Equivariant analogues of Zuckerman functors

0.35 MB·

by  Pavle Pandžić

EQUIVARIANT ANALOGUES OF ZUCKERMAN FUNCTORS
4
0
0 Pavle Pandˇzic´
2

n Department of Mathematics, University of Zagreb, Croatia
a
J

9 Abstract. Wereviewthe Beilinson-Ginzburgconstructionof equivariantderivedcategories
2 of Harish-Chandra modules, and introduce analogues of Zuckerman functors in this setting.

They are given by an explicit formula, which works equally well in the case of modules with
]
T a given infinitesimal character. This is important if one wants to apply Beilinson-Bernstein
R localization, as is explained in [MP1]. We also show how to recover the usual Zuckerman
. functors from the equivariant ones by passing to cohomology.
h
t
a
m
[ Introduction

2
v One of the basic problems in representation theory is to understand linear representa-
6 tions of a real semisimple connected Lie group G with finite center. One can instead study
0
0
related problems of understanding g-modules, where g is the complexified Lie algebra of
1
1 G , or (g,K)-modules, where K is the complexification of a maximal compact subgroup
0
0
of G .
4 0
0 To be more specific, let M(g) be the abelian category of modules over g, or equivalently
h/ over the enveloping algebra U(g) of g. Let K be an algebraic group acting on g via inner
t automorphisms, suchthatthedifferentialofthisactionidentifiestheLiealgebrakofK with
a
m a subalgebra of g. Then one can define an abelian category M(g,K) of Harish-Chandra
: modules for the pair (g,K). This category consists of g-modules with an algebraic action of
v
i K satisfying the usual compatibility conditions; see §1. Clearly, there is a natural forgetful
X
functor from M(g,K) into M(g).
r
a In the late seventies, G. Zuckerman proposed a homological construction of Harish-
Chandra modules based on the following observation. Let T be a closed subgroup of K.
Then there is a natural forgetful functor from M(g,K) into M(g,T). This functor has a
rightadjointfunctorΓ , calledtheZuckerman functor. Roughlyspeaking, Γ attaches
K,T K,T
to a Harish-Chandra module in M(g,T) its largest Harish-Chandra submodule for (g,K).
Zuckerman functors are left exact and usually zero on “interesting” modules. Therefore,
one uses the machinery of homological algebra and considers their right derived functors.
Using them, one can construct “a lot of” (g,K)-modules. For example, if g and K are
related to G as above, then all irreducible (g,K)-modules are submodules of modules
0
obtained in this way for certain specific choices of (g,T)-modules for various suitable
1991 Mathematics Subject Classification. Primary 22E46.
1
2 PAVLE PANDZˇIC´
(reductive) T. See for example [Vo] or [KV]. We review the Zuckerman construction
briefly in §1; a more detailed review is contained in [MP1], §1.
As usual, if one wants to study homological algebra of an abelian category A, the nat-
ural setting is the derived category D(A) of A. In our case, one can consider the bounded
derived categories Db(M(g,T)) and Db(M(g,K)) of complexes of Harish-Chandra mod-
ules, the forgetful functor from Db(M(g,K)) into Db(M(g,T)) and its right adjoint, the
derived Zuckerman functor RΓ from Db(M(g,T)) into Db(M(g,K)). One can recover
K,T
the “classical” derived Zuckerman functors by applying RΓ to a single module and then
K,T
taking cohomology.
