GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS:
PROJECTIVE FUNCTORS IN THE CATEGORY Fquad
CHRISTINE VESPA
Abstract. In this paper, we continue the study of the category of functors
Fquad, associated to F2-vector spaces equipped with a nondegenerate qua-
dratic form, initiated in [?] and [?]. We define a filtration of the standard
projective objects in Fquad; this refines to give a decomposition into inde-
composable factors of the two first standard projective objects in Fquad: PH0
and PH1 . As an application of these two decompositions, we give a complete
description of the polynomial functors of the category Fquad.
Mathematics Subject Classification: 18A25, 16D90, 20C20.
Keywords: functor categories; quadratic forms over F2; Mackey functors; rep-
resentations of orthogonal groups over F2.
Introduction
In the paper [?] we defined the category of functors Fquad from a category having
as objects the nondegenerate F2-quadratic spaces to the category E of F2-vector
spaces, where F2 is the field with two elements. The motivation for the construction
of this category is to obtain an analogous framework for the orthogonal groups over
F2, to that which exists for the general linear groups. We recall that the category
F of functors from the category Ef of finite dimensional F2-vector spaces to the
category E of all F2-vector spaces is a very useful tool for the study of the stable
cohomology of the general linear groups with suitable coefficients (see [?]). Another
motivation, in topology, for the study of the category F is the connection which
exists between this category and unstable modules over the Steenrod algebra (see
[?]). In order to have a good understanding of the category Fquad, we seek to
classify its simple objects. We constructed in [?] two families of simple objects in
Fquad. The first one is obtained by the fully-faithful, exact functor ι : F → Fquad,
defined in [?], which preserves simple objects. By [?], the simple objects in F are
in one-to-one correspondence with the irreducible representations of finite general
linear groups over F2. The second family is obtained by the fully-faithful, exact
functor κ : Fiso → Fquad, which preserves simple objects, where Fiso is equivalent
to the product of the categories of modules over the orthogonal groups of possibly
degenerate quadratic forms. In [?], we constructed two families of simple objects
in the category Fquad which are neither in the image of ι nor in the image of κ.
These simple objects are subfunctors of the tensor product between an object in
the image of ι and an object in the image of κ. We proved that these simple objects
in Fquad are the composition factors of two particular mixed functors, defined in
Date: November 11, 2018.
http://arxiv.org/abs/0704.0502v1
2 CHRISTINE VESPA
The aim of this paper is to begin a programme to obtain a complete classification
of the simple objects in Fquad. Accordingly, we seek to decompose the projective
generators of this category into indecomposable factors and to obtain the simple
factors of these indecomposable factors. This paper begins the study of the standard
projective objects in the category Fquad. Although explicit decompositions of all
the projective generators are not provided in this paper, we give several useful
tools, results and examples for the realization of this programme. Furthermore,
we deduce from the results contained in this paper several interesting consequences
for the structure of the category Fquad. In work in progress, we obtain a general
decomposition of standard projective object PH of Fquad which is indexed by the
subspaces of H . Here we present explicit decompositions of the standard projective
objects associated to “small” quadratic spaces, since these decompositions play a
fundamental rôle in the category Fquad (for example, for the description of the
polynomial functors of Fquad). Furthermore, recall that the decompositions of the
injective standard IF
of the category F and thus, by duality, that of the projective
standard PF
, is fundamental for the comprehension of the other injective standards
of F . Hence, the decompositions of the two smaller projective standard of Fquad
represent an important step in the understanding of the category Fquad.
We briefly summarize the contents of this paper. After some recollections on
the category Fquad, where we recall the definitions of the isotropic functors and
the mixed functors, we define a filtration of the standard projective objects PV in
Fquad:
0 ⊂ P
V ⊂ P
V ⊂ . . . ⊂ P
(dim(V )−1)
V ⊂ P
(dim(V ))
V = PV .
We obtain a general description of the two extremities of this filtration.
Theorem. Let V be a nondegenerate F2-quadratic space.
(1) There is a natural equivalence: P
V ≃ ι(P
ǫ(V )
), where ι : F → Fquad,
ǫ is the functor that forgets the quadratic form and PF
ǫ(V )
is the standard
projective object in F associated to the vector space ǫ(V ).
(2) The functor P
V is a direct summand of PV .
Proposition. Let V be a nondegenerate F2-quadratic space, we have a natural
equivalence:
PV /P
(dim(V )−1)
V ≃ κ(isoV )
where κ : Fiso → Fquad and isoV is an isotropic functor in Fiso.
An important consequence of the Theorem concerning the functor P
V is given
in following result.
Theorem. The category ι(F) is a thick subcategory of Fquad.
Then by an explicit study of the filtration of the functors PH0 and PH1 we
obtain the following fundamental decompositions of these two standard projective
functors.
Theorem. (1) The standard projective object PH0 admits the following decom-
position:
PH0 = ι(P
⊕2)⊕ (Mix0,1
⊕2 ⊕Mix1,1)⊕ κ(isoH0)
where Mix0,1 and Mix1,1 are two mixed functors and isoH0 is an isotropic
functor.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 3
(2) The standard projective object PH1 admits the following decomposition:
PH1 = ι(P
⊕2)⊕Mix1,1
⊕3 ⊕ κ(isoH1)
where Mix1,1 is a mixed functor and isoH1 is an isotropic functor.
These decompositions have several interesting consequences. Firstly, thanks to
this theorem we can complete the study of the functors Mix0,1 and Mix1,1 started
in [?] by the following result.
Proposition. The functors Mix0,1 and Mix1,1 are indecomposable.
We want to emphasize that the complete structure of the direct summands of
the decompositions of PH0 and PH1 is understood. The structure of the isotropic
functors is given in [?], those of the mixed functors Mix0,1 and Mix1,1 is the main
result of [?] and is completed by the previous proposition and those of PF
⊕2 follows
from [?]. Then, these decompositions give rise to a classification of the simple
functors S of Fquad such that S(H0) 6= 0 or S(H1) 6= 0.
Proposition. The isomorphism classes of non-constant simple functors of Fquad
such that either S(H0) 6= 0 or S(H1) 6= 0 are:
ι(Λ1), ι(Λ2), ι(S(2,1)), κ(iso(x,0)), κ(iso(x,1)), RH0 , RH1 , SH1
where RH0 , RH1 and SH1 are the simple functors introduced in Corollary 1.7.
These decompositions also allow us to derive some homological calculations in
the category Fquad.
Proposition. For n a natural number, we have:
ExtnFquad(RH0 , RH0) ≃ F2 and Ext
Fquad
(RH1 , RH1) ≃ F2
where RH0 and RH1 are the simple functors introduced in Corollary 1.7.
Finally, after having introduced the notion of polynomial functor for the category
Fquad, which generalizes that for F , we obtain the following result as an application
of the classification of the simple functors S of Fquad such that S(H0) 6= 0 or
S(H1) 6= 0 and of the thickness of the subcategory ι(F) of Fquad.
Theorem. The polynomial functors of Fquad are in the image of the functor ι :
F → Fquad.
Most of the results of this paper are contained in the Ph.D. thesis of the author
1. The category Fquad: some recollections
We recall in this section some definitions and results about the category Fquad
obtained in [?].
Let Eq be the category having as objects finite dimensional F2-vector spaces
equipped with a non degenerate quadratic form and with morphisms linear maps
that preserve the quadratic forms. By the classification of quadratic forms over
the field F2 (see, for instance, [?]) we know that only spaces of even dimension can
be nondegenerate and, for a fixed even dimension, there are two non-equivalent
nondegenerate spaces, which are distinguished by the Arf invariant. We will denote
by H0 (resp. H1) the nondegenerate quadratic space of dimension two such that
Arf(H0) = 0 (resp. Arf(H1) = 1). The orthogonal sum of two nondegenerate
4 CHRISTINE VESPA
quadratic spaces (V, qV ) and (W, qW ) is, by definition, the quadratic space (V ⊕
W, qV⊕W ) where qV ⊕W (v, w) = qV (v) + qW (w). Recall that the spaces H0⊥H0
and H1⊥H1 are isomorphic. Observe that the morphisms of Eq are injective linear
maps and this category does not admit push-outs or pullbacks. There exists a
pseudo push-out in Eq that allows us to generalize the construction of the category
of co-spans of Bénabou [?] and thus to define the category Tq in which there exist
retractions.
Definition 1.1. The category Tq is the category having as objects those of Eq and,
for V and W objects in Tq, HomTq (V,W ) is the set of equivalence classes of dia-
grams in Eq of the form V
←− W for the equivalence relation generated by
the relation R defined as follows: V
−→ X1
←− W R V
−→ X2
←− W if there
exists a morphism α of Eq such that α ◦ f = u and α ◦ g = v. The composition is
defined using the pseudo push-out. The morphism of HomTq (V,W ) represented by
the diagram V
←−W will be denoted by [V
←−W ].
Remark 1.2. A morphism of HomTq (V,W ) is represented by a diagram of the
form: V
−→ W⊥W ′
←−−W , where iW is the canonical inclusion. In the following,
we will use this representation of a morphism, without further comment.
By definition, the category Fquad is the category of functors from Tq to E . Hence
Fquad is abelian and has enough projective objects. By the Yoneda lemma, for any
object V of Tq, the functor PV = F2[HomTq (V,−)] is a projective object and there
is a natural isomorphism: HomFquad(PV , F ) ≃ F (V ), for all objects F of Fquad.
The set of functors {PV |V ∈ S}, named the standard projective objects in Fquad,
is a set of projective generators of Fquad, where S is a set of representatives of
isometry classes of nondegenerate quadratic spaces.
There is a forgetful functor ǫ : Tq → E
f in Fquad, defined by ǫ(V ) = O(V ) and
−→W⊥W ′
←−W ]) = pg ◦ O(f)
where pg is the orthogonal projection from W⊥W
′ to W and O : Eq → E
f is the
functor which forgets the quadratic form. By the fullness of the functor ǫ and an
argument of essential surjectivity, we obtain the following theorem.
Theorem 1.3. [?] There is a functor ι : F → Fquad, which is exact, fully faithful
and preserves simple objects.
In order to define another subcategory of Fquad, we consider the category E
having as objects finite dimensional F2-vector spaces equipped with a (possibly
degenerate) quadratic form and with morphisms injective linear maps which pre-
serve the quadratic forms. The category Edegq admits pullbacks; consequently the
category of spans Sp(Edegq ) ([?]) is defined. By definition, the category Fiso is the
category of functors from Sp(Edegq ) to E . As in the case of the category Fquad, the
category Fiso is abelian and has enough projective objects: by the Yoneda lemma,
for any object V of Sp(Edegq ), the functor QV = F2[HomSp(Edegq )(V,−)] is a projec-
tive object in Fiso. We define a particular family of functors of Fiso, the isotropic
functors, which form a set of projective generators and injective cogenerators of
Fiso. The category Fiso is related to Fquad by the following theorem.
Theorem 1.4. [?] There is a functor κ : Fiso → Fquad, which is exact, fully-faithful
and preserves simple objects.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 5
We obtain the classification of the simple objects of the category Fiso from the
following theorem.
Theorem 1.5. [?] There is a natural equivalence of categories
Fiso ≃
F2[O(V )]−mod
where S is a set of representatives of isometry classes of quadratic spaces (possibly
degenerate) and O(V ) is the orthogonal group.
The object ofFiso which corresponds, by this equivalence, to the module F2[O(V )]
is the isotropic functor isoV , defined in [?]. Recall that, as a vector space, isoV (W )
is isomorphic to the subspace of QV (W ) generated by the elements [V
←− V →W ].
A straightforward consequence of the classification of simple objects of Fiso given
in Theorem 1.5 is given in the following corollary. Recall that, by definition, an
object F of Fquad is finite if it has a finite composition series with simple subquo-
tients.
Corollary 1.6. The isotropic functors κ(isoV ) are finite in the category Fquad.
In section 3, we will require the composition series for the isotropic functors asso-
ciated to some small quadratic spaces. For α ∈ {0, 1}, let (x, α) be the degenerate
quadratic space of dimension one generated by x such that q(x) = α. Since the or-
thogonal groups O(x, 0) and O(x, 1) are trivial and O(H0) ≃ S2 and O(H1) ≃ S3,
we deduce from Theorem 1.5 and 1.4, the following corollary.
Corollary 1.7. (1) The functors κ(iso(x,0)) and κ(iso(x,1)) are simple in Fquad.
(2) The functor κ(isoH0) is indecomposable. We have the following non-split
short exact sequence:
0→ RH0 → κ(isoH0)→ RH0 → 0
where RH0 is the functor obtained from the trivial representation of O(H0).
(3) The functor κ(isoH1) admits the following decomposition:
κ(isoH1) = FH1 ⊕ (SH1)
where SH1 is the functor obtained from the natural representation of O(H1)
and FH1 is an indecomposable functor for which we have the following non-
split short exact sequence:
0→ RH1 → FH1 → RH1 → 0
where RH1 is the functor obtained from the trivial representation of O(H1).
In [?], we define a new family of functors of Fquad, named the mixed functors
and we decompose two particular functors of this family: the functors Mix0,1 and
Mix1,1. We recall the following description of these functors.
Proposition 1.8. [?] For α ∈ {0, 1}, the functors Mixα,1 : Tq → E are defined by
Mixα,1(V ) = F2[SV ]
where SV = {(v1, v2) |v1 ∈ V, v2 ∈ V, q(v1 + v2) = α, B(v1, v2) = 1} and
Mixα,1([V
−→W⊥L
←−−W ])[(v1, v2)] =
[(pW ◦ f(v1), pW ◦ f(v2))] if f(v1 + v2) ∈W
0 otherwise
where pW is the orthogonal projection.
6 CHRISTINE VESPA
In [?], for a positive integer n, we defined subfunctors Lnα of ι(Λ
n)⊗ κ(iso(x,α)),
where Λn is the nth exterior power and we proved that these functors are simple.
The functor L1α is equivalent to the functor κ(iso(x,α)). We obtain the following
result.
Theorem 1.9. [?] Let α be an element in {0, 1}.
(1) The functor Mixα,1 is infinite.
(2) There exists a subfunctor Σα,1 of Mixα,1 such that we have the following
short exact sequence
0→ Σα,1 → Mixα,1 → Σα,1 → 0.
(3) The functor Σα,1 is uniserial with unique composition series given by the
decreasing filtration given by the subfunctors kdΣα,1 of Σα,1:
. . . ⊂ kdΣα,1 ⊂ . . . ⊂ k1Σα,1 ⊂ k0Σα,1 = Σα,1.
(a) The head of Σα,1 (i.e. Σα,1/k1Σα,1) is isomorphic to the functor
κ(iso(x,α)) where iso(x,α) is a simple object in Fiso.
(b) For d > 0
kdΣα,1/kd+1Σα,1 ≃ L
where Ld+1α is a simple object of the category Fquad that is neither in
the image of ι nor in the image of κ.
The functor Ld+1α is a subfunctor of ι(Λ
d+1)⊗κ(iso(x,α)), where Λ
is the (d+ 1)st exterior power functor.
2. Filtration of the standard projective functors PV of Fquad
In this section, we define a filtration of the standard projective functors PV of
Fquad. This construction gives rise to an essential tool to obtain, in section 3,
the direct decompositions of the projective objects PH0 and PH1 of Fquad, into
indecomposable summands.
After defining this filtration, we will deduce general results about the projective
PV of Fquad. In Theorem 2.6 we prove that the rank zero part is a direct summand
of PV and we identify this functor. This result allows us to prove that ι(F) is a
thick subcategory of Fquad. We will also show that the top quotient of this filtration
is isomorphic to κ(isoV ), where isoV is the isotropic functor.
2.1. Definition of the filtration. We recall that a morphism in Tq from V to W ,
where V and W are nondegenerate quadratic spaces, is represented by a diagram
V → X ←W.
Definition 2.1. A morphism [V → X ← W ] in Tq has rank equal to i if the
pullback in Edegq of the diagram V → X ←W is a quadratic space of dimension i.
Notation 2.2. We denote by Hom
(V,W ) the subset of HomTq (V,W ) of mor-
phisms of rank less than or equal to i.
We have the following proposition:
Proposition 2.3. For W an object in Tq, the following subvector space of PV (W ):
V (W ) = F2[Hom
(V,W )]
defines a subfunctor P
V of PV .
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 7
Proof. It is sufficient to verify that for all morphisms f = [W → Y ← Z] of Tq
and g = [V → X ← W ] of Hom
(V,W ), the composition f ◦ g has rank less than
or equal to i. The composition f ◦ g is represented by the following commutative
diagram:
P ′ //
V // X // X⊥W
where P and P ′ are the pullbacks and X⊥
Y is the pseudo push-out defined in [?].
Consequently: f ◦ g = [V → X⊥
Y ← Z]. Since [V → X ← W ] is an element of
V (W ), we know that the dimension of P is less than or equal to i. We deduce
from the injectivity of the morphisms of Edegq , that P
′ has dimension smaller than
or equal to i. �
The following lemma is a straightforward consequence of Definition 2.1.
Lemma 2.4. There exists a natural equivalence: P
(dim(V ))
V ≃ PV .
We deduce the following proposition.
Proposition 2.5. The functors P
V , for i = 0, . . . , dim(V ), define an increasing
filtration of the functor PV .
Proof. The inclusion of vector spaces P
V (W ) ⊂ P
(i+1)
V (W ) is clear, forW an object
in Tq. Consequently, P
V is a subfunctor of P
(i+1)
V by the proposition 2.3. �
2.2. Extremities of the filtration. In the previous section, we have obtained,
for all objects V of Tq, the following filtration of the functor PV :
0 ⊂ P
V ⊂ P
V ⊂ . . . ⊂ P
(dim(V )−1)
V ⊂ P
(dim(V ))
V = PV .
The aim of this section is to study the two extremities of this filtration, namely,
the functor P
V and the quotient PV /P
(dim(V )−1)
2.2.1. The functor P
V . For V an object in Tq, we recall that the functor P
ǫ(V ) of
F defined by PFǫ(V )(−) = F2[HomEf (ǫ(V ),−)], where ǫ : Tq → E
f is the forgetful
functor of Fquad, is projective, by the Yoneda lemma. The aim of this paragraph
is to prove the following theorem:
Theorem 2.6. Let V be an object of Tq.
(1) There is a natural equivalence: P
V ≃ ι(P
ǫ(V )
), where ι : F → Fquad is the
functor given in Theorem 1.3.
(2) The functor P
V is a direct summand of PV .
Before proving this result, we give the following useful characterization of the
morphisms of rank zero, which is a straightforward consequence of the definition of
the rank of a morphism.
8 CHRISTINE VESPA
Lemma 2.7. Let V be an object in Tq. A morphism T = [V
−→ W⊥W ′
←−− W ]
has rank zero if and only if pW ′ ◦ α is an injective linear map, where pW ′ is the
orthogonal projection from W⊥W ′ to W ′.
For V and W objects in Tq, the forgetful functor ǫ : Tq → E
f gives rise to a map
HomTq(V,W )→ HomEf (ǫ(V ), ǫ(W )). By passage to the vector spaces freely gener-
ated by these sets and by functoriality of ǫ, we deduce the existence of a morphism
from PV to ι(P
ǫ(V )
). As the functors P
V are subfunctors of PV , we obtain a mor-
phism f from P
V to ι(P
ǫ(V )
). Consequently, to prove Theorem 2.6, it is sufficient
to prove the following proposition.
Proposition 2.8. The map P
V (W )
−−→ PFǫ(V )(ǫ(W )) is an isomorphism for V
and W objects in Tq.
The surjectivity of fW relies on the following lemma, which is an improved
version of the fullness of the forgetful functor ǫ given in [?].
Lemma 2.9. Let (V, qV ) and (W, qW ) be two objects of Tq and
f ∈ HomEf (ǫ(V, qV ), ǫ(W, qW )) a linear map, then there exists a morphism
T = [V
−→W⊥Y
←−−W ] of rank zero such that ǫ(T ) = f .
