D-module lecture note 12: Equivariant D-modules
Dongkwan Kim
April 15, 2015
We always assume that k is characteristic 0. Thus every algebraic group over k is
smooth by Cartier’s theorem [2].
Let X be a (smooth) algebraic variety and G be an affine algebraic group acting on
X. It means we have an action map ρ : G × X → X of varieties such that the following
diagram commutes;
G×G×X
idG ×ρ
m×idX
ρ
G×X
/G×X
ρ
/X
where m : G × G → G is the multiplication map. (We know that it should satisfy some
compatibility condition, which is called ”associativity”.) Now we recall the definition of an
equivariant sheaf. We denote by π : G × X → X the projection on the second factor.
Definition 1. A quasi-coherent sheaf M on X is called G-equivariant if it is endowed with
an isomorphism of OG×M = OG OM -modules ϕ : ρ∗ M → π ∗ M = OG M which makes
the following diagram commute:
(idG × ρ)∗ ρ∗ M
(idG ×ρ)∗ ϕ
/ (idG × ρ)∗ (OG M ) = OG ρ∗ M
idOG ×ϕ
OG OO G M
'
(m × idG )∗ ρ∗ M
(m×idX
)∗ ϕ
/ (m × idX )∗ (OG M ) = m∗ OG M
Note that it is the diagram of OG×G×X -modules.
This definition seems somewhat complicated, but if we regard ϕ as an action map of
G on M , such as (g, m) 7→ g · m, then it is clear that the diagram above actually means
(g1 g2 )m = g1 (g2 m), which is the associativity condition. (Also, one can reinterpret all
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the maps as the OG -comodule structure on OX and M , then it is the usual associativity
condition.)
We can also define an equivariant DX -module as follows.
Definition 2. A DX -module M is a weakly G-equivariant D-module on X if it is endowed
with an equivariant structure ϕ : ρ∗ M → π ∗ M = OG M of π ∗ DX = OG DX -modules.
It seems reasonable to call it just a G-equivariant DX -module since we used the DX module structure on M , which is not a part of the definition of quasi-coherent sheaves. But
note that we do not use the ”full power” of the action of G, since DG -module structures on
both ρ∗ M and π ∗ M are not used. (Note that the pull-back of O-modules and D-modules
are ”the same”.) So if ϕ is also an isomorphism as an DG×X = DG DX -module, then it
would give us a stronger and maybe more proper condition of equivariant D-modules.
Now we unravel the condition that ϕ is also DG -linear. First note that the morphism
ρ# : OX → OG×X is given by f 7→ [(g, x) 7→ f (g −1 x)]. Now ϕ is defined as follows;
ϕ : OG×X ⊗ρ M → OG M : f ⊗ m 7→ [g 7→ f (g, •)(gm)]
where f (g, •) is regarded as an element in OX . (We use ⊗ρ to avoid confusion.) To check
it is well defined, for a ∈ OX we have
ϕ(1 ⊗ am)(g) = g(am) = (ga)(gm)
ϕ(ρ# (a) ⊗ m)(g) = ρ# (a)(g, •)(gm) = a(g −1 •)(gm) = (ga)(gm)
Now we check the condition that this is DG -linear. Indeed, note that DG is generated
by OG and g, where g is the Lie algebra of G. Thus it suffices to check only g-linearity. Now
v ∈ g acts on ρ∗ M by v(f ⊗ m) = (vf ) ⊗ m + f ⊗ ρ∗ (v)m where ρ∗ (v) is the push-forward
of the vector field with coefficient in OX , i.e. ρ∗ : g → V ec(X). Meanwhile, v ∈ g acts on
π ∗ M by v(f ⊗ m) = (vf ) ⊗ m, since its image on X is zero. Thus we have
ϕ(v(f ⊗ m))(g) = (vf )(g, •)(gm) + f (g, •)(gρ∗ (v)m)
v(ϕ(f ⊗ m))(g) = (vf )(g, •)(gm) + f (g, •)”v(gm)”
where ”v(gm)” is the differentiation of the action map on m, i.e. em : G → M : g 7→ gm,
with respect to the tangent direction v. In other words, it is (vem )(g).
Therefore, we see that ϕ is DG -linear if and only if the following two actions coincide.
