Modular Categories associated to Unipotent Groups
Tanmay Deshpande

Abstract
Let G be a unipotent algebraic group over an algebraically closed field k of characteristic p > 0
and let l 6= p be another prime. Let e be a minimal idempotent in DG (G), the Ql -linear triangulated braided monoidal category of G-equivariant (for the conjugation action) Ql -complexes
on G under convolution (with compact support) of complexes. Then, by a construction due
to Boyarchenko and Drinfeld, we can associate to G and e a modular category MG,e . In thispaper, we prove that the modular categories that arise in this way from unipotent groups are
precisely those in the class C±
p.

1

Introduction

Let G be a unipotent group over an algebraically closed field k of characteristic p > 0. We fix
the field k and all algebraic groups and schemes we consider will be assumed to be over k. By
an algebraic group, we mean a smooth group scheme of finite type over the field k. A unipotent
group is an algebraic group (over k) that is isomorphic to a closed subgroup of the algebraic group
U Ln (k) of unipotent upper triangular matrices of size n for some positive integer n.
Let us also fix a prime number l 6= p. For a k-scheme X we have the Ql -linear triangulated
symmetric monoidal category D(X) := Dcb (X, Ql ) of constructible Ql -complexes on X under tensor
product of complexes. For any integer n, we have the degree shift by n functor (·)[n] : D(X) →D(X) as well as the n-th Tate twist functor (·)(n) : D(X) → D(X). If we have an action α :G × X → X of the unipotent group G on X, we have the G-equivariant Ql -linear triangulated
cateogry DG (X). We recall (from [BD08, §1.3]) that an object of DG (X) is a pair (M, φ) whereM ∈ D(X) and φ : α∗ M → p∗2 M is an isomorphism (known as the G-equivariant structure)satisfying certain compatibility conditions, where p2 : G × X → X is the second projection. We
have the degree shift and Tate twist functors on DG (X) as well.
For the unipotent group G, the Ql -linear triangulated category D(G) is also monoidal under convolution with compact support. We recall (from [BD08, §1.4]) that for M, N ∈ D(G), we define
the convolution with compact support as M ∗ N := µ! (M  N ) where µ : G × G → G is the group
multiplication and µ! denotes (derived) pushforward with compact support. Here the external tensor product M  N := p∗1 M ⊗ p∗2 N where pi : G × G → G for i = 1, 2 are the two projections. We
also have the Ql -linear triangulated braided monoidal category DG (G) of G-equivariant complexes
for the conjugation action of G on itself. For M ∈ D(G) and (N, φ) ∈ DG (G), we have braiding
isomorphisms βM,N : M ∗ N → N ∗ M defined in [BD08, Defn. A.43] which provide the braided
structure on DG (G). The category DG (G) is equipped with a twist θ, an automorphism of theidentity functor on DG (G) obtained using the G-equivariant structure of the objects of DG (G) (see1[Bo, §4.9],[BD06, §5.5] or [BD08, §1.5.4, Appendix A.5]). This gives DG (G) the structure of a ribbon r-category. We refer to [BD06, §5.5], [BD08, §1 and Appendix A.5] for more on the structureof the categories D(G) and DG (G).
Let e ∈ DG (G) be a minimal idempotent. This means that 0 6= e ∼= e ∗ e and for any idempotent
e0 ∈ DG (G), e ∗ e0 is either isomorphic to e or 0. We refer to [BD08, Defn. 1.11, also Thm. 1.49] forthe precise definition and some subtler points. Associated to the minimal idempotent e, we havethe Ql -linear triangulated Hecke subcategory eDG (G), which is a braided monoidal category withunit object e. We recall from loc. cit. that eDG (G) is the full subcategory of DG (G) consistingof objects M ∈ DG (G) such that M ∼= e ∗ N for some N ∈ DG (G) (or equivalently, such that
M ∼
e
∗
M
).
In
[BD08],
Boyarchenko
and Drinfeld define an L-packet of character sheaves on G=
and a modular category MG,e ⊂ eDG (G) corresponding to each such minimal idempotent e. We
recall this in Theorem 1.2 below.
Remark 1.1. All monoidal categories that we consider in this paper will be Ql -linear. In particular
all the fusion categories that we talk about are assumed to be over the field Ql . We recall that afusion category over Ql is a Ql -linear semisimple abelian rigid tensor category with finitely many
simple objects (up to isomorphism), finite dimensional Hom spaces and with unit object being
simple. A spherical structure on a fusion category is a certain additional structure (see [DGNO,§2.4.3]) which, in particular, allows us to define categorical dimensions d(X) ∈ Ql of objects X in
the fusion category. A modular category is a non-degenerate braided spherical category. We refer
to [ENO02] and [DGNO] for more on the notions of fusion categories, spherical structures, braided
fusion categories, ribbon structures and modular categories.
Theorem 1.2. ([BD08]) Let G be a unipotent group over the field k. Let e ∈ DG (G) be a minimal
idempotent. Then the following hold:(i) Let Mperv
G,e ⊂ eDG (G) denote the full subcategory consisting of objects of eDG (G) whose underlying Ql -complex is a perverse sheaf on G. Then Mperv
G,e is a semisimple abelian category
with finitely many simple objects. The (isomorphism classes of ) simple objects of Mperv
G,e aresaid to form the L-packet of character sheaves associated to the minimal idempotent e.
(ii) There exists an integer ne ∈ {0, 1, · · · , dim(G)} such that e ∈ Mperv
G,e [ne ]. Moreover, the shifted
[n
]
⊂
eD
(G)
is
closedunder
convolution
and the twist θ insubcategory MG,e := Mperv
eG
G,e
DG (G) induces the structure of a modular category on MG,e . Moreover, we have Db (MG,e ) ∼=eDG (G).
eDefinition 1.3. The number de := dim(G)−n∈ 12 Z is said to be the functional dimension of the2minimal idempotent e.
Thus if we have a unipotent group G, and a minimal idempotent e ∈ DG (G), we have the associatedmodular category MG,e . In this paper, we characterize all modular categories that arise in this way
from unipotent groups, namely these modular categories are precisely those in the class C±
p . Westate the technical definition and some properties of this class of modular categories in Appendix
C. (See also [DGNO2].) We briefly recall the definition of the class C+
p below.
A pointed fusion category is one in which all simple objects are invertible. Consequently, the setof isomorphism classes of simple objects is a finite group Γ. All pointed fusion categories (overQl ) are of the form VecωΓ , the category of finite dimensional Γ-graded Ql -vector spaces with usual∗
tensor product, but with the associativity constraint defined by a 3-cocycle ω ∈ H 3 (Γ, Ql ). A