Proof. As the quadratic space V is nondegenerate, we know that it has even di-
mension. We write dim(V ) = 2n. We prove the result by induction on n.
To start the induction, let (V, qV ) be a nondegenerate quadratic space of di-
mension two, with symplectic basis {a, b} and f : V → W be a linear map. The
following linear map preserves the quadratic form:
g1 : V → W⊥H1⊥H0⊥H0 ≃W⊥Span(a1, b1)⊥Span(a0, b0)⊥Span(a
a 7−→ f(a) + (q(a) + q(f(a)))a1 + a0
b 7−→ f(b) + (q(b) + q(f(a)))a1 + (1 +B(f(a), f(b)))b0 + a
Consequently, the morphism: T = V
−→W⊥H1⊥H0 ←֓ W , is a morphism of rank
zero of Tq such that ǫ(T ) = f .
Let Vn be a nondegenerate quadratic space of dimension 2n, {a1, b1, . . . , an, bn}
be a symplectic basis of Vn and fn : Vn →W be a linear map. By induction, there
exists a map:
gn : Vn → W⊥Y
ai 7−→ fn(ai) + yi
bi 7−→ fn(bi) + zi
where yi and zi, for all integers i between 1 and n, are elements of Y . The map
gn preserves the quadratic form and the morphism T = [Vn
−→W⊥Y ←֓ W ] is of
rank zero and verifies ǫ([Vn
−→W⊥Y ←֓ W ]) = fn.
Let Vn+1 be a nondegenerate quadratic space of dimension 2(n + 1),
{a1, b1, . . . , an, bn, an+1, bn+1} a symplectic basis of Vn+1 and fn+1 : Vn+1 → W
a linear map. To define the map gn+1, we will consider the restriction of fn+1 to
Vn and extend the map gn given by the inductive assumption. For that, we need
the following space: E ≃ W⊥W ′⊥H⊥n0 ⊥H
0 ⊥H1⊥H0⊥H0 for which we specify
the notations for a basis:
E ≃W⊥W ′⊥(⊥ni=1Span(a
0))⊥(⊥
i=1Span(A
0))⊥Span(A1, B1)
⊥Span(C0, D0)⊥Span(E0, F0).
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 9
The following map:
gn+1 : V → W⊥W
′⊥H⊥n0 ⊥H
0 ⊥H1⊥H0⊥H0
ai 7−→ fn+1(ai) + yi + a
0 for i between 1 and n
bi 7−→ fn+1(bi) + zi +A
an+1 7−→ fn+1(an+1) + (q(an+1) + q(fn+1(an+1)))A1 + C0
i=1 B(fn+1(ai), fn+1(an+1))b
i=1 B(fn+1(bi), fn+1(an+1))B
bn+1 7−→ fn+1(bn+1) + (q(bn+1) + q(fn+1(bn+1)))A1
+(1 +B(fn+1(an+1), fn+1(bn+1)))D0
i=1 B(fn+1(ai), fn+1(bn+1))b
i=1 B(fn+1(bi), fn+1(bn+1))B
0 + E0
preserves the quadratic form. Furthermore, the morphism
T = [Vn+1
−−−→W⊥W ′⊥H⊥n0 ⊥H
0 ⊥H1⊥H0⊥H0 ←֓ W ]
is of rank zero and satisfies: ǫ(T ) = fn+1, which completes the inductive step.
The proof of the injectivity of fW relies on the following result, which can be
regarded as Witt’s theorem for degenerate quadratic forms.
Theorem 2.10. Let V be a nondegenerate quadratic space, D and D′ subquadratic
spaces (possibly degenerate) of V and f : D → D′ an isometry between these two
quadratic spaces. Then, there exists an isometry f : V → V such that the following
diagram is commutative:
f // V
// D′.
Proof. For a proof of this result, we refer the reader to [?] §4, theorem 1. �
Proof of the injectivity of fW . The natural map f is induced by the natural map
(V,−)→ HomEf (ǫ(V ), ǫ(−))
by passage to the vector spaces freely generated by these sets. So, f is injective if
and only if this natural map is injective. Consequently, it is sufficient to verify that,
for T = [V
−→ W⊥W ′
←−− W ] and T ′ = [V
−→ W⊥W ′′
←−−W ] two generators of
V (W ) such that
(2.10.1) pW ◦ O(α) = p
W ◦ O(α
we have T = T ′.
Let {a1, b1, . . . , an, bn} be a symplectic basis of V . We deduce from 2.10.1 that,
for all i ∈ {1, . . . , n}, we have:
(2.10.2) α(ai) = wi + w
i, α(bi) = xi + x
(2.10.3) α′(ai) = wi + w
i , α
′(bi) = xi + x
10 CHRISTINE VESPA
where, for all i ∈ {1, . . . , n}, wi and xi are in W , w
i and x
i are in W
′ and
x′′i and w
i are in W
′′. By Lemma 2.7, since the morphisms are of rank zero,
{w′1, x
1, . . . , w
n} and {w
1 , x
1 , . . . , w
n} are two linearly independent families
of vectors.
We will denote by W ′ = Span(w′1, x
1, . . . , w
n) (respectively
W ′′ = Span(w′′1 , x
1 , . . . , w
n)) the subquadratic space (possibly degenerate), of
W ′ (respectively W ′′) and we define the linear map f : W ′ → W ′′ by f(w′i) = w
and f(x′i) = x
i for all i ∈ {1, . . . , n}.
Since α and α′ preserve the quadratic forms, we deduce from the relations 2.10.2
and 2.10.3 that f preserves the quadratic form. Hence, we can apply Theorem
2.10 to the nondegenerate space W ′⊥W ′′, which gives a morphism f : W ′⊥W ′′ →
W ′⊥W ′′ of Eq, such that, the restriction of this morphism to W
′ coincides with f .
We deduce the commutativity of the following diagram:
α̃ //
α̃′ ++WWW
W W⊥(W ′⊥W ′′)
W⊥(W ′⊥W ′′)
where α̃ = iW⊥W ′ ◦ α and α̃′ = iW⊥W ′′ ◦ α
′. Consequently, we obtain the equality
T = T ′ since, by inclusion, we have
T = [V
−→W⊥W ′
←−−W ] = [V
−→W⊥W ′⊥W ′′
←−−W ]
T ′ = [V
−→W⊥W ′′
←−−W ] = [V
−→ W⊥W ′⊥W ′′
←−−W ].
Notation 2.11. For V and W two objects of Eq, and f a morphism of HomEf (ǫ(V ), ǫ(W )),
we denote by tf the morphism of Hom
(V,W ) corresponding to f and by [tf ] the
canonical generator of P
V (W ) obtained from tf . To simplify the notation, we will
denote the morphism tIdV of Hom
(V, V ) by eV .
We deduce from the first point of Theorem 2.6 the following corollary.
Corollary 2.12. For V , W and X objects of Eq, f : ǫ(W )→ ǫ(X) and g : ǫ(V )→
ǫ(W ) morphisms of Ef , we have:
tf ◦ tg = tf◦g
where tf , tg and tf◦g are respectively the morphisms of Hom
(W,X), Hom
(V,W )
and Hom
(V,X) associated to the linear maps f, g and f ◦ g.
We can apply this result to the idempotents of the ring of endomorphisms
End(PV ), to obtain the following proposition.
Proposition 2.13. The canonical generator [eV ] of P
V is an idempotent of the
ring of endomorphisms End(PV ) such that PV .[eV ] ≃ P
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 11
Proof. The canonical generator [eV ] is an idempotent of End(PV ) by Corollary 2.12.
By definition of the rank filtration PV .[eV ] ⊂ P
V and, for a canonical generator
[tf ] of P
V , we have [tf ] = [tf ] · [eV ]. �
The idempotent [eV ] plays a central rôle in the proof of the thickness of the
subcategory ι(F) in Fquad, which is the subject of the following paragraph. For
that, the following result is necessary.
Lemma 2.14. Let V and W be objects of Tq, the functor ι induces an isomorphism:
HomF (P
ǫ(V ), P
ǫ(W ))
−→ HomFquad(P
V , P
where ǫ : Tq → E is the forgetful functor.
Proof. By Proposition 2.13 and Theorem 2.6 we have the following equivalences:
HomFquad(P
V , P
W ) ≃ P
W (eV )P
W (V ) ≃ P
W (V )
≃ ι(PFǫ(W ))(V ) ≃ P
ǫ(W )(ǫ(V )) ≃ HomF (P
ǫ(V ), P
ǫ(W )).
To conclude this paragraph, we give the following property of eV which will be
useful in section 4 concerning the polynomial functors of Fquad.
Lemma 2.15. For V and W two objects of Tq, we have: eV⊥W = eV⊥eW , where
⊥ : Tq × Tq → Tq is the functor induced by the orthogonal sum.
Proof. This is a straightforward consequence of Proposition 2.8. �
2.2.2. The category ι(F) is a thick subcategory of Fquad. The aim of this paragraph
is to prove the following result.
Theorem 2.16. The category ι(F) is a thick subcategory of Fquad, where
ι : F → Fquad is the functor defined in Theorem 1.3.
To prove this theorem, we need the following general result about the precom-
position functor which is proved in the Appendix of [?].
Proposition 2.17. Let C and D be two small categories, A be an abelian category,
F : C → D be a functor and −◦F : Func(D,A)→ Func(C,A) be the precomposition
functor, where Func(C,A) is the category of functors from C to A. If F is full and
essentially surjective, then any subobject (respectively quotient) of an object in the
image of the precomposition functor is isomorphic to an object in the image of the
precomposition functor.
Proof of Theorem 2.16. • The subcategory ι(F) of Fquad is full by Theorem
• Let FF be an object in F and G a subobject of ι(FF ). Let F ′ be the
category of functors from Ef−(even) to E , where Ef−(even) is the full sub-
category of Ef having as objects the F2-vector spaces of even dimension.
The categories F and F ′ are equivalent [?]. The functor ǫ : Eq → E
factorizes through the inclusion Ef−(even) →֒ Ef . This induces a functor
ǫ′ : Eq → E
f−(even) which is full and essentially surjective. Consequently,
we can use Proposition 2.17 to obtain: G ≃ ι(GF ). Similarly, we obtain
the result for the quotient.
12 CHRISTINE VESPA
• Let GF and HF be objects of F , we set G = ι(GF ) and H = ι(HF ). For a
short exact sequence: 0 → G → F → H → 0, we have to prove that there
exists a functor FF in F such that F = ι(FF ).
Let P1 → P0 → G
F → 0 and Q1 → Q0 → H
F → 0 be projective
presentations of GF and HF in F , we have the following commutative
diagram
0 // ι(P1) //
ι(P1)⊕ ι(Q1) //
ι(Q1) //
0 // ι(P0) //
ι(P0)⊕ ι(Q0) //
ι(Q0) //
0 // G
0 0 0
where the columns are projective resolutions in Fquad, by the horseshoe
lemma. By Lemma 2.14, the morphism ι(P1)⊕ ι(Q1)→ ι(P0)⊕ ι(Q0) is in-
duced by a morphism of F denoted by f . Consequently,
F ∼= ι(Coker(f)) ∈ ι(F).
By Theorem 2.16, we deduce from Lemma 2.14, the following characterization
of the simple functors of F in Fquad which will be used in section 4 of this paper
concerning the polynomial functors of Fquad.
Lemma 2.18. (1) Let F be a functor of Fquad, then F is in the image of the
functor ι : F → Fquad if and only if, for all objects V in Tq,
F (eV )F (V ) = F (V ).
(2) Let S be a simple object in Fquad, then S is in the image of the functor
ι : F → Fquad if and only if there exists an object W in Tq such that
S(eW )S(W ) 6= 0.
Proof. (1) The forward implication is a consequence of the following fact: for a
functor F in the image of ι, HomFquad(P
V , F ) = F (eV )F (V ) = F (V ). The
reverse implication relies on the fact that the condition
F (eV )F (V ) = F (V ) implies that F is a quotient of a sum of projective
objects of the form P
V . Since the category ι(F) is thick in Fquad by
Theorem 2.16, we obtain the result.
(2) Observe that, if S(eW )S(W ) 6= 0, we have HomFquad(P
W , S) 6= 0, thus
S is a quotient of P
W by simplicity of S. Lemma 2.14 implies that there
exists a one-to-one correspondance between the indecomposable factors of
V and those of P
ǫ(V ). We deduce that the simple quotients of P
V arise
from F . Consequently, S is in the image of the functor ι.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 13
2.2.3. The quotient PV /P
(dim(V )−1)
V . The aim of this paragraph is to prove the
following result:
Proposition 2.19. Let V be an object in Tq, we have a natural equivalence:
PV /P
(dim(V )−1)
V ≃ κ(isoV )
where isoV is an isotropic functor and κ : Fiso → Fquad is the functor given in
Theorem 1.4.
To prove this proposition, we need the following notation and result:
Notation 2.20. Denote by σf the natural map PV
−→ κ(isoV ) which corre-
sponds to the canonical generator [V
−→ V ] of isoV (V ) by the equivalence
Hom(PV , κ(isoV )) ≃ isoV (V ) ≃ F2[O(V )] given by the Yoneda lemma.
Lemma 2.21. The functor κ(isoV ) of Fquad is a quotient of the functor PV =
F2[HomTq(V,−)].
Proof. The natural map PV
−→ κ(isoV ) is surjective: a pre-image of the canonical
generator [V
−→ W ] of κ(isoV )(W ) by (σf )W , is the morphism
g◦f−1
−−−−→W
←−W ]. �
A formal consequence of the previous lemma is given in the following result.
Lemma 2.22. The functor κ(isoV ) of Fquad is a quotient of the functor PV /P
(dim(V )−1)
Proof. By definition of the filtration and by the previous lemma, we have the dia-
gram:
0 // P
(dim(V )−1)
i // PV
// // PV /P
(dim(V )−1)
κ(isoV )
where i is the canonical inclusion of P
(dim(V )−1)
V in PV . By definition of σId,
we have σId ◦ i = 0, from which we deduce the existence of the surjection τ :
PV /P
(dim(V )−1)
V → κ(isoV ). �
We will prove below that this natural map is an isomorphism. It is sufficient to
prove the following result.
Proposition 2.23. For V and W two objects of Tq, we have an isomorphism
(PV /P
(dim(V )−1)
V )(W ) ≃ κ(isoV )(W ).
The proof of this proposition relies on the following lemma.
Lemma 2.24. For a non-zero canonical generator of (PV /P
(dim(V )−1)
V )(W ) repre-
sented by the morphism T = [V
−→ W⊥W ′
←−− W ] of Tq, we have g(V ) ⊂ W and
T = [V
←−W ], where g = iW ◦ f .
14 CHRISTINE VESPA
Proof. By definition of the filtration, for V andW two objects of Tq, the vector space
(PV /P
(dim(V )−1)
V )(W ) is generated by Hom
[dim(V )]
(V,W ) where Hom
[dim(V )]
(V,W )
is the set of morphisms from V to W whose the pullback D in Edegq is a quadratic
space such that dim(D) = dim(V ). We deduce from the existence of a monomor-
phism from D to V and from the equality of the dimensions, that D and V are
isometric. Consequently, for the morphism T of the statement, we have, by defi-
nition of the pullback, g(V ) ⊂ W . Thus, we have T = [V
←− W ], where
g = iW ◦ f , by the equivalence relation defined over the morphisms of Tq in Defini-
tion 1.1.
Proof of Proposition 2.23. The natural map τ obtained in the proof of Lemma 2.22
defines, for W an object in Tq, the linear map
τW : (PV /P
(dim(V )−1)
V )(W ) → κ(isoV )(W )
T = [V
←−W ] 7−→ [V
−→ W ]
which is clearly an isomorphism. �
3. Decomposition of the standard projective functors PH0 and PH1
On abelian categories, the decompositions into direct summands of a functor
F of Fquad correspond to decompositions into orthogonal idempotents of 1 in the
ring EndFquad(F ) (see for example [?]). One of the difficulties of the category
Fquad lies in the fact that the rings of endomorphisms of projectives PV and their
representations are not well-understood. The decompositions of projectives PV ,
obtained in work in preparation, using a refinement of the rank filtration will allow
us to understand the structure of these rings better.
In this section, we obtain the decompositions into indecomposable factors of the
projective objects PH0 and PH1 by an explicit study of the filtration defined in
section 2. This section concludes by several consequences of these decompositions.
In particular, we give a classification of the “small” simple functors of Fquad, which
is an essential ingredient in the following section about the polynomial functors of
Fquad.
3.1. Decomposition of PH0 . To obtain the decomposition of the functor PH0 into
indecomposable factors, we give an explicit description of the subquotients of the
filtration; then, we prove that the filtration splits for this functor and we identify
the factors of this decomposition.
3.1.1. Explicit description of the subquotients of the filtration. The aim of this para-
graph is to give a basis of the vector spaces P
(V ), P
(V ) and PH0/P
for V a given object in Tq.
We deduce from Theorem 2.6 and Notation 2.11, the following result.
Lemma 3.1. A basis B
(V ) is given by the set:
= {[tf ] for f ∈ HomEf (F2
⊕2, ǫ(V ))}.
By definition of the filtration, a canonical generator of P
(V ), represented
by the morphism T = [H0
−→ V⊥L
←− V ] of Tq, satisfies the following property:
I = f(H0) ∩ i(V ) is a quadratic space of dimension one.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 15
Lemma 3.2. Let T = [H0
−→ V⊥L
←− V ] be a morphism of Tq which represents a
canonical generator of P
(V ), and {a0, b0} be a symplectic basis of H0, then
the map f in T has one of the three following forms.
(1) If I = (f(a0), 0), the map f : H0 → V⊥L is defined by:
f(a0) = v and f(b0) = w + l
for v and w elements of V satisfying q(v) = 0 and B(v, w) = 1 and l a
non-zero element of L.
(2) If I = (f(b0), 0), the map f : H0 → V⊥L is defined by:
f(a0) = v + l and f(b0) = w
for v and w elements of V satisfying q(w) = 0 and B(v, w) = 1 and l a
non-zero element of L.
(3) If I = (f(a0 + b0), 1), the map f : H0 → V⊥L is defined by:
f(a0) = v + l and f(b0) = w + l
for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1 and l
a non-zero element of L.
Proof. The quadratic space H0 has three subspaces of dimension one which are:
Span(a0) and Span(b0) isometric to (x, 0) and Span(a0 + b0) isometric to (x, 1).
These three subspaces give rise to each one of the maps f defined in the statement.
Notation 3.3. The morphisms [H0
−→ V⊥L
←− V ], where f is one of the mor-
phisms described in the point (1) (respectively (2) and (3)) of the previous lemma,
will be known as type A (respectively B and C) morphisms.
We have the following proposition.
Proposition 3.4. For T = [H0
−→ V⊥L
←− V ] and T ′ = [H0
−→ V⊥L′
←− V ]
morphisms of Tq which represent canonical generators of P
(V ), the follow-
ing properties are equivalent.
(1) The morphisms T and T ′ of HomTq (H0, V ) have the same type and satisfy
the relation pV ◦ f = p
V ◦ f
(2) The morphisms T and T ′ of HomTq (H0, V ) are equal.
The proof of the implication (2)⇒ (1) relies on the following technical lemma.
Lemma 3.5. Let T = [V
←− W ] and T ′ = [V
−→ X ′
←− W ′] be morphisms
of HomTq (V,W ). If T = T
′, then g(V ) + h(W ) ≃ g′(V ) + h′(W ) in Edegq .
Proof. By definition of the equivalence relation given in Definition 1.1, it is sufficient
to prove that, for two morphisms T and T ′ such that TRT ′, we have g(V )+h(W ) ≃
g′(V ) + h′(W ).