First, we have one derived from the G-action on X, denoted by ρ∗ (v)m. Indeed, if we
look at the infinitesimal action of G on X, we see that we have a well-defined morphism
ρ∗ : g → V ec(X) ⊂ DX . Equivalently, if we differentiate G → Aut(OX ), then we have
g → End(OX ), but it actually factors through g → V ec(X) ⊂ D(X), thus we have
ρ∗
ψ1 : g −→ V ec(X) ⊂ D(X) → End(M )
Second, there is also an action derived from the G-action on M itself, denoted by ”v(gm)”.
Indeed, since we have a morphism G → Aut(M ), by differentiation we have g → End(M ).
We will call this map ψ2 : g → End(M ).
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Definition 3. If M is weakly G-equivariant and such ϕ is also DG -linear, i.e. ψ1 = ψ2 ,
then we call M a (strongly) G-equivariant D-module.
In other words, we have the following commutative diagram;
/ Aut(OX )
G
g
ρ∗
/ DX
/ End(OX )
+
2 Endk (M )
ψ1
g
'
/ Endk (M )
ψ2
/ Autk (M )
G
Note that for φ1 , we only used the DX -module structure on M , not the G-module one. In
contrast, for φ2 we only used the G-module structure, not the other one.
Example 4. Let X = {pt} = Spec k. Then the category of k = DX -modules is isomorphic
to that of k-vector spaces. Thus a weakly G-equivariant DX -module is the same as a
G-equivariant OX -module, which is a locally algebraic representation of G (or an OG comodule.) But if we impose the condition that this is indeed (strongly) G-equivariant,
then since the morphism ρ∗ : g → DX is trivial, the action of g on M is also trivial. It means
G0 , the identity component, acts trivially on M , thus it is equal to a G/G0 -representation.
Remark. Thus note that in general these two actions are not equal. Indeed, the (strong)
G-equivariance is not a structure of O-modules, but an additional property, since ψ1 is
defined by the D-module structure. Meanwhile, if we have a D-module M and G is connected, then we can ”define” ψ2 : g → End(M ) to be ψ1 and ”integrate” it to have an
action of G on M , which makes M (strongly) G-equivariant. (If X is not affine, one needs
to be careful about how to ”integrate” the infinitesimal action.) Thus the category of
(strongly) G-equivariant DX -modules for connected G is a full subcategory of the category
of all DX -modules.
By the previous remark, ψ1 6= ψ2 in general. How different are they from each other?
Lemma 5. For v ∈ g, ψ1 (v) − ψ2 (v) ∈ EndDX (M ).
Proof. For any w ∈ DX and m ∈ M , first we have
ψ1 (v)(wm) = [ρ∗ (v), w]m + wψ1 (v)m,
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where [ρ∗ (v), w] = ∂v [g 7→ g(w) = gwg −1 ] by definition of ψ1 . Meanwhile, we have
[ψ2 (v), w]m = ψ2 (v)(wm) − wψ2 (v)m
= ∂v (g 7→ gwm) − ∂v (g 7→ wgm)
= ∂v (g 7→ (gw − wg)m)
but ∂v (g 7→ (gw − wg)m) = ∂v (g 7→ gwg −1 m). Thus we have [ρ∗ (v), w]m = [ψ2 (v), w]m,
but it means that
(ψ1 (v) − ψ2 (v))(wm) = w(ψ1 (v) − ψ2 (v))m,
from which the result follows.
Now we let λ = ψ1 −ψ2 : g → EndDX (M ). If EndDX M is abelian, e.g. M is irreducible,
then λ factors through g /[g, g] → EndDX (M ). Thus if g is perfect, i.e. g = [g, g], and if
G is connected, then every weakly G-equivariant DX -module is (strongly) G-equivariant.
Also, say if M is irreducible and G is connected, we can fix any weakly G-equivariant
DX -module to be (strongly) G-equivariant by mulitplying a suitable character of G whose
derivative is λ : g → k ' EndDX (M ).
Example 6. We will classify irreducible G = C∗ -equivariant D-modules on X = A1C =
Spec C[x], where the action is standard. Let M be such a module. Then we can decompose
M = ⊕n∈Z Mn where G acts on each Mn by tn . Thus if we differentiate this action we see
that 1 ∈ g = C acts on m ∈ Mn by ψ2 (1) · m = nm.
Meanwhile, g ∈ C∗ acts on C[x] by g · xi = g −i xi . Thus by differentiation we have
ρ∗ : g → End(C[x]) such that for 1 ∈ g, we have ρ∗ (1) · xi = −ixi . In other words, the
image of ρ∗ is spanned by the Euler field x∂ ∈ DX .
Now if we want M to be (strongly) G-equivariant, then we see that the Euler field should
act on Mn by −n. In other words, x∂ is diagonalizable on M with integer eigenvalues.
