cl
is the sheaf of closed
1-forms. This exact sequences produces a map H
0
(Ω
1
M
) → H
2
(O
∗
M
). One can
check that applying the map to the canonical 1-form on M = T
∗
X one gets
the class of the Azumaya algebra D
X
.
2.2.5. Splitting on the zero section. By a well known observation
2
the
small differential operators, i.e., the restriction D
X,0
of D
X
to X
(1)
⊆ T
∗
X
(1)
,
form a sheaf of matrix algebras. In the notation above, this is the observation
that the action map (Fr
X
)
∗
D
X,0
∼
=
−→ E nd
O
X
(1)
((Fr
X
)
∗
O
X
) is an isomorphism by
2.2.1. Thus Azumaya algebra D
X
splits on X
(1)
, and (Fr
X
)
∗
O
X
is a splitting
bundle. The corresponding equivalence between Coh
X
(1)
and D
X,0
modules
sends F∈Coh
X
(1)
to the sheaf Fr
∗
X
F equipped with a standard flat connection
(the one for which pull-back of a section of F is parallel).
2.2.6. Remark. Let Z ⊂ T
∗
X
(1)
be a closed subscheme, such that the
Azumaya algebra D
X
splits on Z (see Section 5 below for more examples of
this situation); thus we have a splitting vector bundle E
Z
on Z such that
D
X
|
Z
∼
=
−→ End(E
Z
). It is easy to see then that E
Z
is a locally free, rank one
module over A
X
|
Z
, thus it can be thought of as a line bundle on the preimage
Z

of Z in T
∗(1)
X under the map Fr × id : X ×
X
(1)
T
∗
X
(1)
→ T
∗
X
(1)
. In the
particular case when Z maps isomorphically to its image
¯
Z in X the scheme
Z

is identified with the Frobenius neighborhood of
¯
Z in X. The action of D
X
equips the resulting line bundle on Fr N(
¯
Z) with a flat connection. The above
splitting on the zero-section corresponds to the trivial line bundle O
X
with the
standard flat connection.
2.3. Torsors. A torsor

X
π
→ X for a torus T defines a Lie algebroid

T
X
def
= π
∗
(T

X
)
T
with the enveloping algebra

D
X
def
= π
∗
(D

X
)
T
. Let t be the Lie
algebra of T. Locally, any trivialization of the torsor splits the exact sequence
0→t ⊗O
X
→

T
X
→T
X
→ 0 and gives

D
X
∼
=
D⊗ Ut. So the map of the constant
sheaf U(t)
X
into

D
X
, given by the T -action, is a central embedding and

D
X
is
a deformation of D
X
∼
=

D
X
⊗
S(
t
)
k
0
over t
∗
. The center O
T
∗

X
(1)
of D

X
gives
a central subalgebra (π
∗
O
T
∗

X
(1)
)
T
= O

T
∗
X
(1)
of

D
X
. We combine the two
into a map from functions on

T
∗
X
(1)
×
t
∗(1)
t
∗
to Z(

D
X
) (the map t
∗
→ t
∗(1)
is
the Artin-Schreier map AS; the corresponding map on the rings of functions
2
The second author thanks Paul Smith from whom he has learned this observation.
956 ROMAN BEZRUKAVNIKOV, IVAN MIRKOVI
´
C, AND DMITRIY RUMYNIN
S(t
(1)
) → S(t) is given by ι(h)=h
p
− h
[p]
,h∈ t
(1)
). Local trivializations
again show that this is an isomorphism and that

D
X
is an Azumaya algebra
on

T
∗
X
(1)
×
t
∗(1)
t
∗
, which splits on X×
X
(1)
(

T
∗
X
(1)
×
t
∗(1)
t
∗
).
In particular, for any λ ∈ t
∗
, specialization D
λ
X
def
=

D
X
⊗
S(
t
)
k
λ
is an
Azumaya algebra on the twisted cotangent bundle
T
∗
AS(λ)
X
(1)
def
=

T
∗
X
(1)
×
t
∗(1)
AS(λ),
which splits on T
∗,(1)
AS(λ)
X
def
= X×
X
(1)
T
∗
AS(λ)
X
(1)
. For instance, if λ = d(χ)isthe
differential of a character χ of T then AS(λ)=0;thusT
∗
AS(λ)
X = T
∗
X.In
this case D
λ
X
is identified with the sheaf
O
χ
D
X
∼
=
O
χ
⊗D
X
⊗O
χ
−1
of differential
operators on sections of the line bundle O
χ
on X, associated to

X and χ.
By a straightforward generalization of 2.1, 2.2,

A
X
def
= O
X×
X
(1)

T
∗
X
(1)
×
t
∗
(1)
t
∗
embeds into

D
X
. As in 2.2, for a point ζ =(a, ω; λ)ofX×
X
(1)