16 CHRISTINE VESPA
By definition, g(V )+h(W ) is the smallest, possibly degenerate, quadratic space
such that we have a commutative diagram in Edegq , of the form:
**UUU
U g(V ) + h(W )
Similarly, g′(V ) + h′(W ) is the smallest quadratic space such that we have an
analogous commutative diagram. By definition of the relation R, we have the
existence of a morphism δ in Eq such that the following diagram is commutative:
By the consideration of the following commutative diagram in Edegq
where Y = g(V ) + h(W ), we deduce from the minimality of g′(V ) + h′(W ), the
existence of a morphism in Edegq from g
′(V ) + h′(W ) to g(V ) + h(W ) such that the
corresponding diagram is commutative. Then, by minimality of g(V ) + h(W ) for
T , we obtain: g(V ) + h(W ) ≃ g′(V ) + h′(W ). �
Proof of Proposition 3.4. Suppose that the morphisms T and T ′ of HomTq (H0, V )
are of type A such that pV ◦ f = p
V ◦ f
′. We deduce that:
f(a0) = v; f(b0) = w + l and f
′(a0) = v; f
′(b0) = w + l
for v and w elements of V and l (resp. l′) a non-zero element of L (resp. L′).
Since the maps f and f ′ preserve the quadratic forms, we have
q(b0) = q(w) + q(l) = q(w) + q(l
′). We deduce that q(l) = q(l′), thus the map
Span(l)
−→ Span(l′), such that α(l) = l′, preserves the quadratic form. Conse-
quently, we can apply Theorem 2.10 to obtain the existence of a map α : L⊥L′ →
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 17
L⊥L′ such that the following diagram is commutative:
α // L⊥L′
Span(l)
// Span(l′).
We deduce the commutativity of the diagram:
V � _
f ′ **VVV
V⊥(L⊥L′)
V⊥(L⊥L′).
Since T = [H0
−→ V⊥L⊥L′
←− V ] and T ′ = [H0
−→ V⊥L⊥L′
←− V ], by inclusion,
we deduce from the previous diagram that T = T ′.
We reason in the same way, for the morphisms of type B and C.
Conversely, if T = T ′, by Lemma 3.5 f(H0) + iV (V ) ≃ f
′(H0) + i
V (V ). Conse-
quently we deduce from Theorem 2.10 the existence of an isometry β : V⊥L⊥L′→
V⊥L⊥L′ making the following diagram commutative:
V⊥L⊥L′
β // V⊥L⊥L′
f(H0) + iV (V )
// f ′(H0) + i
V (V ).
This yields the commutativity of the following diagram:
f ′ **UUU
U V⊥L⊥L′
V⊥L⊥L′
which implies that β ◦ iV = i
V . Thus, β = IdV⊥β
′ where β′ : L⊥L′ → L⊥L′ is a
morphism of HomEq (L⊥L
′, L⊥L′). Consequently, we have
f(a0) = v + l; f(b0) = w + l
′ and f ′(a0) = v + β
′(l); f ′(b0) = w + β
′(l′)
and we deduce that pV ◦ f = p
V . Furthermore, since β
′ is inversible, for all x in
L⊥L′ we have: x is non-zero if and only if β′(x) is non-zero. Consequently, T and
T ′ have the same type. �
This proposition justifies the following notation.
Notation 3.6. We will denote by Av,w, Bv,w and Cv,w the morphisms of HomTq (H0, V )
respectively of type A, B and C and such that pV ◦ f(a0) = v and pV ◦ f(b0) = w.
18 CHRISTINE VESPA
The following result is a straightforward consequence of Lemma 3.2 and Propo-
sition 3.4.
Lemma 3.7. A basis B
(V ) is given by the set:
= { [Av,w] for v and w elements of V satisfying q(v) = 0 and B(v, w) = 1,
[Bv,w] for v and w elements of V satisfying q(w) = 0 and B(v, w) = 1,
[Cv,w] for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1}
By Proposition 2.19, we have (PH0/P
)(V ) ≃ κ(isoH0)(V ). We deduce the
following result.
Lemma 3.8. A basis B
of PH0/P
(V ) is given by the set:
= {[Df ] for f ∈ HomEq (H0, V )},
where Df is the morphism of Tq represented by the diagram: H0
←− V .
We end this paragraph by the rules of composition for the morphisms tf , Av,w,
Bv,w, Cv,w and Df , summarized in the following proposition. This technical re-
sult will be fundamental in the following paragraph, to prove the splitting of the
filtration.
Lemma 3.9. Let T = V
−→ W⊥L
←−− W be a morphism of HomTq(V,W ). The
following relations are satisfied:
(1) For f a morphism of HomEf (F2
⊕2, ǫ(V )) we have:
T ◦ tf = tϕ◦f .
(2) (a) For v and w elements of V satisfying q(v) = 0 and B(v, w) = 1, we
have:
T ◦Av,w =
Aϕ(v),pW ◦ϕ(w) if ϕ(v) ∈W
tpW ◦(ϕ⊥Id)◦α otherwise.
(b) For v and w elements of V satisfying q(w) = 0 and B(v, w) = 1, we
have:
T ◦Bv,w =
BpW ◦ϕ(v),ϕ(w) if ϕ(w) ∈W
tpW ◦(ϕ⊥Id)◦α otherwise.
(c) For v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1,
we have:
T ◦ Cv,w =
CpW ◦ϕ(v),pW ◦ϕ(w) if ϕ(v + w) ∈W
tpW ◦(ϕ⊥Id)◦α otherwise.
(3) For f a morphism of HomEq (H0, V ), we have:
T ◦Df =
Dϕ◦f if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈ W
Aϕ◦f(a0),pW ◦ϕ◦f(b0) if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W
BpW ◦ϕ◦f(a0),ϕ◦f(b0) if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W
CpW ◦ϕ◦f(a0),pW ◦ϕ◦f(b0) if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W
and ϕ ◦ f(a0 + b0) ∈W
tpW ◦(ϕ⊥Id)◦α if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W
and ϕ ◦ f(a0 + b0) /∈W.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 19
Proof. By definition of the composition in Tq, we have the following diagram:
α // V⊥L′
// W⊥L⊥L′
(1) For a morphism tf , we have α(a0) = f(a0) + l and α(b0) = f(b0) + m.
where {l,m} is a linearly independent family of L′. Consequently:
(ϕ⊥Id) ◦ α(a0) = ϕ ◦ f(a0) + l et (ϕ⊥Id) ◦ α(b0) = ϕ ◦ f(b0) +m.
We deduce that T ◦ tf = tϕ◦f .
(2) For Av,w, we have α(a0) = v and α(b0) = w + l
′, where l′ is a non-zero
element of L′. Consequently:
(ϕ⊥Id) ◦ α(a0) = ϕ(v) et (ϕ⊥Id) ◦ α(b0) = ϕ(w) + l
We have to distinguish two cases:
• if ϕ(v) ∈W , since ϕ preserves quadratic forms, we have q(ϕ(v)) = q(v)
and, since L′ is orthogonal to V , B(ϕ(v), pW ◦ϕ(w)) = B(ϕ(v), ϕ(w)) =
B(v, w). Thus the morphism Aϕ(v),pW ◦ϕ(w) is defined and we have:
T ◦Av,w = Aϕ(v),pW ◦ϕ(w);
• otherwise, ϕ(v) = pW ◦ ϕ(v) + m where m is a non-zero element of
L. Consequently, we obtain a morphism of nul rank and we have:
T ◦Av,w = tpW ◦(ϕ⊥Id)◦α.
The cases Bv,w and Cv,w are similar to the case of Av,w and are left to
the reader.
(3) For the morphism Df , where f is an element of HomEq (H0, V ), we have
α(a0) = f(a0) = v and α(b0) = f(b0) = w where v and w are elements of V .
Consequently: (ϕ⊥Id)◦α(a0) = ϕ(v) and (ϕ⊥Id)◦α(b0) = ϕ(w). Since ϕ◦f
preserves the quadratic forms, we have: q(ϕ ◦ f(a0)) = q(ϕ ◦ f(b0)) = 0
and B(ϕ ◦ f(a0), ϕ ◦ f(b0)) = 1. Thus the morphisms Aϕ◦f(a0),ϕ◦f(b0),
Bϕ◦f(a0),ϕ◦f(b0) and Cϕ◦f(a0),ϕ◦f(b0) are defined.
We have to distinguish four cases:
• if ϕ(v) ∈W and ϕ(w) ∈W then T ◦Df = Dϕ◦f ;
• if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W , we have ϕ ◦ f(a0) = w
′ and
ϕ ◦ f(b0) = w
′′ + l, where l is a non-zero element of L. Consequently,
we obtain a morphism of type A and we have T ◦Df = Aϕ◦f(a0),ϕ◦f(b0);
• if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W , we have ϕ ◦ f(a0) = w
′ + l and
ϕ ◦ f(b0) = w
′′, where l is a non-zero element of L. Consequently, we
obtain a morphism of type B and we have T ◦Df = Bϕ◦f(a0),ϕ◦f(b0);
• if ϕ ◦ f(a0) /∈ W , ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) ∈ W , we have
ϕ◦f(a0) = w
′+ l and ϕ◦f(b0) = w
′′+ l, where l is a non-zero element
of L. Consequently, we obtain a morphism of type C and we have
T ◦Df = Cϕ◦f(a0),ϕ◦f(b0);
• if ϕ ◦ f(a0) /∈ W , ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) /∈ W , we have
ϕ ◦ f(a0) = w
′ + l and ϕ ◦ f(b0) = w
′′ + l′, where l and l′ are non-zero
20 CHRISTINE VESPA
elements of L. Consequently, we obtain a morphism of nul rank and
we have T ◦Df = tpW ◦(ϕ⊥Id)◦α.
3.1.2. Splitting of the filtration for the functor PH0 . In this paragraph, we prove
the following result.
Proposition 3.10. The rank filtration splits for the functor PH0 , namely:
PH0 = P
⊕ PH0/P
Proof. By Theorem 2.6, we have: P
. To prove the proposition,
it is sufficient to prove that PH0 = P
⊕ PH0/P
By definition of the filtration, we have the short exact sequence:
(3.10.1) 0→ P
→ PH0
−→ PH0/P
Let V be an object in Tq, we consider a morphism f of HomEq (H0, V ) and
the generator [Df ] of PH0/P
(V ) associated to f . Since the map f preserves
the quadratic forms, we have: q(f(a0)) = q(f(b0)) = 0, q(f(a0 + b0)) = 1; thus
B(f(a0), f(b0)) = 1. Consequently, the morphisms Af(a0),f(b0), Bf(a0),f(b0) and
Cf(a0),f(b0) of HomTq (H0, V ) are defined. We define a map sV : PH0/P
(V ) →
PH0(V ) by:
sV : PH0/P
(V ) −→ PH0(V )
[Df ] 7→ [Df ] + [Af(a0),f(b0)] + [Bf(a0),f(b0)] + [Cf(a0),f(b0)].
We verify the two following statements.
(1) pV ◦ sV = Id.
For [Df ] a canonical generator of PH0/P
(V ), we have
pV ◦ sV ([Df ]) = [Df ]
since the morphisms Af(a0),f(b0), Bf(a0),f(b0) and Cf(a0),f(b0) have a rank
equal to one.
(2) The maps sV define a natural map.
One verifies that, for a morphism T = V
−→W⊥L
←−−W of HomTq (V,W ),
we have the commutativity of the following diagram:
PH0/P
sV //
PH0/P
PH0(V )
PH0 (T )
PH0/P
sW // PH0(W ).
In order to simplify notation, we will write: A′ = Aϕ◦f(a0),pW ◦ϕ◦f(b0),
B′ = BpW ◦ϕ◦f(a0),ϕ◦f(b0), C
′ = CpW ◦ϕ◦f(a0),pW ◦ϕ◦f(b0) and t
′ = tpW ◦(ϕ⊥Id)◦α.
On the one hand, by Lemma 3.9, we have:
PH0 (T ) ◦ sV ([Df ])
= PH0(T )([Df ] + [Af(a0),f(b0)] + [Bf(a0),f(b0)] + [Cf(a0),f(b0)])
= [T ◦Df ] + [T ◦Af(a0),f(b0)] + [T ◦Bf(a0),f(b0)] + [T ◦ Cf(a0),f(b0)]
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 21
[Dϕ◦f ] +[A
′] +[B′] +[C′] if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈ W
[A′] +[A′] +[t′] +[t′] = 0 if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W
[B′] +[t′] +[B′] +[t′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W
[C′] +[t′] +[t′] +[C′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W
and ϕ ◦ f(a0 + b0) ∈W
[t′] +[t′] +[t′] +[t′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W
and ϕ ◦ f(a0 + b0) /∈W.
[Dϕ◦f ] +[A
′] +[B′] +[C′] if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈W
0 otherwise.
On the other hand, by Lemma 3.9, we have:
PH0/P
(T )([Df ]) =
[Dϕ◦f ] if ϕ ◦ f(a0) ∈W and ϕ ◦ f(b0) ∈W
0 otherwise
since the morphisms A, B, C and t are zero in the quotient PH0/P
(W ).
We deduce,
sW ◦ PH0/P
(T )([Df ]) =
[Dϕ◦f ] + [A
′] + [B′] + [C′] if ϕ ◦ f(a0) ∈ W
and ϕ ◦ f(b0) ∈W
0 otherwise.
Consequently, the maps sV define a natural map which is a section of p. This
gives rise to the splitting of the exact sequence 3.10.1. �
3.1.3. Identification of the direct summands. The aim of this paragraph is to iden-
tify the summands of the decomposition given in Proposition 3.10. We begin by
proving that the morphisms of type A (respectively of type B and C) define a
subfunctor of P
which is a direct summand of this functor.
Lemma 3.11. The functor P
admits the following decomposition into di-
rect summands:
= FA ⊕ FB ⊕ FC
where FA, FB and FC are subfunctors of P
generated by, respectively, the
morphisms of type A, B and C.
Proof. By Lemma 3.7, we have an isomorphism of vector spaces
(V ) = FA(V )⊕ FB(V )⊕ FC(V ),
for all objects V in Tq.
Consequently, it is sufficient to prove that FA, FB and FC are subfunctors of
For FA, we have to verify the commutativity of the diagram
FA(V )
iV //
FA(T )
FA(W )
iW // P
22 CHRISTINE VESPA
where T is a morphism of HomTq(V,W ). Let [Av,w] be a canonical generator of
FA(V ), we have by Lemma 3.9
T ◦Av,w =
Aϕ(v),pW ◦ϕ(w) if ϕ(v) ∈W
tpW ◦(ϕ⊥Id)◦α otherwise.
Consequently,
(T ) ◦ iV ([Av,w]) =
[Aϕ(v),pW ◦ϕ(w)] if ϕ(v) ∈ W
0 otherwise,
since the morphism tpW ◦(ϕ⊥Id)◦α, has nul rank.
We deduce that P
(T ) ◦ iV ([Av,w ]) is in the vector space FA(W ); thus,
FA is a subfunctor of P
In the same way, by the use of values of T ◦Bv,w and T ◦ Cv,w given in Lemma
3.9, we prove that the functors FB and FC are subfunctors of P
In the following lemma, we identify the functors FA, FB and FC with certain
mixed functors defined in [?] and recalled in section 1.
Lemma 3.12. (1) The functors FA and FB are isomorphic to the functor
Mix0,1.
(2) The functor FC is isomorphic to the functor Mix1,1.
Proof. (1) The isomorphism FA ≃ Mix0,1.
Let [Av,w] be a canonical generator of FA(V ), we have, by definition,
B(v, w) = 1. Consequently, the following linear map exists:
σ1V : FA(V ) → Mix0,1(V )
[Av,w] 7−→ [(w, v + w)].
The map σ1V is an isomorphism, whose inverse is given by
(σ1V )
: Mix0,1(V ) → FA(V )
[(v, w)] 7−→ [Av+w,v].
We have to verify that the maps σ1V define a natural map; namely, for
a morphism T = [V
−→ W⊥L
←−− W ], that the following diagram is
commutative:
FA(V )
σ1V //
FA(T )
Mix0,1(V )
Mix0,1(T )
FA(W )
σ1W // Mix0,1(W ).
We have:
Mix0,1(T ) ◦ σ
V ([Av,w ]) = Mix0,1(T )[(w, v + w)]
[(pW ◦ (ϕ(w)), pW ◦ (ϕ(v + w))] if ϕ(v) ∈W
0 otherwise
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 23
by the definition of the mixed functors, and
σ1W ◦ FA(T )([Av,w]) = σ
[Aϕ(v),pW ◦ϕ(w)] if ϕ(v) ∈W
0 otherwise
[(pW ◦ (ϕ(w)), ϕ(v) + pW ◦ (ϕ(w))] if ϕ(v) ∈W
0 otherwise.
When ϕ(v) ∈ W , we have:
[(pW ◦ (ϕ(w)), ϕ(v) + pW ◦ (ϕ(w))] = [(pW ◦ (ϕ(w)), pW ◦ (ϕ(v + w))],
what proves the naturality of σ1.
Since the two following cases are very close to the previous one, we only
give the definition of the isomorphism of vector spaces and we leave the
reader to verify that they define natural equivalences.
(2) The isomorphism FB ≃ Mix1,0.
σ2V : FB(V ) → Mix0,1(V )
Bv,w 7−→ [(v, v + w)].
(3) The isomorphism FC ≃Mix1,1.
σ3V : FC(V ) → Mix1,1(V )
Cv,w 7−→ [(v, w)]
We deduce the following proposition.
Proposition 3.13. The projective functor PH0 admits the following decomposition
into direct summands:
PH0 = ι(P
⊕2))⊕ (Mix0,1
⊕2 ⊕Mix1,1)⊕ κ(isoH0)
where Mix0,1 and Mix1,1 are mixed functors and isoH0 is an isotropic functor.
Proof. This proposition is a straightforward consequence of Proposition 3.10, The-
orem 2.6, Proposition 2.19 and Lemmas 3.11 and 3.12. �
3.2. Decomposition of PH1 . The study of the functor PH1 is analogous to that
of the functor PH0 given in the previous section. Consequently, for the functor PH1 ,
we give only the principal results without proofs.
3.2.1. Explicit description of the subquotients of the filtration. In this paragraph,
we give basis of the vector spaces P
(V ), P
(V ) and PH1/P
(V ) for V
an object in Tq.
Lemma 3.14. A basis B
(V ) is given by the set:
= {tf for f ∈ HomEf (F2
⊕2, ǫ(V ))}.
Lemma 3.15. Let T = [H1
−→ V⊥L
←− V ] be a morphism of Tq which represents
a canonical generator of P
(V ), and {a1, b1} a symplectic basis of H1, then
the map f in T has one of the three following forms.
24 CHRISTINE VESPA
(1) If I = (f(a1), 0) the map f : H1 → V⊥L is defined by:
f(a1) = v et f(b1) = w + l
for v and w elements of V satisfying q(v) = 1 and B(v, w) = 1 and l a
non-zero element of L.
(2) If I = (f(b1), 0) the map f : H1 → V⊥L is defined by:
f(a1) = v + l et f(b1) = w
for v and w elements of V satisfying q(w) = 1 and B(v, w) = 1 and l a
non-zero element of L.
(3) If I = (f(a1 + b1), 1) the map f : H1 → V⊥L is defined by:
f(a1) = v + l et f(b1) = w + l
for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1 and l
a non-zero element of L.
Notation 3.16. The morphisms [H1
−→ V⊥L
←− V ], where f is one of the
morphisms described in the point (1) (respectively (2) and (3)) of the previous lemma
will be known as type E (respectively F and G) morphisms.
The analogous proposition to Proposition 3.4 holds for H1. This justifies the
following notation.
Notation 3.17. Denote by Ev,w, Fv,w and Gv,w the morphisms of HomTq (H1, V )
respectively of type E, F and G and such that pV ◦ f(a1) = v and pV ◦ f(b1) = w.
We deduce the following lemmas.
Lemma 3.18. A basis B
(V ) is given by the set:
= { [Ev,w] for v and w elements of V satisfying q(v) = 1 and B(v, w) = 1,
[Fv,w] for v and w elements of V satisfying q(w) = 1 and B(v, w) = 1,
[Gv,w] for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1}
Lemma 3.19. A basis B
of PH1/P
(V ) is given by the set:
= {[Hf ] for f ∈ HomEq (H1, V )},
where Hf is the morphism of Tq represented by the diagram: H1
←− V .