On the other hand, a powerful algebro-geometric method of studying g-modules is the
localization theory, due to A. Beilinson and J. Bernstein. Let θ be a maximal ideal in the
center Z(g) of the enveloping algebra U(g) and denote by U the quotient of U(g) by the
θ
two-sided ideal generated by θ. As before, we can define the abelian categories M(U )
θ
and M(U ,K) of g-modules, resp. Harish-Chandra modules, with a given infinitesimal
θ
character determined by θ. By the Harish-Chandra homomorphism, θ corresponds to
a Weyl group orbit in the linear dual h∗ of a Cartan subalgebra h of g. To each λ in
h∗, Beilinson and Bernstein attach a sheaf of twisted differential operators D on the
λ
flag variety X of g. They prove that the global sections Γ(X,D ) of D are equal to
λ λ
U , and that the higher cohomology vanishes. They define the localization functor from
θ
the category M(U ) into the category of quasicoherent D -modules M(D ) on X by
θ λ λ
∆ (V) = D ⊗ V. In the case of a regular infinitesimal character, and λ which satisfies a
λ λ Uθ
geometricpositivitycondition, thelocalizationfunctor ∆ isanequivalence ofthe category
λ
M(U ) with M(D ). Its quasi-inverse is the functor of global sections Γ(X,−). Under
θ λ
this equivalence, the category M(U ,K) of Harish-Chandra modules corresponds to the
θ
category M(D ,K) of Harish-Chandra sheaves. Without the positivity condition, the
λ
localizationfunctor induces an equivalence of corresponding derived categories Db(M(U ))
θ
and Db(M(D )). This enables one to apply the machinery of sheaf cohomology theory
λ
to the study of representations. Many results have been obtained using this method, the
most famous one being the proof of Kazhdan-Lusztig conjectures.
It is a natural problem to try to connect these two constructions and “localize” the
Zuckermanfunctorconstruction. Thesolutionisnotobvious, asthehomologicalproperties
of the categories M(g,K) and M(D ) are quite different. Still, Bernstein managed to
λ
find a partial solution in the case when λ satisfies the geometric positivity condition so
that the localization functor is exact. Also, Hecht, Miliˇci´c, Schmid and Wolf in their
Duality Theorem calculated the cohomology of “standard” Harish-Chandra sheaves on the
flag variety in terms of “standard” Zuckerman modules, which can be interpreted as a
particular instance of this problem.
In the meantime, the notion of equivariant derived category of Harish-Chandra modules
(and D-modules) has been introduced by Beilinson and Ginzburg ([BB], [G]; see also
[BL2] and [MP1], §2). An analogous construction for equivariant sheaves has been done
by Bernstein and Lunts. This category can be used as a bridge between the above two
theories. Namely,itallowslocalizationinanaturalway,andontheotherhanditcarriesthe
information about homological algebra of Harish-Chandra modules as well. The reason for
EQUIVARIANT ANALOGUES OF ZUCKERMAN FUNCTORS 3
thebetterbehaviorofthiscategorywithrespect tolocalizationisthefact thatitsdefinition
is based on the notion of “weak Harish-Chandra modules.” These are defined in the same
way as Harish-Chandra modules, but the compatibility conditions are relaxed: the two
actions of k are no longer required to agree. In particular, free U(g)-modules are weak
Harish-Chandra modules for the pair (g,K) (but they are not Harish-Chandra modules).
Similarly, free U -modules are weak (U ,K)-modules, and consequently it is easy to study
θ θ
localizationtheory for such modules. In §2 we review the Beilinson-Ginzburg construction,
including some details that are usually not mentioned.
A technical complication is that the equivariant derived category of Harish-Chandra
modules Db(g,K) is a triangulated category, but not a priori a derived category of any
abelian category. Therefore, its use requires some modifications of standard tools used in
homological algebra. In particular, a generalization of the notion of derived functors to
the case of an arbitrary triangulated category and its localization due to P. Deligne, [De]
is used. In this setting the place of projective and injective resolutions is taken by the
notions of S-projective and S-injective objects in the triangulated category in question;
here S is the localizing class defining the localization. These objects were first defined
by Verdier in [Ve]; he calls them “free on the left” (respectively right). In the setting of
homotopic categories, they were further studied and applied by Spaltenstein in [Sp] and
Bernstein and Lunts in [BL1] and [BL2] under the name of K-projectives and K-injectives.