T
∗
X
(1)
×
t
∗(1)
t
∗
we
define the point module δ
ζ
=

D
X
⊗

A
X
O
ζ
.Ifζ
(1)
=(ω, λ) is the corresponding
point of

T
∗
X
(1)
×
t
∗(1)
t
∗
then we have

D
X
⊗
Z(

D
X
)
O
ζ
(1)
∼
=
−→ End
k
(δ
ζ
).
We finish the section with a technical lemma to be used in Section 5.
2.3.1. Lemma. Let ν = d(η) be an integral character. Define a morphism
τ
ν
from

T
∗
X
(1)
×
t
∗(1)
t
∗
to itself by τ
ν
(x, λ)=(x, λ + ν). Then the Azumaya
algebras

D
X
and τ
∗
ν
(

D
X
) are canonically equivalent.
Proof. Recall that to establish an equivalence between two Azumaya al-
gebras A, A

on a scheme Y (i.e. an equivalence between their categories of
modules) one needs to provide a locally projective module M over A⊗
O
Y
(A

)
op
such that A
∼
=
−→ End
(A

)
op
(M), A

∼
=
−→ End
A
(M). The sheaf π
∗
(D

X
)
T,η
of sec-
tions of π
∗
(D

X
) which transform by the character η under the action of T
carries the structure of such a module.
3. Localization of g-modules to D-modules on the flag variety
This crucial section extends the basic result of [BB], [BrKa] to positive
characteristic.
3.1. The setting. We define relevant triangulated categories of g-modules
and D-modules and functors between them.
3.1.1. Semisimple group G. Let G be a semisimple simply-connected
algebraic group over k. Let B = T · N be a Borel subgroup with the unipotent
radical N and a Cartan subgroup T . Let H be the (abstract) Cartan group of
G so that B gives isomorphism ι
b
=(T
∼
=
−→ B/N
∼
=
H). Let g, b, t, n, h be the
corresponding Lie algebras. The weight lattice Λ = X
∗
(H) contains the set
LOCALIZATION IN CHARACTERISTIC P
957
of roots Δ and of positive roots Δ
+
. Roots in Δ
+
are identified with T -roots
in g/b via the above “b-identification” ι
b
. Also, Λ contains the root lattice Q
generated by Δ, the dominant cone Λ
+
⊆ Λ and the semi-group Q
+
generated
by Δ
+
. Let I ⊆ Δ
+
be the set of simple roots. For a root α ∈ Δ let α→ˇα ∈
ˇ
Δ
be the corresponding coroot.
Similarly, ι
b
identifies N
G
(T )/T with the Weyl group W ⊆ Aut(H). Let
W
aff
def
= W  Q ⊆ W

aff
def
= W  Λ be the affine Weyl group and the extended
affine Weyl group. We have the standard action of W on Λ, w : λ → w(λ)=
w·λ, and the ρ-shift gives the dot-action w : λ → w•λ = w•
ρ
λ
def
= w(λ + ρ) − ρ
which is centered at −ρ, where ρ is the half sum of positive roots. Both actions
extend to W

aff
so that μ ∈ Λ acts by the pμ-translation. We will indicate the
dot-action by writing (W, •), this is really the action of the ρ-conjugate
ρ
W of
the subgroup W ⊆ W

aff
.
Any weight ν ∈ Λ defines a line bundle O
B,ν
= O
ν
on the flag variety
B
∼
=
G/B, and a standard G-module V
ν
def
=H
0
(B, O
ν
+
) with extremal weight ν.
Here ν
+
denotes the dominant W -conjugate of ν (notice that a dominant
weight corresponds to a semi-ample line bundle in our normalization). We
will also write O
ν
instead of π
∗
(O
ν
) for a scheme X equipped with a map
π : X →B(e.g. a subscheme of

g
∗
).
We let N⊂g
∗
denote the nilpotent cone, i.e. the zero set of invariant
polynomials of positive degree.
3.1.2. Restrictions on the characteristic p. Let h be the maximum of
Coxeter numbers of simple components of G.IfG is simple then h = ρ, ˇα
0
+1
where ˇα
0
is the highest coroot. We mostly work under the assumption p>h,
though some intermediate statements are proved under weaker assumptions; a
straightforward extension of the main Theorem 3.2 with weaker assumptions
on p is recorded in the sequel paper [BMR2]. The main result is obtained for a
regular Harish-Chandra central character, and the most interesting case is that
of an integral Harish-Chandra central character; integral regular characters
exist only for p ≥ h, hence our choice of restrictions
3
on p.
Recall that a prime is called good if it does not coincide with a coefficient
of a simple root in the highest root [SS, §4], and p is very good if it is good
and G does not contain a factor isomorphic to SL(mp) [Sl, 3.13]. We will need
a crude observation that p>h⇒ very good ⇒ good.
For p very good g carries a nondegenerate invariant bilinear form; also g
is simple provided that G is simple [Ja, 6.4]. We will occasionally identify g
and g
∗
as G-modules. This will identify the nilpotent cones N in g and g
∗
.
3
The case p = h is excluded because for G =SL(p), p = h is not very good and
g

∼
=
g
∗
as
G-modules.