The rules of composition of morphisms Ev,w, Fv,w , Gv,w and Hf are similar to
those given for Av,w, Bv,w, Cv,w and Df in Lemma 3.9. The details can be provided
by the reader.
3.2.2. Splitting of the filtration for the functor PH1 .
Proposition 3.20. The rank filtration splits for the functor PH1 , namely:
PH1 = P
⊕ PH1/P
Proof. One verifies that the map sV : PH1/P
(V )→ PH1(V ) given by:
sV : PH1/P
(V ) −→ PH1(V )
[Hf ] 7→ [Hf ] + [Ef(a1),f(b1)] + [Ff(a1),f(b1)] + [Gf(a1),f(b1)].
defines a natural map which is a section of the projection PH1 → PH1/P
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 25
3.2.3. Identification of the direct summands. We have the following lemma.
Lemma 3.21. The functor P
admits the following decomposition into di-
rect summands
= FE ⊕ FF ⊕ FG
where FE, FF and FG are subfunctors of P
generated by, respectively, the
morphisms of type E, F and G.
In the following lemma, we identify the functors FE , FF and FG with mixed
functor.
Lemma 3.22. The functors FE , FF and FG are equivalent to the functor Mix1,1.
We deduce the following decomposition.
Proposition 3.23. The projective functor PH1 admits the following decomposition
into direct summands:
PH1 = ι(P
)⊕Mix1,1
⊕3 ⊕ κ(isoH1)
where Mix1,1 is a mixed functor and isoH1 is an isotropic functor.
3.3. Consequences of decompositions of functors PH0 and PH1 . In this sec-
tion, we draw the conclusions of the decompositions of PH0 and PH1 given in Propo-
sitions 3.13 and 3.23. We deduce the indecomposability of the functors Mix0,1 and
Mix1,1, we study the projectivity of the first isotropic functors in Fquad and we give
the classification of the “small” simple objects of Fquad.
3.3.1. Indecomposability of functors Mix0,1 and Mix1,1. The aim of this paragraph
is to prove the following result:
Proposition 3.24. The functors Mix0,1 and Mix1,1 are indecomposable.
The proof of this proposition relies on the following obvious lemma.
Lemma 3.25. If the functor F of Fquad decomposes as a direct sum: F1⊕ . . .⊕Fn,
then the projections πi : F → Fi and the inclusions ji : Fi → F induce idempotents
ei = ji ◦ πi in the ring End(F ).
Proof of Proposition 3.24. (1) By the Yoneda lemma, we have:
Hom(PH0 ,Mix0,1) = Mix0,1(H0).
By a calculation, we obtain that the dimension of the space Mix0,1(H0) is
equal to 4. According to Proposition 3.13, the order of multiplicity of the
summand Mix0,1 in the decomposition of PH0 is equal to 2. Consequently,
the dimension of the vector space E := Hom(Mix0,1,Mix0,1) is 2. We have
the following basis: {Id, τ} where the map τ is given by: τ([(u, v)]) =
([(v, u)]). Consequently, E = ({0, Id, τ, Id + τ},+, ◦), as a ring, and it is
easy to see that this ring does not admit a non-trivial idempotent.
(2) Similarly, we have Hom(PH0 ,Mix1,1) = Mix1,1(H0) and
dim(Mix1,1(H0)) = 2. The order of multiplicity of the summand Mix1,1
in the decomposition of PH0 is equal to 1. We deduce that the ring
Hom(Mix1,1,Mix1,1) does not admit a non-trivial idempotent.
26 CHRISTINE VESPA
We deduce from this proposition the following result, which complements The-
orem 1.9, obtained in [?]:
Corollary 3.26. The short exact sequence 0 → Σα,1 → Mixα,1 → Σα,1 → 0 does
not split.
3.3.2. Projectivity of certain isotropic functors in Fquad. The decompositions given
in Propositions 3.13 and 3.23 allow us to study the projectivity of isotropic functors
in Fquad. Corollary 4.37 in [?] shows that the set of functors {isoV |V ∈ S} is a
set of projective generators of Fiso, where S is a set of representatives of isometry
classes of (possibly degenerate) quadratic spaces.
Since the functor κ(isoH0) (respectively κ(isoH1) ) is a direct summand of the
functor PH0 (respectively PH1) we have the following result.
Proposition 3.27. The functors κ(isoH0) and κ(isoH1) are projective in the cat-
egory Fquad.
We deduce from Corollary 1.6 and the previous proposition, the following result.
Corollary 3.28. The category Fquad contains non-constant finite, projective ob-
jects.
This corollary constitutes one of the new features of the category Fquad compared
to F . Recall that, according to Corollary B7 in [?], due to Lionel Schwartz, the
category F does not contain non-constant finite projective functors.
Recall that, the functor κ(iso(x,0)) is the top composition factor of Mix0,1 and
κ(iso(x,1)) is that of Mix1,1. We have the following result.
Proposition 3.29. The projective cover of κ(iso(x,0)) (respectively κ(iso(x,1))) is
the functor Mix0,1 (respectively Mix1,1). In particular, the functors κ(iso(x,0)) and
κ(iso(x,1)) are not projective in Fquad.
Proof. Since κ(iso(x,0))(H0) 6= {0} and κ(iso(x,1))(H0) 6= {0}, if these two functors
were projective, they would be direct summands of the functor PH0 . We deduce
from Proposition 3.13, that these functors are not projective. �
Remark 3.30. Propositions 3.27 and 3.29 let us conjecture that, for a nondegen-
erate F2-quadratic space H, κ(isoH) is a projective functor in Fquad and, for a
degenerate quadratic space D, κ(isoD) is not a projective functor in Fquad and its
projective cover is a generalized mixed functor. This result will be the subject of
future work.
3.3.3. Classification of simple objects S of Fquad such that either S(H0) 6= 0 or
S(H1) 6= 0. If S is a simple object in Fquad, such that S(H0) 6= 0, the Yoneda lemma
implies that Hom(PH0 , S) = S(H0) 6= 0. Consequently, there exists a morphism of
Fquad from PH0 to S which is an epimorphism, by simplicity of S. We deduce
from the decompositions given in Proposition 3.13 and 3.23, from Corollary 1.7
concerning the functors isoH0 and isoH1 and from the study of the functors Mix0,1
and Mix1,1 done in [?] and recalled in section 1, the following result.
Proposition 3.31. The isomorphism classes of non-constant simple functors of
Fquad such that either S(H0) 6= 0 or S(H1) 6= 0 are:
ι(Λ1), ι(Λ2), ι(S(2,1)), κ(iso(x,0)), κ(iso(x,1)), RH0 , RH1 , SH1
where RH0 , RH1 and SH1 are the simple functors introduced in Corollary 1.7.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 27
3.3.4. Extension groups in Fquad. By Theorem 1.4 and 1.5 we obtain an exact, fully-
faithful functor
V ∈S F2[O(V )]−mod
−→ Fquad, where S is a set of representatives
of isometry classes of quadratic spaces (possibly degenerate). Consequently, for
M and N two F2[O(V )]− modules, this functor induces a morphism of extension
groups:
F2[O(V )]−mod
(M,N)
(κ̃)∗
−−−→ Ext∗Fquad(κ̃(M), κ̃(N)).
We have the following proposition.
Proposition 3.32. For V ∈ {H0, H1}, the morphism (κ̃)∗ is an isomorphism.
The proof of this proposition relies on the following lemma.
Lemma 3.33. For V ∈ {H0, H1}, if P is a finite projective F2[O(V )]-module, κ̃(P )
is projective in Fquad.
Proof. If P is a finite projective F2[O(V )]-module, there exists a F2[O(V )]-module
Q such that P ⊕Q ≃ F2[O(V )]
⊕N . We deduce from the exactness of κ that κ̃(P ⊕
Q) ≃ κ̃(P ) ⊕ κ̃(Q). Since κ̃(F2[O(V )]) = κ(isoV ) and the functors κ(isoH0) and
κ(isoH1) are projective, by Proposition 3.27, we obtain that κ̃(P ) is projective. �
Proof of Proposition 3.32. Let M and N be F2[O(V )]-modules for V ∈ {H0, H1}
and P• → M be a projective resolution of M . Lemma 3.33 implies that κ̃(P•)
is a projective resolution of κ̃(M). The functor κ̃ induces a morphism of cochain
complexes
HomF2[O(V )]−mod(P•, N)→ HomFquad(κ̃(P•), κ̃(N))
which induces the morphism (κ̃)∗ in cohomology. Since the functor κ̃ is fully-
faithful the previous morphism is an isomorphism and so induces an isomorphism
in cohomology.
We deduce the following corollary:
Corollary 3.34. For n a natural number, we have:
ExtnFquad(RH0 , RH0) ≃ F2 and Ext
Fquad
(RH1 , RH1) ≃ F2
where RH0 and RH1 are the simple functors introduced in Corollary 1.7.
Proof. Let ǫ be an element in {0, 1}. For V = Hǫ, by Corollary 1.7 (1), we have
κ̃(F2) = RHǫ . So, applying Proposition 3.32 to M = N = F2 we obtain:
Ext∗Fquad(RHǫ , RHǫ) ≃ Ext
F2[O(Hǫ)]−mod
(F2,F2) = H
∗(O(Hǫ),F2).
Since O(H0) ≃ S2 ≃ C2 and O(H1) ≃ S3 ≃ GL2(F2) we know by classical results
of cohomology of groups that
Hn(O(H0),F2) = H
n(O(H1),F2) = F2.
Remark 3.35. This corollary exhibits an important difference between the cate-
gories F and Fquad; recall that in F Ext
F (S, S) = 0 for all simple objects S of F
(see [?]).
28 CHRISTINE VESPA
4. Application: the polynomial functors of Fquad
In this section, having generalized the notion of polynomial functor to the cat-
egory Fquad, we prove, by induction, that the polynomial functors of Fquad are in
the image of the functor ι : F → Fquad.
4.1. Definition of polynomial functors of Fquad.
4.1.1. The difference functors of Fquad. We define the difference functors of Fquad
which generalize the notion of difference functor of F . Recall that, according to
[?], the difference functor ∆ : F → F is the functor given by
∆F (V ) := Ker(F (V ⊕ F2)
F (p)
−−−→ F (V )),
for F an object in F , V an object in Ef and p : V ⊕ F2 → V the projection.
Definition 4.1. The difference functors ∆H0 : Fquad → Fquad and ∆H1 : Fquad →
Fquad are the functors defined by:
∆H0F (V ) := Ker(F (V⊥H0)
F (T0)
−−−−→ F (V )),
∆H1F (V ) := Ker(F (V⊥H1)
F (T1)
−−−−→ F (V )),
for F an object in Fquad, V an object in Tq, and Ti = [V⊥Hi
−→ V⊥Hi
←− V ] for
i ∈ {0, 1} .
We have the following result:
Lemma 4.2. The functors ∆H0 and ∆H1 are exact.
4.1.2. Definition of polynomial functors. Before giving the definition of polynomial
functors in Fquad, let us recall that of polynomial functors of F ( [?]). For an object
F of F , F is a polynomial functor of degree 0 if and only if ∆F = 0 and, for an
integer d, F is polynomial of degree at most d+ 1 if and only if ∆F is polynomial
of degree at most d.
Definition 4.3. Let F be an object in Fquad:
(1) the functor F is polynomial of degree 0 if and only if ∆H0F = ∆H1F = 0;
(2) for an integer d, the functor F is polynomial of degree at most d+1, if and
only if ∆H0F and ∆H1F are polynomial of degree at most d.
The following proposition allows us to simplify the definition of a polynomial
functor of degree 0.
Lemma 4.4. Let F be an object in Fquad. The functor ∆H0F is zero if and only
if the functor ∆H1F is zero.
Proof. If ∆H0F = 0, we have, for all objects V of Tq, F (V ) ≃ F (V⊥H0). Conse-
quently,
F (V ) ≃ F (V⊥H0) ≃ F (V⊥H0⊥H0) ≃ F (V⊥H1⊥H1),
where the last isomorphism is obtained from the isomorphism H0⊥H0 ≃ H1⊥H1,
recalled in section 1. We deduce from the existence of morphisms F (V ) →֒ F (V⊥H1)
and F (V ⊥H1) →֒ F (V⊥H1⊥H1), induced by the inclusions and from the isomor-
phism between F (V ) and F (V⊥H1⊥H1), that
F (V ) ≃ F (V⊥H1) ≃ F (V⊥H1⊥H1).
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 29
Therefore ∆H1F = 0.
The proof of the converse is similar. �
4.2. Study of polynomial functors of Fquad. The aim of this section is to prove
the following result:
Theorem 4.5. The polynomial functors of Fquad are in the image of the functor
ι : F → Fquad.
We will prove this theorem by induction over the degree of the polynomial func-
tors.
4.2.1. Polynomial functors of degree zero of Fquad. In this paragraph, we start the
induction. The proof of the following result relies, in an essential way, on the
classification of simple functors S of Fquad such that S(H0) 6= 0 or S(H1) 6= 0,
obtained in Proposition 3.31.
Lemma 4.6. Let S be a simple functor of Fquad, S is a polynomial functor of
degree zero if and only if S is the constant functor F2.
Proof. In order to prove the direct implication, we have to distinguish two cases.
(1) If S(H0) = S(H1) = 0.
By the classification of the nondegenerate quadratic spaces over F2, if W
is a space of minimal dimension, satisfying S(W ) 6= 0, we have the existence
of an element ǫ of {0, 1} and a nondegenerate quadratic space V which is
non-zero, such that: W ≃ Hǫ⊥V. Since W is of minimal dimension, we
have S(V ) = 0. This implies:
∆HǫS(V ) = S(Hǫ⊥V ) 6= 0.
We deduce the result in this case.
(2) If S(H0) 6= 0 or S(H1) 6= 0.
In this case, we use the classification of simple functors S of Fquad such
that S(H0) 6= 0 or S(H1) 6= 0 obtained in Proposition 3.31. By an explicit
calculation for all the functors S obtained in this classification, we obtain
that the functors ∆H0S are non-zero except for the constant functor S = F2.
The converse is trivial.
4.2.2. Proof of Theorem 4.5. To prove Theorem 4.5, we need the following result
where the idempotents [eV ], obtained in Proposition 2.13, play a crucial rôle.
Proposition 4.7. Let S be a non-trivial simple functor of Fquad which is not in
the image of the functor ι : F → Fquad, then, one of the functors ∆H0S or ∆H1S
is not in the image of the functor ι : F → Fquad.
Proof. Let W be a nondegenerate quadratic space of minimal dimension, such that
S(W ) 6= 0. We distinguish the two following cases.
(1) If dim(W ) = 2.
By an explicit calculation for all the functors S of the classification given
in Proposition 3.31, we obtain the result.
30 CHRISTINE VESPA
(2) If dim(W ) > 2.
There exists a nondegenerate quadratic space V , possibly trivial, and an
element ǫ of {0, 1}, such that: W ≃ H0⊥Hǫ⊥V. Suppose that ∆H0S and
∆H1S are in the image of the functor ι, we prove, below, that this implies
that S is in the image of ι. By Lemma 2.18 it is sufficient to show the
existence of an object W in Tq such that: S(eW )S(W ) 6= 0. By Lemma
2.15, we have: eW = eH0⊥eHǫ⊥eV .
Since W is assumed to be a space of minimal dimension such that
S(W ) 6= 0, we have S(H0⊥V ) = S(Hǫ⊥V ) = 0. This implies that
∆H0S(Hǫ⊥V ) ≃ S(W )(4.7.1)
∆HǫS(H0⊥V ) ≃ S(W ).(4.7.2)
These isomorphisms are natural and, for (4.7.1), the action of EndTq(Hǫ⊥V )
on ∆H0S(Hǫ⊥V ) corresponds to the restriction of the action of EndTq (W )
on S(W ). In the same way, for (4.7.2), the action of EndTq (H0⊥V ) on
∆HǫS(H0⊥V ) corresponds to the restriction of the action of EndTq (W ) on
S(W ). Suppose that ∆H0S and ∆H1S are in the image of ι. We deduce
that:
S(1H0⊥eHǫ⊥eV )S(W ) = ∆H0S(eHǫ⊥eV )∆H0S(Hǫ⊥V )
= ∆H0S(Hǫ⊥V ) = S(W )
where the first equality comes from the action described previously, the
second is a consequence of Lemma 2.15 and the third is given by 4.7.1. In
the same way, we obtain:
S(eH0⊥1Hǫ⊥eV )S(W ) = ∆HǫS(eH0⊥eV )∆HǫS(H0⊥V )
= ∆HǫS(H0⊥V ) = S(W ).
We deduce that: S(1H0⊥eHǫ⊥eV ) ◦ S(eH0⊥1Hǫ⊥eV )S(W ) = S(W ). Since
S(1H0⊥eHǫ⊥eV ) ◦ S(eH0⊥1Hǫ⊥eV ) = S(eW ) by Lemma 2.15, we have:
S(eW )S(W ) 6= 0, as required.
The proof of the theorem relies, also, on the following lemmas.
Lemma 4.8. (1) A functor F of Fquad which takes values in finite vector
spaces and such that ∆HǫF is finite, is finite.
(2) A polynomial functor F of Fquad which takes values in finite vector spaces
is finite.
Proof. (1) The functor
Fquad
−→ E × Fquad
F 7→ (F (0),∆HǫF )
is exact and faithful by Lemma 4.2. Hence, if γ(F ) is finite then F is finite.
(2) If F takes values in finite vector spaces, so does ∆HǫF . Consequently we
can apply the first point recursively to obtain the result.
Lemma 4.9. A finite object F of Fquad whose composition factors are in the image
of the functor ι is in the image of the functor ι.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 31
Proof. This result is a straightforward consequence of the thickness of the subcat-
egory ι(F) in Fquad given in Theorem 2.16. �
Proof of Theorem 4.5. Since a functor of Fquad is colimit of its subfunctors which
take values in finite vector spaces, we deduce from Lemma 4.8, that it is sufficient
to prove the result for the finite polynomial functors of Fquad. Furthermore, since
the functors ∆H0 and ∆H1 are exact by Lemma 4.2, it is sufficient to consider the
case of a simple functor S. We prove the theorem by induction over the polynomial
degree.
If S is polynomial of degree 0, according to Lemma 4.2, S is in the image of the
functor ι.
Suppose that all simple polynomial functors of Fquad and of degree d are in the
image of the functor ι and consider a simple polynomial functor S of Fquad such
that deg(S) = d + 1. By the definition of polynomial functor in Fquad given in
4.3, the functors ∆H0S and ∆H1S are polynomial of degree d. We deduce that all
composition factors of ∆H0S and ∆H1S are polynomial of degree smaller than or
equal to d and, by induction, we obtain that they are in the image of ι. Since S is a
simple functor, it is a quotient of a standard projective functor PV . Consequently
S takes its values in finite dimensional vector spaces. Therefore, ∆H0S and ∆H1S
take their values in finite dimensional vector spaces. We deduce from Lemma 4.8
that the functors ∆H0S and ∆H1S are finite, and, by Lemma 4.9, we obtain that
∆H0S and ∆H1S are in the image of the functor ι. Consequently, by Proposition
4.7, S is in the image of the functor ι. �
Ecole Polytechnique Fédérale de Lausanne, Institut de Géométrie, Algèbre et
Topologie, Lausanne, Switzerland.
E-mail address: christine.vespa@epfl.ch
	Introduction
	1. The category Fquad: some recollections
	2. Filtration of the standard projective functors PV of Fquad
	2.1. Definition of the filtration
	2.2. Extremities of the filtration
	3. Decomposition of the standard projective functors PH0 and PH1
	3.1. Decomposition of PH0
	3.2. Decomposition of PH1
	3.3. Consequences of decompositions of functors PH0 and PH1
	4. Application: the polynomial functors of Fquad
	4.1. Definition of polynomial functors of Fquad
	4.2. Study of polynomial functors of Fquad

  In this paper, we continue the study of the category of functors Fquad,
associated to F_2-vector spaces equipped with a nondegenerate quadratic form,
initiated in two previous papers of the author. We define a filtration of the
standard projective objects in Fquad; this refines to give a decomposition into
indecomposable factors of the two first standard projective objects in Fquad.