One can find an account of this material in the Appendix of [P2], together with a summary
of some mostly well-known properties of adjoint functors, with emphasis on their use in
homological algebra.
eq
§3 is devoted to the construction of the equivariant analogues Γ of the derived
K,T
eq
Zuckerman functors. Γ is defined as the right adjoint of the forgetful functor from the
K,T
equivariantderived categoryD(g,K)into D(g,T). Analogously, one candefine a variant of
eq
this functor Γ : D(U ,T) −→ D(U ,K). It is shown how both of them can be described
K,T θ θ
by thesameexplicitformula, involvingthestandard complex N(k)ofk. The maintechnical
result here is that N(k) isa K-projective equivariant (k,T)-complex; see §3.2. Furthermore,
it is proved that in the case of Γeq : D(g,T) −→ D(g,K), when applied to a single Harish-
K,T
Chandra module V, this formula gives a complex with cohomology modules equal to the
“classical” derived Zuckerman functors of V. In fact, in this case the formula reduces to
the Duflo-Vergne formula 1.3.3; see [DV] and [MP1], 1.6. All of the above is also true on
the level of the bounded equivariant derived categories.
It is explained in [MP1] how these results can be used to localize the Zuckerman con-
struction.
Finally, let me mention that, as indicated in [MP1], one can use localizationof the Zuck-
erman construction to obtain a formula for the cohomology of standard Harish-Chandra
sheaves in terms of the derived Zuckerman functors. As a special case one can recover
the Duality theorem of Hecht, Miliˇci´c, Schmid and Wolf from [HMSW]. The details will
appear in [MP2].
Most of this paper comes out of my 1995 University of Utah Ph.D. thesis, which was
written under guidance of my advisor Dragan Miliˇci´c. I am very much indebted to him
for his generous help and many suggestions and ideas he supplied me with. I would also
4 PAVLE PANDZˇIC´
like to thank David Vogan for several useful conversations.
1. Harish-Chandra modules and Zuckerman functors
1.1. Harish-Chandra pairs and modules. Let K be a complex algebraic group and k
its Lie algebra. We recall that an algebraic representation of K is a vector space V with
a K-action, which is a union of K-invariant finite-dimensional subspaces V , so that the
i
action on each V is given by an algebraic group morphism K −→ GL(V ).
i i
Let A be an associative algebra with identity, with an algebraic action of K
φ : K −→ Aut(A),
and a Lie algebra homomorphism ψ from k into A such that the following conditions hold:
(A1) ψ is K-equivariant, i.e.,
ψ ◦Adk = φ(k)◦ψ for all k ∈ K;
(A2) The differential of the action φ is the same as the action of k on A given by inner
derivations ψ(ξ), ξ ∈ k, i.e.,
L(φ)(ξ)(a) = [ψ(ξ),a]
for any ξ ∈ k and a ∈ A.
These conditions are related, but in a general situation none of them implies the other.
A pair (A,K) as above is called a Harish-Chandra pair. The main examples come from
a similar notion of a “classical” Harish-Chandra pair, when A is replaced by a Lie algebra
g. Let g be a complex Lie algebra, and K a (usually reductive) algebraic group over C,
acting algebraically on g via a morphism φ : K −→ Int(g) of algebraic groups. Assume
further there is a K-equivariant embedding ψ of the Lie algebra k of K into g, such that
the differential of φ is equal to ad◦ψ. Then (g,K) is called a (classical) Harish-Chandra
pair.
In this situation, (U(g),K) is a Harish-Chandra pair as above: φ defines an action of K
on U(g) in an obvious way, and ψ defines an embedding of k into U(g); the compatibility
conditions (A1) and (A2) clearly hold. An important special case of this is when G is a
real semisimple Lie group, g its complexified Lie algebra, and K the complexification of
the maximal compact subgroup of G. The map φ is just the adjoint action and ψ is the
inclusion of k into U(g).
Another set of examples is obtained from a complex Lie group G; let g be the Lie
algebra of G, and K be any algebraic subgroup of G, like a Borel subgroup or its unipotent
radical; K can also be G itself. The maps φ and ψ are, as before, the adjoint action and
the inclusion.