Now suppose M is irreducible and v ∈ M be a nonzero vector such that x∂v = nv for some
n ∈ Z. In the first PSet, Ex 3., we saw that DX /(x∂ − n)DX is either in Ext1 (O, δ0 ) or
Ext1 (δ0 , O). Thus it means M is either O or δ.
In sum, the category of (strongly) G-equivariant DX -modules is the Serre subcategory
generated by O and δ. (Remark: in the lecture note, he said only four kinds of modules are indecomposable; O, δ, nontrivial elements in Ext1 (O, δ0 ), or those in Ext1 (δ0 , O).
However, In PSet 2, Ex 3. c), we saw that there exists a D-module Mlog(x) with the socle
filtration whose successive quotients are the alternating sequence of O and δ. This is also
indecomposable since its socle is irreducible. We’ll see.)
Exercise 7. This is an exercise in [1]. Let X be a variety with a free action of G, thus
it is a principal G-bundle over Y = X/G. Let π : X → Y be the usual projection. Then
π ∗ : Ml (DY ) → Ml (DX ) is an equivalence of Ml (DY ) onto MlG (DX ), the category of
(strongly) G-equivariant D-modules on X.
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Exercise 8. Here is another exercise. Let L be a line bundle on Y , X be the set of nonzero
vectors in the total space of L, and G = C∗ , so Y = X/G. Now consider MlG (DX )m , which
is the category of weakly G-equivariant D-modules where two Lie algebra actions differs
by m-th power of the generating character of C∗ , i.e. λ(t) = ψ1 (t) − ψ2 (t) = nt. Also, let
Ml (DY (L⊗m )) be the category of modules over the twisted differential operators (acting
'
on L⊗m .) Then π ∗ is an equivalence Ml (DY (L⊗m )) −
→ MlG (DX )m .
Remark. In the argument above, we secretly assumed that X is smooth, thus a DX module over X is a quasi-coherent sheaf on X. If X is not smooth, then in general we can
embed X to a smooth variety and use Kashiwara’s theorem. Note that the criterion above
still makes sense for arbitrary X. Or, one can use !-crystals.
Now we consider the case when G has finitely many orbits on X. Then we have the
following remarkable theorem.
Theorem 9. Let X be affine and G act on X. Suppose its orbits are finitely many. Then
any finitely generated G-equivariant DX -module is holonomic.
Proof. First note that we may assume G is connected since G has finitely many components. Let M be such a module, and pick a finite number of generators m1 , · · · , mn ∈ M .
Then since the action of G on M is (locally) algebraic, we can find a finite dimensional
subrepresentation E ⊂ M which contains all mi . Now we define Fi M := (Fi DX )E. By
assumption it is a filtration on M , and also it is good since E is finite dimensional. Now
for m ∈ E and v ∈ g, ψ1 (v)m = ψ2 (v)m ∈ E since ψ2 is defined using the G-action
on M and E is G-invariant. Therefore ψ1 (v) ∈ Ann gr M since ψ1 (v) is of degree 1 but
it preserves degree when
√ multiplied on m ∈ E. In other words, the singular support
SS(M ) = Spec OT ∗ X / Ann gr M ⊂ V (im(ψ1 : g → OT ∗ X )), where the latter one is the
(finite) union of conormal bundles of G-orbits. But they are Lagrangian, i.e. of dimension
the same as dim X. Thus we get the result.
Now we note that the (strongly) C∗ -equivariant DA1 -modules O and δ are holonomic.
C
Also we see that the singular supports of these modules are contained in ”the cotangent
space of 0” and ”the zero section of T ∗ X on C \ 0”.
Example 10. Suppose X = G/H and G is connected. Then a G-equivariant DX -module
on X is the same as a π0 (H)-module. Indeed, let M be such a module. Then since the action
of h ⊂ g on X is trivial, it acts trivially on M , thus we have an H/H 0 = π0 (H)-module
structure on M . Conversely, if M is a π0 (H)-module then using DX -module structure on
M we can define the G-action on M by integration. The fact that H 0 acts trivially on M ,
or h acts trivially on M , ensures that this integration is well-defined.
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References
[1] Etingof, Pavel. Lecture 12 (D-module lecture note). http://www-math.mit.edu/
~etingof/769lect12.pdf
[2] Milne, J.S. Algebraic Groups: An introduction to the theory of algebraic group schemes
over fields. (available on http://www.jmilne.org/math/)
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