As an application of these two decompositions, we give a complete description
of the polynomial functors of the category Fquad.

Introduction
In the paper [?] we defined the category of functors Fquad from a category having
as objects the nondegenerate F2-quadratic spaces to the category E of F2-vector
spaces, where F2 is the field with two elements. The motivation for the construction
of this category is to obtain an analogous framework for the orthogonal groups over
F2, to that which exists for the general linear groups. We recall that the category
F of functors from the category Ef of finite dimensional F2-vector spaces to the
category E of all F2-vector spaces is a very useful tool for the study of the stable
cohomology of the general linear groups with suitable coefficients (see [?]). Another
motivation, in topology, for the study of the category F is the connection which
exists between this category and unstable modules over the Steenrod algebra (see
[?]). In order to have a good understanding of the category Fquad, we seek to
classify its simple objects. We constructed in [?] two families of simple objects in
Fquad. The first one is obtained by the fully-faithful, exact functor ι : F → Fquad,
defined in [?], which preserves simple objects. By [?], the simple objects in F are
in one-to-one correspondence with the irreducible representations of finite general
linear groups over F2. The second family is obtained by the fully-faithful, exact
functor κ : Fiso → Fquad, which preserves simple objects, where Fiso is equivalent
to the product of the categories of modules over the orthogonal groups of possibly
degenerate quadratic forms. In [?], we constructed two families of simple objects
in the category Fquad which are neither in the image of ι nor in the image of κ.
These simple objects are subfunctors of the tensor product between an object in
the image of ι and an object in the image of κ. We proved that these simple objects
in Fquad are the composition factors of two particular mixed functors, defined in
Date: November 11, 2018.
http://arxiv.org/abs/0704.0502v1
2 CHRISTINE VESPA
The aim of this paper is to begin a programme to obtain a complete classification
of the simple objects in Fquad. Accordingly, we seek to decompose the projective
generators of this category into indecomposable factors and to obtain the simple
factors of these indecomposable factors. This paper begins the study of the standard
projective objects in the category Fquad. Although explicit decompositions of all
the projective generators are not provided in this paper, we give several useful
tools, results and examples for the realization of this programme. Furthermore,
we deduce from the results contained in this paper several interesting consequences
for the structure of the category Fquad. In work in progress, we obtain a general
decomposition of standard projective object PH of Fquad which is indexed by the
subspaces of H . Here we present explicit decompositions of the standard projective
objects associated to “small” quadratic spaces, since these decompositions play a
fundamental rôle in the category Fquad (for example, for the description of the
polynomial functors of Fquad). Furthermore, recall that the decompositions of the
injective standard IF
of the category F and thus, by duality, that of the projective
standard PF
, is fundamental for the comprehension of the other injective standards
of F . Hence, the decompositions of the two smaller projective standard of Fquad
represent an important step in the understanding of the category Fquad.
We briefly summarize the contents of this paper. After some recollections on
the category Fquad, where we recall the definitions of the isotropic functors and
the mixed functors, we define a filtration of the standard projective objects PV in
Fquad:
0 ⊂ P
V ⊂ P
V ⊂ . . . ⊂ P
(dim(V )−1)
V ⊂ P
(dim(V ))
V = PV .
We obtain a general description of the two extremities of this filtration.
Theorem. Let V be a nondegenerate F2-quadratic space.
(1) There is a natural equivalence: P
V ≃ ι(P
ǫ(V )
), where ι : F → Fquad,
ǫ is the functor that forgets the quadratic form and PF
ǫ(V )
is the standard
projective object in F associated to the vector space ǫ(V ).
(2) The functor P
V is a direct summand of PV .
Proposition. Let V be a nondegenerate F2-quadratic space, we have a natural
equivalence:
PV /P
(dim(V )−1)
V ≃ κ(isoV )
where κ : Fiso → Fquad and isoV is an isotropic functor in Fiso.
An important consequence of the Theorem concerning the functor P
V is given
in following result.
Theorem. The category ι(F) is a thick subcategory of Fquad.
Then by an explicit study of the filtration of the functors PH0 and PH1 we
obtain the following fundamental decompositions of these two standard projective
functors.
Theorem. (1) The standard projective object PH0 admits the following decom-
position:
PH0 = ι(P
⊕2)⊕ (Mix0,1
⊕2 ⊕Mix1,1)⊕ κ(isoH0)
where Mix0,1 and Mix1,1 are two mixed functors and isoH0 is an isotropic
functor.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 3
(2) The standard projective object PH1 admits the following decomposition:
PH1 = ι(P
⊕2)⊕Mix1,1
⊕3 ⊕ κ(isoH1)
where Mix1,1 is a mixed functor and isoH1 is an isotropic functor.
These decompositions have several interesting consequences. Firstly, thanks to
this theorem we can complete the study of the functors Mix0,1 and Mix1,1 started
in [?] by the following result.
Proposition. The functors Mix0,1 and Mix1,1 are indecomposable.
We want to emphasize that the complete structure of the direct summands of
the decompositions of PH0 and PH1 is understood. The structure of the isotropic
functors is given in [?], those of the mixed functors Mix0,1 and Mix1,1 is the main
result of [?] and is completed by the previous proposition and those of PF
⊕2 follows
from [?]. Then, these decompositions give rise to a classification of the simple
functors S of Fquad such that S(H0) 6= 0 or S(H1) 6= 0.
Proposition. The isomorphism classes of non-constant simple functors of Fquad
such that either S(H0) 6= 0 or S(H1) 6= 0 are:
ι(Λ1), ι(Λ2), ι(S(2,1)), κ(iso(x,0)), κ(iso(x,1)), RH0 , RH1 , SH1
where RH0 , RH1 and SH1 are the simple functors introduced in Corollary 1.7.
These decompositions also allow us to derive some homological calculations in
the category Fquad.
Proposition. For n a natural number, we have:
ExtnFquad(RH0 , RH0) ≃ F2 and Ext
Fquad
(RH1 , RH1) ≃ F2
where RH0 and RH1 are the simple functors introduced in Corollary 1.7.
Finally, after having introduced the notion of polynomial functor for the category
Fquad, which generalizes that for F , we obtain the following result as an application
of the classification of the simple functors S of Fquad such that S(H0) 6= 0 or
S(H1) 6= 0 and of the thickness of the subcategory ι(F) of Fquad.
Theorem. The polynomial functors of Fquad are in the image of the functor ι :
F → Fquad.
Most of the results of this paper are contained in the Ph.D. thesis of the author
1. The category Fquad: some recollections
We recall in this section some definitions and results about the category Fquad
obtained in [?].
Let Eq be the category having as objects finite dimensional F2-vector spaces
equipped with a non degenerate quadratic form and with morphisms linear maps
that preserve the quadratic forms. By the classification of quadratic forms over
the field F2 (see, for instance, [?]) we know that only spaces of even dimension can
be nondegenerate and, for a fixed even dimension, there are two non-equivalent
nondegenerate spaces, which are distinguished by the Arf invariant. We will denote
by H0 (resp. H1) the nondegenerate quadratic space of dimension two such that
Arf(H0) = 0 (resp. Arf(H1) = 1). The orthogonal sum of two nondegenerate
4 CHRISTINE VESPA
quadratic spaces (V, qV ) and (W, qW ) is, by definition, the quadratic space (V ⊕
W, qV⊕W ) where qV ⊕W (v, w) = qV (v) + qW (w). Recall that the spaces H0⊥H0
and H1⊥H1 are isomorphic. Observe that the morphisms of Eq are injective linear
maps and this category does not admit push-outs or pullbacks. There exists a
pseudo push-out in Eq that allows us to generalize the construction of the category
of co-spans of Bénabou [?] and thus to define the category Tq in which there exist
retractions.
Definition 1.1. The category Tq is the category having as objects those of Eq and,
for V and W objects in Tq, HomTq (V,W ) is the set of equivalence classes of dia-
grams in Eq of the form V
←− W for the equivalence relation generated by
the relation R defined as follows: V
−→ X1
←− W R V
−→ X2
←− W if there
exists a morphism α of Eq such that α ◦ f = u and α ◦ g = v. The composition is
defined using the pseudo push-out. The morphism of HomTq (V,W ) represented by
the diagram V
←−W will be denoted by [V
←−W ].
Remark 1.2. A morphism of HomTq (V,W ) is represented by a diagram of the
form: V
−→ W⊥W ′
←−−W , where iW is the canonical inclusion. In the following,
we will use this representation of a morphism, without further comment.
By definition, the category Fquad is the category of functors from Tq to E . Hence
Fquad is abelian and has enough projective objects. By the Yoneda lemma, for any
object V of Tq, the functor PV = F2[HomTq (V,−)] is a projective object and there
is a natural isomorphism: HomFquad(PV , F ) ≃ F (V ), for all objects F of Fquad.
The set of functors {PV |V ∈ S}, named the standard projective objects in Fquad,
is a set of projective generators of Fquad, where S is a set of representatives of
isometry classes of nondegenerate quadratic spaces.
There is a forgetful functor ǫ : Tq → E
f in Fquad, defined by ǫ(V ) = O(V ) and
−→W⊥W ′
←−W ]) = pg ◦ O(f)
where pg is the orthogonal projection from W⊥W
′ to W and O : Eq → E
f is the
functor which forgets the quadratic form. By the fullness of the functor ǫ and an
argument of essential surjectivity, we obtain the following theorem.
Theorem 1.3. [?] There is a functor ι : F → Fquad, which is exact, fully faithful
and preserves simple objects.
In order to define another subcategory of Fquad, we consider the category E
having as objects finite dimensional F2-vector spaces equipped with a (possibly
degenerate) quadratic form and with morphisms injective linear maps which pre-
serve the quadratic forms. The category Edegq admits pullbacks; consequently the
category of spans Sp(Edegq ) ([?]) is defined. By definition, the category Fiso is the
category of functors from Sp(Edegq ) to E . As in the case of the category Fquad, the
category Fiso is abelian and has enough projective objects: by the Yoneda lemma,
for any object V of Sp(Edegq ), the functor QV = F2[HomSp(Edegq )(V,−)] is a projec-
tive object in Fiso. We define a particular family of functors of Fiso, the isotropic
functors, which form a set of projective generators and injective cogenerators of
Fiso. The category Fiso is related to Fquad by the following theorem.
Theorem 1.4. [?] There is a functor κ : Fiso → Fquad, which is exact, fully-faithful
and preserves simple objects.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 5
We obtain the classification of the simple objects of the category Fiso from the
following theorem.
Theorem 1.5. [?] There is a natural equivalence of categories
Fiso ≃
F2[O(V )]−mod
where S is a set of representatives of isometry classes of quadratic spaces (possibly
degenerate) and O(V ) is the orthogonal group.
The object ofFiso which corresponds, by this equivalence, to the module F2[O(V )]
is the isotropic functor isoV , defined in [?]. Recall that, as a vector space, isoV (W )
is isomorphic to the subspace of QV (W ) generated by the elements [V
←− V →W ].
A straightforward consequence of the classification of simple objects of Fiso given
in Theorem 1.5 is given in the following corollary. Recall that, by definition, an
object F of Fquad is finite if it has a finite composition series with simple subquo-
tients.
Corollary 1.6. The isotropic functors κ(isoV ) are finite in the category Fquad.
In section 3, we will require the composition series for the isotropic functors asso-
ciated to some small quadratic spaces. For α ∈ {0, 1}, let (x, α) be the degenerate
quadratic space of dimension one generated by x such that q(x) = α. Since the or-
thogonal groups O(x, 0) and O(x, 1) are trivial and O(H0) ≃ S2 and O(H1) ≃ S3,
we deduce from Theorem 1.5 and 1.4, the following corollary.
Corollary 1.7. (1) The functors κ(iso(x,0)) and κ(iso(x,1)) are simple in Fquad.
(2) The functor κ(isoH0) is indecomposable. We have the following non-split
short exact sequence:
0→ RH0 → κ(isoH0)→ RH0 → 0
where RH0 is the functor obtained from the trivial representation of O(H0).
(3) The functor κ(isoH1) admits the following decomposition:
κ(isoH1) = FH1 ⊕ (SH1)
where SH1 is the functor obtained from the natural representation of O(H1)
and FH1 is an indecomposable functor for which we have the following non-
split short exact sequence:
0→ RH1 → FH1 → RH1 → 0
where RH1 is the functor obtained from the trivial representation of O(H1).
In [?], we define a new family of functors of Fquad, named the mixed functors
and we decompose two particular functors of this family: the functors Mix0,1 and
Mix1,1. We recall the following description of these functors.
Proposition 1.8. [?] For α ∈ {0, 1}, the functors Mixα,1 : Tq → E are defined by
Mixα,1(V ) = F2[SV ]
where SV = {(v1, v2) |v1 ∈ V, v2 ∈ V, q(v1 + v2) = α, B(v1, v2) = 1} and
Mixα,1([V
−→W⊥L
←−−W ])[(v1, v2)] =
[(pW ◦ f(v1), pW ◦ f(v2))] if f(v1 + v2) ∈W
0 otherwise
where pW is the orthogonal projection.
6 CHRISTINE VESPA
In [?], for a positive integer n, we defined subfunctors Lnα of ι(Λ
n)⊗ κ(iso(x,α)),
where Λn is the nth exterior power and we proved that these functors are simple.
The functor L1α is equivalent to the functor κ(iso(x,α)). We obtain the following
result.
Theorem 1.9. [?] Let α be an element in {0, 1}.
(1) The functor Mixα,1 is infinite.
(2) There exists a subfunctor Σα,1 of Mixα,1 such that we have the following
short exact sequence
0→ Σα,1 → Mixα,1 → Σα,1 → 0.
(3) The functor Σα,1 is uniserial with unique composition series given by the
decreasing filtration given by the subfunctors kdΣα,1 of Σα,1:
. . . ⊂ kdΣα,1 ⊂ . . . ⊂ k1Σα,1 ⊂ k0Σα,1 = Σα,1.
(a) The head of Σα,1 (i.e. Σα,1/k1Σα,1) is isomorphic to the functor
κ(iso(x,α)) where iso(x,α) is a simple object in Fiso.
(b) For d > 0
kdΣα,1/kd+1Σα,1 ≃ L
where Ld+1α is a simple object of the category Fquad that is neither in
the image of ι nor in the image of κ.
The functor Ld+1α is a subfunctor of ι(Λ
d+1)⊗κ(iso(x,α)), where Λ
is the (d+ 1)st exterior power functor.
2. Filtration of the standard projective functors PV of Fquad
In this section, we define a filtration of the standard projective functors PV of
Fquad. This construction gives rise to an essential tool to obtain, in section 3,
the direct decompositions of the projective objects PH0 and PH1 of Fquad, into
indecomposable summands.
After defining this filtration, we will deduce general results about the projective
PV of Fquad. In Theorem 2.6 we prove that the rank zero part is a direct summand
of PV and we identify this functor. This result allows us to prove that ι(F) is a
thick subcategory of Fquad. We will also show that the top quotient of this filtration
is isomorphic to κ(isoV ), where isoV is the isotropic functor.
2.1. Definition of the filtration. We recall that a morphism in Tq from V to W ,
where V and W are nondegenerate quadratic spaces, is represented by a diagram
V → X ←W.
Definition 2.1. A morphism [V → X ← W ] in Tq has rank equal to i if the
pullback in Edegq of the diagram V → X ←W is a quadratic space of dimension i.
Notation 2.2. We denote by Hom
(V,W ) the subset of HomTq (V,W ) of mor-
phisms of rank less than or equal to i.
We have the following proposition:
Proposition 2.3. For W an object in Tq, the following subvector space of PV (W ):
V (W ) = F2[Hom
(V,W )]
defines a subfunctor P
V of PV .
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 7
Proof. It is sufficient to verify that for all morphisms f = [W → Y ← Z] of Tq
and g = [V → X ← W ] of Hom
(V,W ), the composition f ◦ g has rank less than
or equal to i. The composition f ◦ g is represented by the following commutative
diagram:
P ′ //
V // X // X⊥W
where P and P ′ are the pullbacks and X⊥
Y is the pseudo push-out defined in [?].
Consequently: f ◦ g = [V → X⊥
Y ← Z]. Since [V → X ← W ] is an element of
V (W ), we know that the dimension of P is less than or equal to i. We deduce
from the injectivity of the morphisms of Edegq , that P
′ has dimension smaller than
or equal to i. �
The following lemma is a straightforward consequence of Definition 2.1.
Lemma 2.4. There exists a natural equivalence: P
(dim(V ))
V ≃ PV .
We deduce the following proposition.
Proposition 2.5. The functors P
V , for i = 0, . . . , dim(V ), define an increasing
filtration of the functor PV .
Proof. The inclusion of vector spaces P
V (W ) ⊂ P
(i+1)
V (W ) is clear, forW an object
in Tq. Consequently, P
V is a subfunctor of P
(i+1)
V by the proposition 2.3. �
2.2. Extremities of the filtration. In the previous section, we have obtained,
for all objects V of Tq, the following filtration of the functor PV :
0 ⊂ P
V ⊂ P
V ⊂ . . . ⊂ P
(dim(V )−1)
V ⊂ P
(dim(V ))
V = PV .
The aim of this section is to study the two extremities of this filtration, namely,
the functor P
V and the quotient PV /P
(dim(V )−1)
2.2.1. The functor P
V . For V an object in Tq, we recall that the functor P
ǫ(V ) of
F defined by PFǫ(V )(−) = F2[HomEf (ǫ(V ),−)], where ǫ : Tq → E
f is the forgetful
functor of Fquad, is projective, by the Yoneda lemma. The aim of this paragraph
is to prove the following theorem:
Theorem 2.6. Let V be an object of Tq.
(1) There is a natural equivalence: P
V ≃ ι(P
ǫ(V )
), where ι : F → Fquad is the
functor given in Theorem 1.3.
(2) The functor P
V is a direct summand of PV .
Before proving this result, we give the following useful characterization of the
morphisms of rank zero, which is a straightforward consequence of the definition of
the rank of a morphism.
8 CHRISTINE VESPA
Lemma 2.7. Let V be an object in Tq. A morphism T = [V
−→ W⊥W ′
←−− W ]
has rank zero if and only if pW ′ ◦ α is an injective linear map, where pW ′ is the
orthogonal projection from W⊥W ′ to W ′.
For V and W objects in Tq, the forgetful functor ǫ : Tq → E
f gives rise to a map
HomTq(V,W )→ HomEf (ǫ(V ), ǫ(W )). By passage to the vector spaces freely gener-
ated by these sets and by functoriality of ǫ, we deduce the existence of a morphism
from PV to ι(P
ǫ(V )
). As the functors P
V are subfunctors of PV , we obtain a mor-
phism f from P
V to ι(P
ǫ(V )
). Consequently, to prove Theorem 2.6, it is sufficient
to prove the following proposition.
Proposition 2.8. The map P
V (W )
−−→ PFǫ(V )(ǫ(W )) is an isomorphism for V
and W objects in Tq.
The surjectivity of fW relies on the following lemma, which is an improved
version of the fullness of the forgetful functor ǫ given in [?].
Lemma 2.9. Let (V, qV ) and (W, qW ) be two objects of Tq and
f ∈ HomEf (ǫ(V, qV ), ǫ(W, qW )) a linear map, then there exists a morphism
T = [V
−→W⊥Y
←−−W ] of rank zero such that ǫ(T ) = f .
Proof. As the quadratic space V is nondegenerate, we know that it has even di-
mension. We write dim(V ) = 2n. We prove the result by induction on n.