In the above cases, when g is semisimple, one can also take A to be the quotient U
θ
of U(g) corresponding to an infinitesimal character given by the Weyl group orbit θ of an
EQUIVARIANT ANALOGUES OF ZUCKERMAN FUNCTORS 5
element of the dual of a Cartan subalgebra of g. This is particularly important for us, as
it is the setting of the Beilinson-Bernstein localization theory.
Finally, we will occasionally us another simple example: A = C, and K is an arbitrary
algebraic group; φ is the trivial action on C and ψ is 0.
A vector space V is called a weak (A,K)-module or a weak Harish-Chandra module for
the pair (A,K) if:
(HC1) V is an A-module with an action π;
(HC2) V is an algebraic K-module with an action ν;
(HC3) for any a ∈ A and k ∈ K we have
π(φ(k)a) = ν(k)π(a)ν(k)−1;
i.e., the A-action map A⊗V −→ V is K-equivariant.
In future we will denote the representation π ◦ ψ of k on V simply by π. This should
not create confusion; in fact in the most important examples ψ is injective and k can be
identified with a Lie subalgebra of A.
The action ν of K differentiates to an action of k which we denote also by ν. We put
ω(ξ) = ν(ξ)−π(ξ) for ξ ∈ k. The following lemma is [MP1, 2.1].
1.1.1. Lemma. Let V be a weak (A,K)-module. Then
(i) ω is a representation of k on V;
(ii) ω is K-equivariant, i.e.,
ω(Ad(k)ξ) = ν(k)ω(ξ)ν(k)−1
for ξ ∈ k and k ∈ K.
(iii)
[ω(ξ),π(a)]= 0,
for a ∈ A and ξ ∈ k. (cid:3)
We say that a weak (A,K)-module V is an (A,K)-module if ω = 0, i.e., if in addition
to (HC1)-(HC3) it satisfies:
(HC4) π(ξ) = ν(ξ) for all ξ ∈ k.
A morphism α : V −→ W of two weak (A,K)-modules is a linear map which is a morphism
for both A- and K-module structures. We denote by M(A,K) the category of all weak
w
(A,K)-modules, and by M(A,K) its full subcategory of (A,K)-modules. Clearly, these
are abelian categories.
The inclusion functor from M(A,K) into M(A,K) has both adjoints. They are
w
defined as follows. Let U ∈ M(A,K) . Let
w
Uk = {u ∈ U | ω(ξ)u = 0, ξ ∈ k}
6 PAVLE PANDZˇIC´
be the k-invariants in U with respect to ω and let
U = U/ω(k)U
k
be the k-coinvariants of U with respect to ω. Then 1.1.1 implies that the actions of A
and K on U induce actions on Uk and U . It is now clear that Uk is the largest (A,K)-
k
submodule of U, and that U is the largest (A,K)-quotient of U. This immediately implies
k
the following proposition.
1.1.2. Proposition. For any (A,K)-module V
Hom (V,U) = Hom (V,Uk) and
(A,K) (A,K)
Hom (U,V) = Hom (U ,V).
(A,K) (A,K) k
In other words, the functors U 7−→ Uk and U 7−→ U are right respectively left adjoint to
k
the forgetful functor from M(A,K) into M(A,K) . (cid:3)
w
1.2. Change of algebras. Let (A,K) and (B,K) be two Harish-Chandra pairs. Denote
by φ , ψ and φ , ψ the corresponding maps. Assume that there is a morphism of
A A B B
algebras γ : B −→ A such that
γ ◦φ (k) = φ (k)◦γ, k ∈ K,
A B
i.e., γ is K-equivariant. If V is a weak (A,K)-module, we can view it as a weak (B,K)-
module, where the action of B is given by π˜(b) = π(γ(b)) for b ∈ B. The equivariance
condition is preserved because of the equivariance of γ. This defines an exact functor
For = For from the category M(A,K) into the category M(B,K) . This forgetful
B,A w w
functor has both adjoints. We will need the left adjoint, which we now describe.