To start the induction, let (V, qV ) be a nondegenerate quadratic space of di-
mension two, with symplectic basis {a, b} and f : V → W be a linear map. The
following linear map preserves the quadratic form:
g1 : V → W⊥H1⊥H0⊥H0 ≃W⊥Span(a1, b1)⊥Span(a0, b0)⊥Span(a
a 7−→ f(a) + (q(a) + q(f(a)))a1 + a0
b 7−→ f(b) + (q(b) + q(f(a)))a1 + (1 +B(f(a), f(b)))b0 + a
Consequently, the morphism: T = V
−→W⊥H1⊥H0 ←֓ W , is a morphism of rank
zero of Tq such that ǫ(T ) = f .
Let Vn be a nondegenerate quadratic space of dimension 2n, {a1, b1, . . . , an, bn}
be a symplectic basis of Vn and fn : Vn →W be a linear map. By induction, there
exists a map:
gn : Vn → W⊥Y
ai 7−→ fn(ai) + yi
bi 7−→ fn(bi) + zi
where yi and zi, for all integers i between 1 and n, are elements of Y . The map
gn preserves the quadratic form and the morphism T = [Vn
−→W⊥Y ←֓ W ] is of
rank zero and verifies ǫ([Vn
−→W⊥Y ←֓ W ]) = fn.
Let Vn+1 be a nondegenerate quadratic space of dimension 2(n + 1),
{a1, b1, . . . , an, bn, an+1, bn+1} a symplectic basis of Vn+1 and fn+1 : Vn+1 → W
a linear map. To define the map gn+1, we will consider the restriction of fn+1 to
Vn and extend the map gn given by the inductive assumption. For that, we need
the following space: E ≃ W⊥W ′⊥H⊥n0 ⊥H
0 ⊥H1⊥H0⊥H0 for which we specify
the notations for a basis:
E ≃W⊥W ′⊥(⊥ni=1Span(a
0))⊥(⊥
i=1Span(A
0))⊥Span(A1, B1)
⊥Span(C0, D0)⊥Span(E0, F0).
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 9
The following map:
gn+1 : V → W⊥W
′⊥H⊥n0 ⊥H
0 ⊥H1⊥H0⊥H0
ai 7−→ fn+1(ai) + yi + a
0 for i between 1 and n
bi 7−→ fn+1(bi) + zi +A
an+1 7−→ fn+1(an+1) + (q(an+1) + q(fn+1(an+1)))A1 + C0
i=1 B(fn+1(ai), fn+1(an+1))b
i=1 B(fn+1(bi), fn+1(an+1))B
bn+1 7−→ fn+1(bn+1) + (q(bn+1) + q(fn+1(bn+1)))A1
+(1 +B(fn+1(an+1), fn+1(bn+1)))D0
i=1 B(fn+1(ai), fn+1(bn+1))b
i=1 B(fn+1(bi), fn+1(bn+1))B
0 + E0
preserves the quadratic form. Furthermore, the morphism
T = [Vn+1
−−−→W⊥W ′⊥H⊥n0 ⊥H
0 ⊥H1⊥H0⊥H0 ←֓ W ]
is of rank zero and satisfies: ǫ(T ) = fn+1, which completes the inductive step.
The proof of the injectivity of fW relies on the following result, which can be
regarded as Witt’s theorem for degenerate quadratic forms.
Theorem 2.10. Let V be a nondegenerate quadratic space, D and D′ subquadratic
spaces (possibly degenerate) of V and f : D → D′ an isometry between these two
quadratic spaces. Then, there exists an isometry f : V → V such that the following
diagram is commutative:
f // V
// D′.
Proof. For a proof of this result, we refer the reader to [?] §4, theorem 1. �
Proof of the injectivity of fW . The natural map f is induced by the natural map
(V,−)→ HomEf (ǫ(V ), ǫ(−))
by passage to the vector spaces freely generated by these sets. So, f is injective if
and only if this natural map is injective. Consequently, it is sufficient to verify that,
for T = [V
−→ W⊥W ′
←−− W ] and T ′ = [V
−→ W⊥W ′′
←−−W ] two generators of
V (W ) such that
(2.10.1) pW ◦ O(α) = p
W ◦ O(α
we have T = T ′.
Let {a1, b1, . . . , an, bn} be a symplectic basis of V . We deduce from 2.10.1 that,
for all i ∈ {1, . . . , n}, we have:
(2.10.2) α(ai) = wi + w
i, α(bi) = xi + x
(2.10.3) α′(ai) = wi + w
i , α
′(bi) = xi + x
10 CHRISTINE VESPA
where, for all i ∈ {1, . . . , n}, wi and xi are in W , w
i and x
i are in W
′ and
x′′i and w
i are in W
′′. By Lemma 2.7, since the morphisms are of rank zero,
{w′1, x
1, . . . , w
n} and {w
1 , x
1 , . . . , w
n} are two linearly independent families
of vectors.
We will denote by W ′ = Span(w′1, x
1, . . . , w
n) (respectively
W ′′ = Span(w′′1 , x
1 , . . . , w
n)) the subquadratic space (possibly degenerate), of
W ′ (respectively W ′′) and we define the linear map f : W ′ → W ′′ by f(w′i) = w
and f(x′i) = x
i for all i ∈ {1, . . . , n}.
Since α and α′ preserve the quadratic forms, we deduce from the relations 2.10.2
and 2.10.3 that f preserves the quadratic form. Hence, we can apply Theorem
2.10 to the nondegenerate space W ′⊥W ′′, which gives a morphism f : W ′⊥W ′′ →
W ′⊥W ′′ of Eq, such that, the restriction of this morphism to W
′ coincides with f .
We deduce the commutativity of the following diagram:
α̃ //
α̃′ ++WWW
W W⊥(W ′⊥W ′′)
W⊥(W ′⊥W ′′)
where α̃ = iW⊥W ′ ◦ α and α̃′ = iW⊥W ′′ ◦ α
′. Consequently, we obtain the equality
T = T ′ since, by inclusion, we have
T = [V
−→W⊥W ′
←−−W ] = [V
−→W⊥W ′⊥W ′′
←−−W ]
T ′ = [V
−→W⊥W ′′
←−−W ] = [V
−→ W⊥W ′⊥W ′′
←−−W ].
Notation 2.11. For V and W two objects of Eq, and f a morphism of HomEf (ǫ(V ), ǫ(W )),
we denote by tf the morphism of Hom
(V,W ) corresponding to f and by [tf ] the
canonical generator of P
V (W ) obtained from tf . To simplify the notation, we will
denote the morphism tIdV of Hom
(V, V ) by eV .
We deduce from the first point of Theorem 2.6 the following corollary.
Corollary 2.12. For V , W and X objects of Eq, f : ǫ(W )→ ǫ(X) and g : ǫ(V )→
ǫ(W ) morphisms of Ef , we have:
tf ◦ tg = tf◦g
where tf , tg and tf◦g are respectively the morphisms of Hom
(W,X), Hom
(V,W )
and Hom
(V,X) associated to the linear maps f, g and f ◦ g.
We can apply this result to the idempotents of the ring of endomorphisms
End(PV ), to obtain the following proposition.
Proposition 2.13. The canonical generator [eV ] of P
V is an idempotent of the
ring of endomorphisms End(PV ) such that PV .[eV ] ≃ P
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 11
Proof. The canonical generator [eV ] is an idempotent of End(PV ) by Corollary 2.12.
By definition of the rank filtration PV .[eV ] ⊂ P
V and, for a canonical generator
[tf ] of P
V , we have [tf ] = [tf ] · [eV ]. �
The idempotent [eV ] plays a central rôle in the proof of the thickness of the
subcategory ι(F) in Fquad, which is the subject of the following paragraph. For
that, the following result is necessary.
Lemma 2.14. Let V and W be objects of Tq, the functor ι induces an isomorphism:
HomF (P
ǫ(V ), P
ǫ(W ))
−→ HomFquad(P
V , P
where ǫ : Tq → E is the forgetful functor.
Proof. By Proposition 2.13 and Theorem 2.6 we have the following equivalences:
HomFquad(P
V , P
W ) ≃ P
W (eV )P
W (V ) ≃ P
W (V )
≃ ι(PFǫ(W ))(V ) ≃ P
ǫ(W )(ǫ(V )) ≃ HomF (P
ǫ(V ), P
ǫ(W )).
To conclude this paragraph, we give the following property of eV which will be
useful in section 4 concerning the polynomial functors of Fquad.
Lemma 2.15. For V and W two objects of Tq, we have: eV⊥W = eV⊥eW , where
⊥ : Tq × Tq → Tq is the functor induced by the orthogonal sum.
Proof. This is a straightforward consequence of Proposition 2.8. �
2.2.2. The category ι(F) is a thick subcategory of Fquad. The aim of this paragraph
is to prove the following result.
Theorem 2.16. The category ι(F) is a thick subcategory of Fquad, where
ι : F → Fquad is the functor defined in Theorem 1.3.
To prove this theorem, we need the following general result about the precom-
position functor which is proved in the Appendix of [?].
Proposition 2.17. Let C and D be two small categories, A be an abelian category,
F : C → D be a functor and −◦F : Func(D,A)→ Func(C,A) be the precomposition
functor, where Func(C,A) is the category of functors from C to A. If F is full and
essentially surjective, then any subobject (respectively quotient) of an object in the
image of the precomposition functor is isomorphic to an object in the image of the
precomposition functor.
Proof of Theorem 2.16. • The subcategory ι(F) of Fquad is full by Theorem
• Let FF be an object in F and G a subobject of ι(FF ). Let F ′ be the
category of functors from Ef−(even) to E , where Ef−(even) is the full sub-
category of Ef having as objects the F2-vector spaces of even dimension.
The categories F and F ′ are equivalent [?]. The functor ǫ : Eq → E
factorizes through the inclusion Ef−(even) →֒ Ef . This induces a functor
ǫ′ : Eq → E
f−(even) which is full and essentially surjective. Consequently,
we can use Proposition 2.17 to obtain: G ≃ ι(GF ). Similarly, we obtain
the result for the quotient.
12 CHRISTINE VESPA
• Let GF and HF be objects of F , we set G = ι(GF ) and H = ι(HF ). For a
short exact sequence: 0 → G → F → H → 0, we have to prove that there
exists a functor FF in F such that F = ι(FF ).
Let P1 → P0 → G
F → 0 and Q1 → Q0 → H
F → 0 be projective
presentations of GF and HF in F , we have the following commutative
diagram
0 // ι(P1) //
ι(P1)⊕ ι(Q1) //
ι(Q1) //
0 // ι(P0) //
ι(P0)⊕ ι(Q0) //
ι(Q0) //
0 // G
0 0 0
where the columns are projective resolutions in Fquad, by the horseshoe
lemma. By Lemma 2.14, the morphism ι(P1)⊕ ι(Q1)→ ι(P0)⊕ ι(Q0) is in-
duced by a morphism of F denoted by f . Consequently,
F ∼= ι(Coker(f)) ∈ ι(F).
By Theorem 2.16, we deduce from Lemma 2.14, the following characterization
of the simple functors of F in Fquad which will be used in section 4 of this paper
concerning the polynomial functors of Fquad.
Lemma 2.18. (1) Let F be a functor of Fquad, then F is in the image of the
functor ι : F → Fquad if and only if, for all objects V in Tq,
F (eV )F (V ) = F (V ).
(2) Let S be a simple object in Fquad, then S is in the image of the functor
ι : F → Fquad if and only if there exists an object W in Tq such that
S(eW )S(W ) 6= 0.
Proof. (1) The forward implication is a consequence of the following fact: for a
functor F in the image of ι, HomFquad(P
V , F ) = F (eV )F (V ) = F (V ). The
reverse implication relies on the fact that the condition
F (eV )F (V ) = F (V ) implies that F is a quotient of a sum of projective
objects of the form P
V . Since the category ι(F) is thick in Fquad by
Theorem 2.16, we obtain the result.
(2) Observe that, if S(eW )S(W ) 6= 0, we have HomFquad(P
W , S) 6= 0, thus
S is a quotient of P
W by simplicity of S. Lemma 2.14 implies that there
exists a one-to-one correspondance between the indecomposable factors of
V and those of P
ǫ(V ). We deduce that the simple quotients of P
V arise
from F . Consequently, S is in the image of the functor ι.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 13
2.2.3. The quotient PV /P
(dim(V )−1)
V . The aim of this paragraph is to prove the
following result:
Proposition 2.19. Let V be an object in Tq, we have a natural equivalence:
PV /P
(dim(V )−1)
V ≃ κ(isoV )
where isoV is an isotropic functor and κ : Fiso → Fquad is the functor given in
Theorem 1.4.
To prove this proposition, we need the following notation and result:
Notation 2.20. Denote by σf the natural map PV
−→ κ(isoV ) which corre-
sponds to the canonical generator [V
−→ V ] of isoV (V ) by the equivalence
Hom(PV , κ(isoV )) ≃ isoV (V ) ≃ F2[O(V )] given by the Yoneda lemma.
Lemma 2.21. The functor κ(isoV ) of Fquad is a quotient of the functor PV =
F2[HomTq(V,−)].
Proof. The natural map PV
−→ κ(isoV ) is surjective: a pre-image of the canonical
generator [V
−→ W ] of κ(isoV )(W ) by (σf )W , is the morphism
g◦f−1
−−−−→W
←−W ]. �
A formal consequence of the previous lemma is given in the following result.
Lemma 2.22. The functor κ(isoV ) of Fquad is a quotient of the functor PV /P
(dim(V )−1)
Proof. By definition of the filtration and by the previous lemma, we have the dia-
gram:
0 // P
(dim(V )−1)
i // PV
// // PV /P
(dim(V )−1)
κ(isoV )
where i is the canonical inclusion of P
(dim(V )−1)
V in PV . By definition of σId,
we have σId ◦ i = 0, from which we deduce the existence of the surjection τ :
PV /P
(dim(V )−1)
V → κ(isoV ). �
We will prove below that this natural map is an isomorphism. It is sufficient to
prove the following result.
Proposition 2.23. For V and W two objects of Tq, we have an isomorphism
(PV /P
(dim(V )−1)
V )(W ) ≃ κ(isoV )(W ).
The proof of this proposition relies on the following lemma.
Lemma 2.24. For a non-zero canonical generator of (PV /P
(dim(V )−1)
V )(W ) repre-
sented by the morphism T = [V
−→ W⊥W ′
←−− W ] of Tq, we have g(V ) ⊂ W and
T = [V
←−W ], where g = iW ◦ f .
14 CHRISTINE VESPA
Proof. By definition of the filtration, for V andW two objects of Tq, the vector space
(PV /P
(dim(V )−1)
V )(W ) is generated by Hom
[dim(V )]
(V,W ) where Hom
[dim(V )]
(V,W )
is the set of morphisms from V to W whose the pullback D in Edegq is a quadratic
space such that dim(D) = dim(V ). We deduce from the existence of a monomor-
phism from D to V and from the equality of the dimensions, that D and V are
isometric. Consequently, for the morphism T of the statement, we have, by defi-
nition of the pullback, g(V ) ⊂ W . Thus, we have T = [V
←− W ], where
g = iW ◦ f , by the equivalence relation defined over the morphisms of Tq in Defini-
tion 1.1.
Proof of Proposition 2.23. The natural map τ obtained in the proof of Lemma 2.22
defines, for W an object in Tq, the linear map
τW : (PV /P
(dim(V )−1)
V )(W ) → κ(isoV )(W )
T = [V
←−W ] 7−→ [V
−→ W ]
which is clearly an isomorphism. �
3. Decomposition of the standard projective functors PH0 and PH1
On abelian categories, the decompositions into direct summands of a functor
F of Fquad correspond to decompositions into orthogonal idempotents of 1 in the
ring EndFquad(F ) (see for example [?]). One of the difficulties of the category
Fquad lies in the fact that the rings of endomorphisms of projectives PV and their
representations are not well-understood. The decompositions of projectives PV ,
obtained in work in preparation, using a refinement of the rank filtration will allow
us to understand the structure of these rings better.
In this section, we obtain the decompositions into indecomposable factors of the
projective objects PH0 and PH1 by an explicit study of the filtration defined in
section 2. This section concludes by several consequences of these decompositions.
In particular, we give a classification of the “small” simple functors of Fquad, which
is an essential ingredient in the following section about the polynomial functors of
Fquad.
3.1. Decomposition of PH0 . To obtain the decomposition of the functor PH0 into
indecomposable factors, we give an explicit description of the subquotients of the
filtration; then, we prove that the filtration splits for this functor and we identify
the factors of this decomposition.
3.1.1. Explicit description of the subquotients of the filtration. The aim of this para-
graph is to give a basis of the vector spaces P
(V ), P
(V ) and PH0/P
for V a given object in Tq.
We deduce from Theorem 2.6 and Notation 2.11, the following result.
Lemma 3.1. A basis B
(V ) is given by the set:
= {[tf ] for f ∈ HomEf (F2
⊕2, ǫ(V ))}.
By definition of the filtration, a canonical generator of P
(V ), represented
by the morphism T = [H0
−→ V⊥L
←− V ] of Tq, satisfies the following property:
I = f(H0) ∩ i(V ) is a quadratic space of dimension one.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 15
Lemma 3.2. Let T = [H0
−→ V⊥L
←− V ] be a morphism of Tq which represents a
canonical generator of P
(V ), and {a0, b0} be a symplectic basis of H0, then
the map f in T has one of the three following forms.
(1) If I = (f(a0), 0), the map f : H0 → V⊥L is defined by:
f(a0) = v and f(b0) = w + l
for v and w elements of V satisfying q(v) = 0 and B(v, w) = 1 and l a
non-zero element of L.
(2) If I = (f(b0), 0), the map f : H0 → V⊥L is defined by:
f(a0) = v + l and f(b0) = w
for v and w elements of V satisfying q(w) = 0 and B(v, w) = 1 and l a
non-zero element of L.
(3) If I = (f(a0 + b0), 1), the map f : H0 → V⊥L is defined by:
f(a0) = v + l and f(b0) = w + l
for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1 and l
a non-zero element of L.
Proof. The quadratic space H0 has three subspaces of dimension one which are:
Span(a0) and Span(b0) isometric to (x, 0) and Span(a0 + b0) isometric to (x, 1).
These three subspaces give rise to each one of the maps f defined in the statement.
Notation 3.3. The morphisms [H0
−→ V⊥L
←− V ], where f is one of the mor-
phisms described in the point (1) (respectively (2) and (3)) of the previous lemma,
will be known as type A (respectively B and C) morphisms.
We have the following proposition.
Proposition 3.4. For T = [H0
−→ V⊥L
←− V ] and T ′ = [H0
−→ V⊥L′
←− V ]
morphisms of Tq which represent canonical generators of P
(V ), the follow-
ing properties are equivalent.
(1) The morphisms T and T ′ of HomTq (H0, V ) have the same type and satisfy
the relation pV ◦ f = p
V ◦ f
(2) The morphisms T and T ′ of HomTq (H0, V ) are equal.
The proof of the implication (2)⇒ (1) relies on the following technical lemma.
Lemma 3.5. Let T = [V
←− W ] and T ′ = [V
−→ X ′
←− W ′] be morphisms
of HomTq (V,W ). If T = T
′, then g(V ) + h(W ) ≃ g′(V ) + h′(W ) in Edegq .
Proof. By definition of the equivalence relation given in Definition 1.1, it is sufficient
to prove that, for two morphisms T and T ′ such that TRT ′, we have g(V )+h(W ) ≃
g′(V ) + h′(W ).
16 CHRISTINE VESPA
By definition, g(V )+h(W ) is the smallest, possibly degenerate, quadratic space
such that we have a commutative diagram in Edegq , of the form:
**UUU
U g(V ) + h(W )
Similarly, g′(V ) + h′(W ) is the smallest quadratic space such that we have an
analogous commutative diagram. By definition of the relation R, we have the
existence of a morphism δ in Eq such that the following diagram is commutative:
By the consideration of the following commutative diagram in Edegq
where Y = g(V ) + h(W ), we deduce from the minimality of g′(V ) + h′(W ), the
existence of a morphism in Edegq from g
′(V ) + h′(W ) to g(V ) + h(W ) such that the
corresponding diagram is commutative. Then, by minimality of g(V ) + h(W ) for
T , we obtain: g(V ) + h(W ) ≃ g′(V ) + h′(W ). �
Proof of Proposition 3.4. Suppose that the morphisms T and T ′ of HomTq (H0, V )
are of type A such that pV ◦ f = p
V ◦ f
′. We deduce that:
f(a0) = v; f(b0) = w + l and f
′(a0) = v; f
′(b0) = w + l
for v and w elements of V and l (resp. l′) a non-zero element of L (resp. L′).
Since the maps f and f ′ preserve the quadratic forms, we have
q(b0) = q(w) + q(l) = q(w) + q(l
′). We deduce that q(l) = q(l′), thus the map
Span(l)
−→ Span(l′), such that α(l) = l′, preserves the quadratic form. Conse-
quently, we can apply Theorem 2.10 to obtain the existence of a map α : L⊥L′ →
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 17
L⊥L′ such that the following diagram is commutative:
α // L⊥L′
Span(l)
// Span(l′).
We deduce the commutativity of the diagram:
V � _
f ′ **VVV
V⊥(L⊥L′)
V⊥(L⊥L′).
Since T = [H0
−→ V⊥L⊥L′
←− V ] and T ′ = [H0
−→ V⊥L⊥L′
←− V ], by inclusion,
we deduce from the previous diagram that T = T ′.
We reason in the same way, for the morphisms of type B and C.
Conversely, if T = T ′, by Lemma 3.5 f(H0) + iV (V ) ≃ f
′(H0) + i
V (V ). Conse-
quently we deduce from Theorem 2.10 the existence of an isometry β : V⊥L⊥L′→
V⊥L⊥L′ making the following diagram commutative:
V⊥L⊥L′
β // V⊥L⊥L′
f(H0) + iV (V )
// f ′(H0) + i
V (V ).
This yields the commutativity of the following diagram:
f ′ **UUU
U V⊥L⊥L′
V⊥L⊥L′
which implies that β ◦ iV = i
V . Thus, β = IdV⊥β
′ where β′ : L⊥L′ → L⊥L′ is a
morphism of HomEq (L⊥L
′, L⊥L′). Consequently, we have
f(a0) = v + l; f(b0) = w + l
′ and f ′(a0) = v + β
′(l); f ′(b0) = w + β
′(l′)
and we deduce that pV ◦ f = p
V . Furthermore, since β
′ is inversible, for all x in
L⊥L′ we have: x is non-zero if and only if β′(x) is non-zero. Consequently, T and
T ′ have the same type. �
This proposition justifies the following notation.
Notation 3.6. We will denote by Av,w, Bv,w and Cv,w the morphisms of HomTq (H0, V )
respectively of type A, B and C and such that pV ◦ f(a0) = v and pV ◦ f(b0) = w.
18 CHRISTINE VESPA
The following result is a straightforward consequence of Lemma 3.2 and Propo-
sition 3.4.
Lemma 3.7. A basis B
(V ) is given by the set:
= { [Av,w] for v and w elements of V satisfying q(v) = 0 and B(v, w) = 1,
[Bv,w] for v and w elements of V satisfying q(w) = 0 and B(v, w) = 1,
[Cv,w] for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1}
By Proposition 2.19, we have (PH0/P
)(V ) ≃ κ(isoH0)(V ). We deduce the
following result.
Lemma 3.8. A basis B
of PH0/P
(V ) is given by the set:
= {[Df ] for f ∈ HomEq (H0, V )},
where Df is the morphism of Tq represented by the diagram: H0
←− V .
We end this paragraph by the rules of composition for the morphisms tf , Av,w,
Bv,w, Cv,w and Df , summarized in the following proposition. This technical re-
sult will be fundamental in the following paragraph, to prove the splitting of the
filtration.
Lemma 3.9. Let T = V
−→ W⊥L
←−− W be a morphism of HomTq(V,W ). The
following relations are satisfied:
(1) For f a morphism of HomEf (F2
⊕2, ǫ(V )) we have:
T ◦ tf = tϕ◦f .
(2) (a) For v and w elements of V satisfying q(v) = 0 and B(v, w) = 1, we
have:
T ◦Av,w =
Aϕ(v),pW ◦ϕ(w) if ϕ(v) ∈W
tpW ◦(ϕ⊥Id)◦α otherwise.
(b) For v and w elements of V satisfying q(w) = 0 and B(v, w) = 1, we
have:
T ◦Bv,w =
BpW ◦ϕ(v),ϕ(w) if ϕ(w) ∈W
tpW ◦(ϕ⊥Id)◦α otherwise.
(c) For v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1,
we have:
T ◦ Cv,w =
CpW ◦ϕ(v),pW ◦ϕ(w) if ϕ(v + w) ∈W
tpW ◦(ϕ⊥Id)◦α otherwise.
(3) For f a morphism of HomEq (H0, V ), we have:
T ◦Df =
Dϕ◦f if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈ W
Aϕ◦f(a0),pW ◦ϕ◦f(b0) if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W
BpW ◦ϕ◦f(a0),ϕ◦f(b0) if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W
CpW ◦ϕ◦f(a0),pW ◦ϕ◦f(b0) if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W
and ϕ ◦ f(a0 + b0) ∈W
tpW ◦(ϕ⊥Id)◦α if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W
and ϕ ◦ f(a0 + b0) /∈W.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 19
Proof. By definition of the composition in Tq, we have the following diagram:
α // V⊥L′
// W⊥L⊥L′
(1) For a morphism tf , we have α(a0) = f(a0) + l and α(b0) = f(b0) + m.
where {l,m} is a linearly independent family of L′. Consequently:
(ϕ⊥Id) ◦ α(a0) = ϕ ◦ f(a0) + l et (ϕ⊥Id) ◦ α(b0) = ϕ ◦ f(b0) +m.
We deduce that T ◦ tf = tϕ◦f .
(2) For Av,w, we have α(a0) = v and α(b0) = w + l
′, where l′ is a non-zero
element of L′. Consequently:
(ϕ⊥Id) ◦ α(a0) = ϕ(v) et (ϕ⊥Id) ◦ α(b0) = ϕ(w) + l
We have to distinguish two cases:
• if ϕ(v) ∈W , since ϕ preserves quadratic forms, we have q(ϕ(v)) = q(v)
and, since L′ is orthogonal to V , B(ϕ(v), pW ◦ϕ(w)) = B(ϕ(v), ϕ(w)) =
B(v, w). Thus the morphism Aϕ(v),pW ◦ϕ(w) is defined and we have:
T ◦Av,w = Aϕ(v),pW ◦ϕ(w);
• otherwise, ϕ(v) = pW ◦ ϕ(v) + m where m is a non-zero element of
L. Consequently, we obtain a morphism of nul rank and we have:
T ◦Av,w = tpW ◦(ϕ⊥Id)◦α.
The cases Bv,w and Cv,w are similar to the case of Av,w and are left to
the reader.
(3) For the morphism Df , where f is an element of HomEq (H0, V ), we have
α(a0) = f(a0) = v and α(b0) = f(b0) = w where v and w are elements of V .
Consequently: (ϕ⊥Id)◦α(a0) = ϕ(v) and (ϕ⊥Id)◦α(b0) = ϕ(w). Since ϕ◦f
preserves the quadratic forms, we have: q(ϕ ◦ f(a0)) = q(ϕ ◦ f(b0)) = 0
and B(ϕ ◦ f(a0), ϕ ◦ f(b0)) = 1. Thus the morphisms Aϕ◦f(a0),ϕ◦f(b0),
Bϕ◦f(a0),ϕ◦f(b0) and Cϕ◦f(a0),ϕ◦f(b0) are defined.
We have to distinguish four cases:
• if ϕ(v) ∈W and ϕ(w) ∈W then T ◦Df = Dϕ◦f ;
• if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W , we have ϕ ◦ f(a0) = w
′ and
ϕ ◦ f(b0) = w
′′ + l, where l is a non-zero element of L. Consequently,
we obtain a morphism of type A and we have T ◦Df = Aϕ◦f(a0),ϕ◦f(b0);
• if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W , we have ϕ ◦ f(a0) = w
′ + l and
ϕ ◦ f(b0) = w
′′, where l is a non-zero element of L. Consequently, we
obtain a morphism of type B and we have T ◦Df = Bϕ◦f(a0),ϕ◦f(b0);
• if ϕ ◦ f(a0) /∈ W , ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) ∈ W , we have
ϕ◦f(a0) = w
′+ l and ϕ◦f(b0) = w
′′+ l, where l is a non-zero element
of L. Consequently, we obtain a morphism of type C and we have
T ◦Df = Cϕ◦f(a0),ϕ◦f(b0);
• if ϕ ◦ f(a0) /∈ W , ϕ ◦ f(b0) /∈ W and ϕ ◦ f(a0 + b0) /∈ W , we have
ϕ ◦ f(a0) = w
′ + l and ϕ ◦ f(b0) = w
′′ + l′, where l and l′ are non-zero
20 CHRISTINE VESPA
elements of L. Consequently, we obtain a morphism of nul rank and
we have T ◦Df = tpW ◦(ϕ⊥Id)◦α.
3.1.2. Splitting of the filtration for the functor PH0 . In this paragraph, we prove
the following result.
Proposition 3.10. The rank filtration splits for the functor PH0 , namely:
PH0 = P
⊕ PH0/P
Proof. By Theorem 2.6, we have: P
. To prove the proposition,
it is sufficient to prove that PH0 = P
⊕ PH0/P
By definition of the filtration, we have the short exact sequence:
(3.10.1) 0→ P
→ PH0
−→ PH0/P
Let V be an object in Tq, we consider a morphism f of HomEq (H0, V ) and
the generator [Df ] of PH0/P
(V ) associated to f . Since the map f preserves
the quadratic forms, we have: q(f(a0)) = q(f(b0)) = 0, q(f(a0 + b0)) = 1; thus
B(f(a0), f(b0)) = 1. Consequently, the morphisms Af(a0),f(b0), Bf(a0),f(b0) and
Cf(a0),f(b0) of HomTq (H0, V ) are defined. We define a map sV : PH0/P
(V ) →
PH0(V ) by:
sV : PH0/P
(V ) −→ PH0(V )
[Df ] 7→ [Df ] + [Af(a0),f(b0)] + [Bf(a0),f(b0)] + [Cf(a0),f(b0)].
We verify the two following statements.
(1) pV ◦ sV = Id.
For [Df ] a canonical generator of PH0/P
(V ), we have
pV ◦ sV ([Df ]) = [Df ]
since the morphisms Af(a0),f(b0), Bf(a0),f(b0) and Cf(a0),f(b0) have a rank
equal to one.
(2) The maps sV define a natural map.
One verifies that, for a morphism T = V
−→W⊥L
←−−W of HomTq (V,W ),
we have the commutativity of the following diagram:
PH0/P
sV //
PH0/P
PH0(V )
PH0 (T )
PH0/P
sW // PH0(W ).
In order to simplify notation, we will write: A′ = Aϕ◦f(a0),pW ◦ϕ◦f(b0),
B′ = BpW ◦ϕ◦f(a0),ϕ◦f(b0), C
′ = CpW ◦ϕ◦f(a0),pW ◦ϕ◦f(b0) and t
′ = tpW ◦(ϕ⊥Id)◦α.
On the one hand, by Lemma 3.9, we have:
PH0 (T ) ◦ sV ([Df ])
= PH0(T )([Df ] + [Af(a0),f(b0)] + [Bf(a0),f(b0)] + [Cf(a0),f(b0)])
= [T ◦Df ] + [T ◦Af(a0),f(b0)] + [T ◦Bf(a0),f(b0)] + [T ◦ Cf(a0),f(b0)]
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 21
[Dϕ◦f ] +[A
′] +[B′] +[C′] if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈ W
[A′] +[A′] +[t′] +[t′] = 0 if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) /∈ W
[B′] +[t′] +[B′] +[t′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) ∈ W
[C′] +[t′] +[t′] +[C′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W
and ϕ ◦ f(a0 + b0) ∈W
[t′] +[t′] +[t′] +[t′] = 0 if ϕ ◦ f(a0) /∈ W and ϕ ◦ f(b0) /∈ W
and ϕ ◦ f(a0 + b0) /∈W.
[Dϕ◦f ] +[A
′] +[B′] +[C′] if ϕ ◦ f(a0) ∈ W and ϕ ◦ f(b0) ∈W
0 otherwise.
On the other hand, by Lemma 3.9, we have:
PH0/P
(T )([Df ]) =
[Dϕ◦f ] if ϕ ◦ f(a0) ∈W and ϕ ◦ f(b0) ∈W
0 otherwise
since the morphisms A, B, C and t are zero in the quotient PH0/P
(W ).
We deduce,
sW ◦ PH0/P
(T )([Df ]) =
[Dϕ◦f ] + [A
′] + [B′] + [C′] if ϕ ◦ f(a0) ∈ W
and ϕ ◦ f(b0) ∈W
0 otherwise.
Consequently, the maps sV define a natural map which is a section of p. This
gives rise to the splitting of the exact sequence 3.10.1. �
3.1.3. Identification of the direct summands. The aim of this paragraph is to iden-
tify the summands of the decomposition given in Proposition 3.10. We begin by
proving that the morphisms of type A (respectively of type B and C) define a
subfunctor of P
which is a direct summand of this functor.
Lemma 3.11. The functor P
admits the following decomposition into di-
rect summands:
= FA ⊕ FB ⊕ FC
where FA, FB and FC are subfunctors of P
generated by, respectively, the
morphisms of type A, B and C.
Proof. By Lemma 3.7, we have an isomorphism of vector spaces
(V ) = FA(V )⊕ FB(V )⊕ FC(V ),
for all objects V in Tq.
Consequently, it is sufficient to prove that FA, FB and FC are subfunctors of
For FA, we have to verify the commutativity of the diagram
FA(V )
iV //
FA(T )
FA(W )
iW // P
22 CHRISTINE VESPA
where T is a morphism of HomTq(V,W ). Let [Av,w] be a canonical generator of
FA(V ), we have by Lemma 3.9
T ◦Av,w =
Aϕ(v),pW ◦ϕ(w) if ϕ(v) ∈W
tpW ◦(ϕ⊥Id)◦α otherwise.
Consequently,
(T ) ◦ iV ([Av,w]) =
[Aϕ(v),pW ◦ϕ(w)] if ϕ(v) ∈ W
0 otherwise,
since the morphism tpW ◦(ϕ⊥Id)◦α, has nul rank.
We deduce that P
(T ) ◦ iV ([Av,w ]) is in the vector space FA(W ); thus,
FA is a subfunctor of P
In the same way, by the use of values of T ◦Bv,w and T ◦ Cv,w given in Lemma
3.9, we prove that the functors FB and FC are subfunctors of P
In the following lemma, we identify the functors FA, FB and FC with certain
mixed functors defined in [?] and recalled in section 1.
Lemma 3.12. (1) The functors FA and FB are isomorphic to the functor
Mix0,1.
(2) The functor FC is isomorphic to the functor Mix1,1.
Proof. (1) The isomorphism FA ≃ Mix0,1.
Let [Av,w] be a canonical generator of FA(V ), we have, by definition,
B(v, w) = 1. Consequently, the following linear map exists:
σ1V : FA(V ) → Mix0,1(V )
[Av,w] 7−→ [(w, v + w)].
The map σ1V is an isomorphism, whose inverse is given by
(σ1V )
: Mix0,1(V ) → FA(V )
[(v, w)] 7−→ [Av+w,v].
We have to verify that the maps σ1V define a natural map; namely, for
a morphism T = [V
−→ W⊥L
←−− W ], that the following diagram is
commutative:
FA(V )
σ1V //
FA(T )
Mix0,1(V )
Mix0,1(T )
FA(W )
σ1W // Mix0,1(W ).
We have:
Mix0,1(T ) ◦ σ
V ([Av,w ]) = Mix0,1(T )[(w, v + w)]
[(pW ◦ (ϕ(w)), pW ◦ (ϕ(v + w))] if ϕ(v) ∈W
0 otherwise
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 23
by the definition of the mixed functors, and
σ1W ◦ FA(T )([Av,w]) = σ
[Aϕ(v),pW ◦ϕ(w)] if ϕ(v) ∈W
0 otherwise
[(pW ◦ (ϕ(w)), ϕ(v) + pW ◦ (ϕ(w))] if ϕ(v) ∈W
0 otherwise.
When ϕ(v) ∈ W , we have:
[(pW ◦ (ϕ(w)), ϕ(v) + pW ◦ (ϕ(w))] = [(pW ◦ (ϕ(w)), pW ◦ (ϕ(v + w))],
what proves the naturality of σ1.
Since the two following cases are very close to the previous one, we only
give the definition of the isomorphism of vector spaces and we leave the
reader to verify that they define natural equivalences.
(2) The isomorphism FB ≃ Mix1,0.
σ2V : FB(V ) → Mix0,1(V )
Bv,w 7−→ [(v, v + w)].
(3) The isomorphism FC ≃Mix1,1.
σ3V : FC(V ) → Mix1,1(V )
Cv,w 7−→ [(v, w)]
We deduce the following proposition.
Proposition 3.13. The projective functor PH0 admits the following decomposition
into direct summands:
PH0 = ι(P
⊕2))⊕ (Mix0,1
⊕2 ⊕Mix1,1)⊕ κ(isoH0)
where Mix0,1 and Mix1,1 are mixed functors and isoH0 is an isotropic functor.
Proof. This proposition is a straightforward consequence of Proposition 3.10, The-
orem 2.6, Proposition 2.19 and Lemmas 3.11 and 3.12. �
3.2. Decomposition of PH1 . The study of the functor PH1 is analogous to that
of the functor PH0 given in the previous section. Consequently, for the functor PH1 ,
we give only the principal results without proofs.
3.2.1. Explicit description of the subquotients of the filtration. In this paragraph,
we give basis of the vector spaces P
(V ), P
(V ) and PH1/P
(V ) for V
an object in Tq.
Lemma 3.14. A basis B
(V ) is given by the set:
= {tf for f ∈ HomEf (F2
⊕2, ǫ(V ))}.
Lemma 3.15. Let T = [H1
−→ V⊥L
←− V ] be a morphism of Tq which represents
a canonical generator of P
(V ), and {a1, b1} a symplectic basis of H1, then
the map f in T has one of the three following forms.
24 CHRISTINE VESPA
(1) If I = (f(a1), 0) the map f : H1 → V⊥L is defined by:
f(a1) = v et f(b1) = w + l
for v and w elements of V satisfying q(v) = 1 and B(v, w) = 1 and l a
non-zero element of L.
(2) If I = (f(b1), 0) the map f : H1 → V⊥L is defined by:
f(a1) = v + l et f(b1) = w
for v and w elements of V satisfying q(w) = 1 and B(v, w) = 1 and l a
non-zero element of L.
(3) If I = (f(a1 + b1), 1) the map f : H1 → V⊥L is defined by:
f(a1) = v + l et f(b1) = w + l
for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1 and l
a non-zero element of L.
Notation 3.16. The morphisms [H1
−→ V⊥L
←− V ], where f is one of the
morphisms described in the point (1) (respectively (2) and (3)) of the previous lemma
will be known as type E (respectively F and G) morphisms.
The analogous proposition to Proposition 3.4 holds for H1. This justifies the
following notation.
Notation 3.17. Denote by Ev,w, Fv,w and Gv,w the morphisms of HomTq (H1, V )
respectively of type E, F and G and such that pV ◦ f(a1) = v and pV ◦ f(b1) = w.
We deduce the following lemmas.
Lemma 3.18. A basis B
(V ) is given by the set:
= { [Ev,w] for v and w elements of V satisfying q(v) = 1 and B(v, w) = 1,
[Fv,w] for v and w elements of V satisfying q(w) = 1 and B(v, w) = 1,
[Gv,w] for v and w elements of V satisfying q(v + w) = 1 and B(v, w) = 1}
Lemma 3.19. A basis B
of PH1/P
(V ) is given by the set:
= {[Hf ] for f ∈ HomEq (H1, V )},
where Hf is the morphism of Tq represented by the diagram: H1
←− V .