Let V be a weak (B,K)-module. We can consider A as a right B-module via the right
multiplication composed with γ. Then we can form the vector space
ind (V) = A⊗ V.
A,B B
One easily checks that this space becomes a weak (A,K)-module if we let A act by left
multiplication in the first factor, and K by the tensor product of the actions φ on A
A
and ν on V. Furthermore, ind is a functor from M(B,K) into M(A,K) . If
V A,B w w
we forget the K-action, then it is a well-known fact that this functor is left adjoint to
the forgetful functor from A-modules to B-modules. However, the standard adjunction
morphisms v 7→ 1⊗v, v ∈ V for a B-module V, and a⊗v 7→ π (a)v, a ∈ A, v ∈ V for an
V
A-module V, are easily checked to be K-equivariant in our present situation. This implies
the following proposition.
EQUIVARIANT ANALOGUES OF ZUCKERMAN FUNCTORS 7
1.2.1. Proposition. Let γ : B −→ A be a K-equivariant morphism of algebras. Let For be
the corresponding forgetful functor from M(A,K) into M(B,K) , and let the functor
w w
ind be defined as above. Then ind is left adjoint to For. (cid:3)
A,B A,B
In case K is reductive, we can use this result to construct projectives in the categories
M(A,K) and M(A,K). Namely, clearly M(K) = M(C,K) and this category is
w w
semisimple, hence all its objects are projective. Furthermore, the inclusion C −→ A given
by λ 7→ λ1 is clearly K-equivariant. Since ForC,A is exact, its left adjoint indA,C preserves
projectives. So for any V ∈ M(A,K)w, indA,C(V) is a projective weak (A,K)-module.
On the other hand, the adjunction morphism ΨV : indA,C(V) −→ V is surjective, since
Ψ (1⊗v) = v for any v ∈ V. So V is a quotient of a projective module.
V
Using this together with 1.1.2, we can similarly prove that M(A,K) also has enough
projectives. Hence we have proved
1.2.2. Proposition. Let K be a reductivealgebraic group. Thenthe categoriesM(A,K)
w
and M(A,K) have enough projectives. (cid:3)
The other adjoint is constructed similarly. It is given by
pro (V) = Hom (A,V)alg
A,B B
for a weak (B,K)-module V, with suitably defined actions. Here Walg denotes the K-
algebraic part of a K-module W which is not necessarily algebraic. We omit the details,
since we do not need the functor pro in this paper.
A,B
Let us note that if we in addition assume that ψ = γ ◦ ψ , then clearly For
A B B,A
sends M(A,K) into M(B,K). Moreover, an easy calculation shows that the ω-action on
ind V is equal to the tensor product of the trivial action on A with ω . It follows that
A,B V
ind sends (B,K)-modules into (A,K)-modules. This implies the following variant of
A,B
1.2.1.
1.2.3. Proposition. Let γ : B −→ A be a K-equivariant morphism of algebras such that
ψ = γ ◦ ψ . Then the functor ind : M(B,K) −→ M(A,K) is left adjoint to the
A B A,B
forgetful functor from M(A,K) into M(B,K). (cid:3)
An analogous result holds for the functor pro .
A,B
1.3. Change of groups and Zuckerman functors. Let (A,K) be a Harish-Chandra
pair, with φ : K −→ Aut(A) and ψ : k −→ A the corresponding morphisms.
K K
Let T be another algebraic group and γ : T −→ K a morphism of algebraic groups. The
main example is the inclusion of a closed subgroup. We denote the differential of γ again
by γ; in applications it will always be injective. It is easily checked that (A,T) becomes a
Harish-Chandra pair if we define φ = φ ◦γ and ψ = ψ ◦γ. Moreover, composing the
T K T K
K-action with γ to get a T-action clearly defines a forgetful functor from M(A,K) into
w
M(A,T) , and also from M(A,K) into M(A,T). We want to describe right adjoints of
w
these forgetful functors.
8 PAVLE PANDZˇIC´
First we treat the case T = {1}. Then, as was proved in [MP1, 2.2] (see also [MP1,
1.2]), the right adjoint of For : M(A,K) −→ M(A) is Ind , given by
w w
Ind (V) = R(K)⊗V = R(K,V).
w
Here V is an A-module, R(K) denotes the space of regular functions on K, and R(K,V)
denotes the space of V-valued regular functions on K. Recall that K acts on Ind (V) by
w
the right regular representation, and a ∈ A acts on F ∈ Ind (V) by
w
(π(a)F)(k) = π (φ (k)a)(F(k)), k ∈ K.
V K
Theadjunctionmorphismsaregivenasfollows. ForV ∈ M(A,K) ,Φ : V −→ Ind (ForV)
w V w
is the matrix coefficient map, given by
Φ (v)(k) = ν (k)v, v ∈ V, k ∈ K.
V V
For W ∈ M(A), Ψ : For(Ind (W)) −→ W is the evaluation at 1.
W w
Composing Ind with the functor of taking k-invariants with respect to ω from 1.1.2,
w
we get the functor Γ which is right adjoint to For : M(A,K) −→ M(A). One can use
K
the functors Ind and Γ to show the existence of enough injectives in M(A,K) and
w K w
M(A,K); see [MP1, 1.3, 2.3]. Note that this is analogous to 1.2.2.
Now we go back to the ‘relative’ case, when the group T is nontrivial. We first want
to construct the right adjoint of For : M(A,K) −→ M(A,T) . Let V be a weak (A,T)-
w w
module. We can consider V as an A-module and construct the weak (A,K)-module
Ind (V) = R(K) ⊗ V as above. We define an action λ of T on Ind (V) as the tensor
w T w
product of the left regular representation of T on R(K) (via γ) with the given action of T
on V. Then it is easy to see that λ commutes with the actions π and ν of A and K.
T
It follows that the space I (V) of λ -invariants in Ind (V) is a weak (A,K)-submodule
w T w
of Ind (V). It is now easy to see that I : M(A,T) −→ M(A,K) is the right adjoint
w w w w
of the forgetful functor; the adjunction morphisms are obtained by restriction from the
adjunction morphisms for Ind .
w
Taking invariants with respect to ω, we get the functor V 7→ I (V)k, which is right
w
adjoint to the forgetful functor For : M(A,K) −→ M(A,T) . Since M(A,T) is a full
w
subcategory of M(A,T) , by restricting the domainofthisfunctor we get the right adjoint
w
of the forgetful functor For : M(A,K) −→ M(A,T); see [P2], A2.1.1. This functor is
denoted by Γ , and called the Zuckerman functor.
K,T
Another construction of Γ was explained in [MP1, §1]; instead of k-invariants with
K,T
respect to the action ω, we can take k-invariants with respect to the action λ , which is the
k
tensor product of the left regular action of k on R(K) with the given action on V. These
two constructions agree because of the following formula which was proved below 2.4 in
[MP1], and which implies that taking k-invariants with respect to ω or λ gives the same
k
result.
EQUIVARIANT ANALOGUES OF ZUCKERMAN FUNCTORS 9
1.3.1. Lemma. For any ξ ∈ k, F ∈ Ind (V) and k ∈ K,
w
(ω(ξ)F)(k) = −(λ (Ad(k)ξ)F)(k). (cid:3)
k
So we have
1.3.2. Theorem. The forgetful functor from M(A,K) to M(A,T) has a right adjoint,
the Zuckerman functor Γ . For an (A,T)-module V,
K,T
Γ (V) = Hom (C,R(K,V)),
K,T k,T
where C is the trivial (k,T)-module, R(K,V) is a (k,T)-module with respect to the ac-
tion λ = (λ ,λ ), and the (A,K)-action on Γ (V) comes from the (A,K)-action on
k T K,T
R(K,V). (cid:3)
Being a right adjoint, Γ isleft exact, and it has right derived functors (since there are
K,T
enough injectives). The following is a version of the Duflo-Vergne formula [DV], which is
an explicit formula for the derived Zuckerman functors holding under certain assumptions.