The rules of composition of morphisms Ev,w, Fv,w , Gv,w and Hf are similar to
those given for Av,w, Bv,w, Cv,w and Df in Lemma 3.9. The details can be provided
by the reader.
3.2.2. Splitting of the filtration for the functor PH1 .
Proposition 3.20. The rank filtration splits for the functor PH1 , namely:
PH1 = P
⊕ PH1/P
Proof. One verifies that the map sV : PH1/P
(V )→ PH1(V ) given by:
sV : PH1/P
(V ) −→ PH1(V )
[Hf ] 7→ [Hf ] + [Ef(a1),f(b1)] + [Ff(a1),f(b1)] + [Gf(a1),f(b1)].
defines a natural map which is a section of the projection PH1 → PH1/P
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 25
3.2.3. Identification of the direct summands. We have the following lemma.
Lemma 3.21. The functor P
admits the following decomposition into di-
rect summands
= FE ⊕ FF ⊕ FG
where FE, FF and FG are subfunctors of P
generated by, respectively, the
morphisms of type E, F and G.
In the following lemma, we identify the functors FE , FF and FG with mixed
functor.
Lemma 3.22. The functors FE , FF and FG are equivalent to the functor Mix1,1.
We deduce the following decomposition.
Proposition 3.23. The projective functor PH1 admits the following decomposition
into direct summands:
PH1 = ι(P
)⊕Mix1,1
⊕3 ⊕ κ(isoH1)
where Mix1,1 is a mixed functor and isoH1 is an isotropic functor.
3.3. Consequences of decompositions of functors PH0 and PH1 . In this sec-
tion, we draw the conclusions of the decompositions of PH0 and PH1 given in Propo-
sitions 3.13 and 3.23. We deduce the indecomposability of the functors Mix0,1 and
Mix1,1, we study the projectivity of the first isotropic functors in Fquad and we give
the classification of the “small” simple objects of Fquad.
3.3.1. Indecomposability of functors Mix0,1 and Mix1,1. The aim of this paragraph
is to prove the following result:
Proposition 3.24. The functors Mix0,1 and Mix1,1 are indecomposable.
The proof of this proposition relies on the following obvious lemma.
Lemma 3.25. If the functor F of Fquad decomposes as a direct sum: F1⊕ . . .⊕Fn,
then the projections πi : F → Fi and the inclusions ji : Fi → F induce idempotents
ei = ji ◦ πi in the ring End(F ).
Proof of Proposition 3.24. (1) By the Yoneda lemma, we have:
Hom(PH0 ,Mix0,1) = Mix0,1(H0).
By a calculation, we obtain that the dimension of the space Mix0,1(H0) is
equal to 4. According to Proposition 3.13, the order of multiplicity of the
summand Mix0,1 in the decomposition of PH0 is equal to 2. Consequently,
the dimension of the vector space E := Hom(Mix0,1,Mix0,1) is 2. We have
the following basis: {Id, τ} where the map τ is given by: τ([(u, v)]) =
([(v, u)]). Consequently, E = ({0, Id, τ, Id + τ},+, ◦), as a ring, and it is
easy to see that this ring does not admit a non-trivial idempotent.
(2) Similarly, we have Hom(PH0 ,Mix1,1) = Mix1,1(H0) and
dim(Mix1,1(H0)) = 2. The order of multiplicity of the summand Mix1,1
in the decomposition of PH0 is equal to 1. We deduce that the ring
Hom(Mix1,1,Mix1,1) does not admit a non-trivial idempotent.
26 CHRISTINE VESPA
We deduce from this proposition the following result, which complements The-
orem 1.9, obtained in [?]:
Corollary 3.26. The short exact sequence 0 → Σα,1 → Mixα,1 → Σα,1 → 0 does
not split.
3.3.2. Projectivity of certain isotropic functors in Fquad. The decompositions given
in Propositions 3.13 and 3.23 allow us to study the projectivity of isotropic functors
in Fquad. Corollary 4.37 in [?] shows that the set of functors {isoV |V ∈ S} is a
set of projective generators of Fiso, where S is a set of representatives of isometry
classes of (possibly degenerate) quadratic spaces.
Since the functor κ(isoH0) (respectively κ(isoH1) ) is a direct summand of the
functor PH0 (respectively PH1) we have the following result.
Proposition 3.27. The functors κ(isoH0) and κ(isoH1) are projective in the cat-
egory Fquad.
We deduce from Corollary 1.6 and the previous proposition, the following result.
Corollary 3.28. The category Fquad contains non-constant finite, projective ob-
jects.
This corollary constitutes one of the new features of the category Fquad compared
to F . Recall that, according to Corollary B7 in [?], due to Lionel Schwartz, the
category F does not contain non-constant finite projective functors.
Recall that, the functor κ(iso(x,0)) is the top composition factor of Mix0,1 and
κ(iso(x,1)) is that of Mix1,1. We have the following result.
Proposition 3.29. The projective cover of κ(iso(x,0)) (respectively κ(iso(x,1))) is
the functor Mix0,1 (respectively Mix1,1). In particular, the functors κ(iso(x,0)) and
κ(iso(x,1)) are not projective in Fquad.
Proof. Since κ(iso(x,0))(H0) 6= {0} and κ(iso(x,1))(H0) 6= {0}, if these two functors
were projective, they would be direct summands of the functor PH0 . We deduce
from Proposition 3.13, that these functors are not projective. �
Remark 3.30. Propositions 3.27 and 3.29 let us conjecture that, for a nondegen-
erate F2-quadratic space H, κ(isoH) is a projective functor in Fquad and, for a
degenerate quadratic space D, κ(isoD) is not a projective functor in Fquad and its
projective cover is a generalized mixed functor. This result will be the subject of
future work.
3.3.3. Classification of simple objects S of Fquad such that either S(H0) 6= 0 or
S(H1) 6= 0. If S is a simple object in Fquad, such that S(H0) 6= 0, the Yoneda lemma
implies that Hom(PH0 , S) = S(H0) 6= 0. Consequently, there exists a morphism of
Fquad from PH0 to S which is an epimorphism, by simplicity of S. We deduce
from the decompositions given in Proposition 3.13 and 3.23, from Corollary 1.7
concerning the functors isoH0 and isoH1 and from the study of the functors Mix0,1
and Mix1,1 done in [?] and recalled in section 1, the following result.
Proposition 3.31. The isomorphism classes of non-constant simple functors of
Fquad such that either S(H0) 6= 0 or S(H1) 6= 0 are:
ι(Λ1), ι(Λ2), ι(S(2,1)), κ(iso(x,0)), κ(iso(x,1)), RH0 , RH1 , SH1
where RH0 , RH1 and SH1 are the simple functors introduced in Corollary 1.7.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 27
3.3.4. Extension groups in Fquad. By Theorem 1.4 and 1.5 we obtain an exact, fully-
faithful functor
V ∈S F2[O(V )]−mod
−→ Fquad, where S is a set of representatives
of isometry classes of quadratic spaces (possibly degenerate). Consequently, for
M and N two F2[O(V )]− modules, this functor induces a morphism of extension
groups:
F2[O(V )]−mod
(M,N)
(κ̃)∗
−−−→ Ext∗Fquad(κ̃(M), κ̃(N)).
We have the following proposition.
Proposition 3.32. For V ∈ {H0, H1}, the morphism (κ̃)∗ is an isomorphism.
The proof of this proposition relies on the following lemma.
Lemma 3.33. For V ∈ {H0, H1}, if P is a finite projective F2[O(V )]-module, κ̃(P )
is projective in Fquad.
Proof. If P is a finite projective F2[O(V )]-module, there exists a F2[O(V )]-module
Q such that P ⊕Q ≃ F2[O(V )]
⊕N . We deduce from the exactness of κ that κ̃(P ⊕
Q) ≃ κ̃(P ) ⊕ κ̃(Q). Since κ̃(F2[O(V )]) = κ(isoV ) and the functors κ(isoH0) and
κ(isoH1) are projective, by Proposition 3.27, we obtain that κ̃(P ) is projective. �
Proof of Proposition 3.32. Let M and N be F2[O(V )]-modules for V ∈ {H0, H1}
and P• → M be a projective resolution of M . Lemma 3.33 implies that κ̃(P•)
is a projective resolution of κ̃(M). The functor κ̃ induces a morphism of cochain
complexes
HomF2[O(V )]−mod(P•, N)→ HomFquad(κ̃(P•), κ̃(N))
which induces the morphism (κ̃)∗ in cohomology. Since the functor κ̃ is fully-
faithful the previous morphism is an isomorphism and so induces an isomorphism
in cohomology.
We deduce the following corollary:
Corollary 3.34. For n a natural number, we have:
ExtnFquad(RH0 , RH0) ≃ F2 and Ext
Fquad
(RH1 , RH1) ≃ F2
where RH0 and RH1 are the simple functors introduced in Corollary 1.7.
Proof. Let ǫ be an element in {0, 1}. For V = Hǫ, by Corollary 1.7 (1), we have
κ̃(F2) = RHǫ . So, applying Proposition 3.32 to M = N = F2 we obtain:
Ext∗Fquad(RHǫ , RHǫ) ≃ Ext
F2[O(Hǫ)]−mod
(F2,F2) = H
∗(O(Hǫ),F2).
Since O(H0) ≃ S2 ≃ C2 and O(H1) ≃ S3 ≃ GL2(F2) we know by classical results
of cohomology of groups that
Hn(O(H0),F2) = H
n(O(H1),F2) = F2.
Remark 3.35. This corollary exhibits an important difference between the cate-
gories F and Fquad; recall that in F Ext
F (S, S) = 0 for all simple objects S of F
(see [?]).
28 CHRISTINE VESPA
4. Application: the polynomial functors of Fquad
In this section, having generalized the notion of polynomial functor to the cat-
egory Fquad, we prove, by induction, that the polynomial functors of Fquad are in
the image of the functor ι : F → Fquad.
4.1. Definition of polynomial functors of Fquad.
4.1.1. The difference functors of Fquad. We define the difference functors of Fquad
which generalize the notion of difference functor of F . Recall that, according to
[?], the difference functor ∆ : F → F is the functor given by
∆F (V ) := Ker(F (V ⊕ F2)
F (p)
−−−→ F (V )),
for F an object in F , V an object in Ef and p : V ⊕ F2 → V the projection.
Definition 4.1. The difference functors ∆H0 : Fquad → Fquad and ∆H1 : Fquad →
Fquad are the functors defined by:
∆H0F (V ) := Ker(F (V⊥H0)
F (T0)
−−−−→ F (V )),
∆H1F (V ) := Ker(F (V⊥H1)
F (T1)
−−−−→ F (V )),
for F an object in Fquad, V an object in Tq, and Ti = [V⊥Hi
−→ V⊥Hi
←− V ] for
i ∈ {0, 1} .
We have the following result:
Lemma 4.2. The functors ∆H0 and ∆H1 are exact.
4.1.2. Definition of polynomial functors. Before giving the definition of polynomial
functors in Fquad, let us recall that of polynomial functors of F ( [?]). For an object
F of F , F is a polynomial functor of degree 0 if and only if ∆F = 0 and, for an
integer d, F is polynomial of degree at most d+ 1 if and only if ∆F is polynomial
of degree at most d.
Definition 4.3. Let F be an object in Fquad:
(1) the functor F is polynomial of degree 0 if and only if ∆H0F = ∆H1F = 0;
(2) for an integer d, the functor F is polynomial of degree at most d+1, if and
only if ∆H0F and ∆H1F are polynomial of degree at most d.
The following proposition allows us to simplify the definition of a polynomial
functor of degree 0.
Lemma 4.4. Let F be an object in Fquad. The functor ∆H0F is zero if and only
if the functor ∆H1F is zero.
Proof. If ∆H0F = 0, we have, for all objects V of Tq, F (V ) ≃ F (V⊥H0). Conse-
quently,
F (V ) ≃ F (V⊥H0) ≃ F (V⊥H0⊥H0) ≃ F (V⊥H1⊥H1),
where the last isomorphism is obtained from the isomorphism H0⊥H0 ≃ H1⊥H1,
recalled in section 1. We deduce from the existence of morphisms F (V ) →֒ F (V⊥H1)
and F (V ⊥H1) →֒ F (V⊥H1⊥H1), induced by the inclusions and from the isomor-
phism between F (V ) and F (V⊥H1⊥H1), that
F (V ) ≃ F (V⊥H1) ≃ F (V⊥H1⊥H1).
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 29
Therefore ∆H1F = 0.
The proof of the converse is similar. �
4.2. Study of polynomial functors of Fquad. The aim of this section is to prove
the following result:
Theorem 4.5. The polynomial functors of Fquad are in the image of the functor
ι : F → Fquad.
We will prove this theorem by induction over the degree of the polynomial func-
tors.
4.2.1. Polynomial functors of degree zero of Fquad. In this paragraph, we start the
induction. The proof of the following result relies, in an essential way, on the
classification of simple functors S of Fquad such that S(H0) 6= 0 or S(H1) 6= 0,
obtained in Proposition 3.31.
Lemma 4.6. Let S be a simple functor of Fquad, S is a polynomial functor of
degree zero if and only if S is the constant functor F2.
Proof. In order to prove the direct implication, we have to distinguish two cases.
(1) If S(H0) = S(H1) = 0.
By the classification of the nondegenerate quadratic spaces over F2, if W
is a space of minimal dimension, satisfying S(W ) 6= 0, we have the existence
of an element ǫ of {0, 1} and a nondegenerate quadratic space V which is
non-zero, such that: W ≃ Hǫ⊥V. Since W is of minimal dimension, we
have S(V ) = 0. This implies:
∆HǫS(V ) = S(Hǫ⊥V ) 6= 0.
We deduce the result in this case.
(2) If S(H0) 6= 0 or S(H1) 6= 0.
In this case, we use the classification of simple functors S of Fquad such
that S(H0) 6= 0 or S(H1) 6= 0 obtained in Proposition 3.31. By an explicit
calculation for all the functors S obtained in this classification, we obtain
that the functors ∆H0S are non-zero except for the constant functor S = F2.
The converse is trivial.
4.2.2. Proof of Theorem 4.5. To prove Theorem 4.5, we need the following result
where the idempotents [eV ], obtained in Proposition 2.13, play a crucial rôle.
Proposition 4.7. Let S be a non-trivial simple functor of Fquad which is not in
the image of the functor ι : F → Fquad, then, one of the functors ∆H0S or ∆H1S
is not in the image of the functor ι : F → Fquad.
Proof. Let W be a nondegenerate quadratic space of minimal dimension, such that
S(W ) 6= 0. We distinguish the two following cases.
(1) If dim(W ) = 2.
By an explicit calculation for all the functors S of the classification given
in Proposition 3.31, we obtain the result.
30 CHRISTINE VESPA
(2) If dim(W ) > 2.
There exists a nondegenerate quadratic space V , possibly trivial, and an
element ǫ of {0, 1}, such that: W ≃ H0⊥Hǫ⊥V. Suppose that ∆H0S and
∆H1S are in the image of the functor ι, we prove, below, that this implies
that S is in the image of ι. By Lemma 2.18 it is sufficient to show the
existence of an object W in Tq such that: S(eW )S(W ) 6= 0. By Lemma
2.15, we have: eW = eH0⊥eHǫ⊥eV .
Since W is assumed to be a space of minimal dimension such that
S(W ) 6= 0, we have S(H0⊥V ) = S(Hǫ⊥V ) = 0. This implies that
∆H0S(Hǫ⊥V ) ≃ S(W )(4.7.1)
∆HǫS(H0⊥V ) ≃ S(W ).(4.7.2)
These isomorphisms are natural and, for (4.7.1), the action of EndTq(Hǫ⊥V )
on ∆H0S(Hǫ⊥V ) corresponds to the restriction of the action of EndTq (W )
on S(W ). In the same way, for (4.7.2), the action of EndTq (H0⊥V ) on
∆HǫS(H0⊥V ) corresponds to the restriction of the action of EndTq (W ) on
S(W ). Suppose that ∆H0S and ∆H1S are in the image of ι. We deduce
that:
S(1H0⊥eHǫ⊥eV )S(W ) = ∆H0S(eHǫ⊥eV )∆H0S(Hǫ⊥V )
= ∆H0S(Hǫ⊥V ) = S(W )
where the first equality comes from the action described previously, the
second is a consequence of Lemma 2.15 and the third is given by 4.7.1. In
the same way, we obtain:
S(eH0⊥1Hǫ⊥eV )S(W ) = ∆HǫS(eH0⊥eV )∆HǫS(H0⊥V )
= ∆HǫS(H0⊥V ) = S(W ).
We deduce that: S(1H0⊥eHǫ⊥eV ) ◦ S(eH0⊥1Hǫ⊥eV )S(W ) = S(W ). Since
S(1H0⊥eHǫ⊥eV ) ◦ S(eH0⊥1Hǫ⊥eV ) = S(eW ) by Lemma 2.15, we have:
S(eW )S(W ) 6= 0, as required.
The proof of the theorem relies, also, on the following lemmas.
Lemma 4.8. (1) A functor F of Fquad which takes values in finite vector
spaces and such that ∆HǫF is finite, is finite.
(2) A polynomial functor F of Fquad which takes values in finite vector spaces
is finite.
Proof. (1) The functor
Fquad
−→ E × Fquad
F 7→ (F (0),∆HǫF )
is exact and faithful by Lemma 4.2. Hence, if γ(F ) is finite then F is finite.
(2) If F takes values in finite vector spaces, so does ∆HǫF . Consequently we
can apply the first point recursively to obtain the result.
Lemma 4.9. A finite object F of Fquad whose composition factors are in the image
of the functor ι is in the image of the functor ι.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS 31
Proof. This result is a straightforward consequence of the thickness of the subcat-
egory ι(F) in Fquad given in Theorem 2.16. �
Proof of Theorem 4.5. Since a functor of Fquad is colimit of its subfunctors which
take values in finite vector spaces, we deduce from Lemma 4.8, that it is sufficient
to prove the result for the finite polynomial functors of Fquad. Furthermore, since
the functors ∆H0 and ∆H1 are exact by Lemma 4.2, it is sufficient to consider the
case of a simple functor S. We prove the theorem by induction over the polynomial
degree.
If S is polynomial of degree 0, according to Lemma 4.2, S is in the image of the
functor ι.
Suppose that all simple polynomial functors of Fquad and of degree d are in the
image of the functor ι and consider a simple polynomial functor S of Fquad such
that deg(S) = d + 1. By the definition of polynomial functor in Fquad given in
4.3, the functors ∆H0S and ∆H1S are polynomial of degree d. We deduce that all
composition factors of ∆H0S and ∆H1S are polynomial of degree smaller than or
equal to d and, by induction, we obtain that they are in the image of ι. Since S is a
simple functor, it is a quotient of a standard projective functor PV . Consequently
S takes its values in finite dimensional vector spaces. Therefore, ∆H0S and ∆H1S
take their values in finite dimensional vector spaces. We deduce from Lemma 4.8
that the functors ∆H0S and ∆H1S are finite, and, by Lemma 4.9, we obtain that
∆H0S and ∆H1S are in the image of the functor ι. Consequently, by Proposition
4.7, S is in the image of the functor ι. �
Ecole Polytechnique Fédérale de Lausanne, Institut de Géométrie, Algèbre et
Topologie, Lausanne, Switzerland.
E-mail address: christine.vespa@epfl.ch
	Introduction
	1. The category Fquad: some recollections
	2. Filtration of the standard projective functors PV of Fquad
	2.1. Definition of the filtration
	2.2. Extremities of the filtration
	3. Decomposition of the standard projective functors PH0 and PH1
	3.1. Decomposition of PH0
	3.2. Decomposition of PH1
	3.3. Consequences of decompositions of functors PH0 and PH1
	4. Application: the polynomial functors of Fquad
	4.1. Definition of polynomial functors of Fquad
	4.2. Study of polynomial functors of Fquad