It generalizes the formula for Γ from 1.3.2. A proof can be found in [MP1, 1.6].
K,T
1.3.3. Theorem. Assume that T is reductive and that A is a flat right U(k)-module for
the right multiplication. Let V be an (A,T)-module. Then
RpΓ (V) = Hp(k,T;Ind (V))
K,T w
for p ∈ Z , where R(K,V) is a (k,T)-module with respect to the action λ. The (A,K)-
+
action on RpΓ (V) comes from the (A,K)-action on R(K,V). (cid:3)
K,T
One can also consider the derived functor RΓ between the corresponding derived
K,T
categories. However, 1.3.3 does not generalize readily to a claim about this functor. The
situation is better in the setting of equivariant derived categories which we define in the
following.
2. Equivariant derived categories
We are going to outline an algebraic construction of equivariant derived categories due
toBeilinsonandGinzburg (see [BB],[G]and[BL2]). Thereisalsoa geometricconstruction
due to Bernstein and Lunts [BL1].
2.1. Equivariant complexes. Let (A,K) be a Harish-Chandra pair. An equivariant
(A,K)-complex is a complex V = V· of weak (A,K)-modules, together with a family of
linear maps i : V −→ V of degree −1, depending linearly on ξ ∈ k, and satisfying the
ξ
following conditions:
(EC1) i π(a) = π(a)i , ξ ∈ k, a ∈ A;
ξ ξ
10 PAVLE PANDZˇIC´
(EC2) i = ν(k)i ν(k)−1, ξ ∈ k, k ∈ K;
Ad(k)ξ ξ
(EC3) i i +i i = 0, ξ,η ∈ k;
ξ η η ξ
(EC4) di +i d = ω(ξ) ξ ∈ k.
ξ ξ
Here π and ν are the actions of A and K on V, ω = ν−π is the action of k from §1.1, and
d is the differential of the complex V.
Note that for any ξ ∈ k, ω(ξ) is a morphism of complexes of A-modules, which is
homotopic to zero, i being the homotopy. It follows that Hp(ω(ξ)) = 0 for any p ∈ Z, so
ξ
the cohomology modules of V are (A,K)-modules.
A morphism between two equivariant (A,K)-complexes is a morphism of complexes of
weak(A,K)-moduleswhichcommutes withthei ’s. Wedenotethecategoryofequivariant
ξ
(A,K)-complexes by C(A,K). It is clearly an abelian category.
2.2. DG algebras and modules. It will be convenient to reformulate the definition of
equivariant complexes using the notion of DG modules over DG algebras. For more about
these see [Il]; see also [BL1] and [BL2]. The details needed here are written out in [P1].
The Liealgebra version ofthe theory isclosely relatedwiththetheory of Liesuperalgebras,
which can be looked up in [Sch].
In the following, a complex will often be identified with the sum of its components
with the natural grading. A differential graded (DG) Lie algebra is a complex of vector
spaces d with a graded bilinear operation [−,−] called the supercommutator, satisfying
the following conditions for any three homogeneous X,Y,Z ∈ d:
(DGLA1) [X,Y] = −(−1)degXdegY[Y,X];
(DGLA2) [X,[Y,Z]]= [[X,Y],Z]+(−1)degXdegY[Y,[X,Z]];
(DGLA3) d[X,Y] = [dX,Y]+(−1)degX[X,dY].
IfweforgetthedifferentialandintroducetheobviousZ -grading,wegetaLiesuperalgebra.
2
A differential graded (DG) module over d is a complex V of vector spaces, with a graded
action of d on V by linear endomorphisms, satisfying the properties
[X,Y]v = XYv −(−1)degXdegYYXv
and
d (Xv) = (d X)v+(−1)degXXd v
V d V