2

pointed fusion category has a unique positive spherical structure (see [DGNO, Ex. 2.26]) and withthis structure all simple objects are of categorical dimension 1. The Drinfeld center Z(D) of sucha pointed (positive spherical) fusion category D is a modular category, which can also be realized
as the category of representations of the so-called twisted Drinfeld double of a finite group. A
modular category C is said to be in the class C+
p if we have an equivalence of modular categoriesC∼= Z(VecωΓ ), where Γ is a finite p-group and VecωΓ is equipped with the positive spherical structure.
(See also Proposition C.2.) The definition of the class C−
p is more technical and we provide it inAppendix C.
The goal of this paper is to prove the following result conjectured by Drinfeld in [BD06]:
Theorem 1.4. (i) Let G be a unipotent group. Let e ∈ DG (G) be a minimal idempotent with1−functional dimension de (∈ 12 Z). Then if de ∈ Z, MG,e ∈ C+
p and if de ∈ 2 + Z, MG,e ∈ Cp .
±(ii) If C ∈ Cp , then there exists a connected unipotent group G with a minimal idempotent e ∈DG (G) such that MG,e ∼= C as modular categories.
We prove Theorem 1.4(i) in §2. In [BD08], Boyarchenko and Drinfeld prove that every minimal
idempotent e ∈ DG (G) can be induced from a Heisenberg idempotent (Definition 2.6) e0 ∈ DG0 (G0 )satisfying a certain geometric Mackey condition, where G0 is some subgroup of G. They also
prove that de = de0 + dim(G/G0 ) and that there is an equivalence MG,e ∼= MG0 ,e0 of modular
categories. In fact, we prove in §2.1 that we may choose e0 to be a special (Definition 2.6) Heisenberg
idempotent. Hence we reduce Theorem 1.4(i) to the case where e is a special Heisenberg idempotent.
In §2.3 we complete the proof of Theorem 1.4(i) by proving it in the case of special Heisenberg
idempotents.
In §3, we prove Theorem 1.4(ii). Let C ∈ C±
p . In §3.1, we show that it is enough to find a possibly
disconnected unipotent group G with a Heisenberg idempotent e ∈ DG (G) such that C ∼= MG,e .
We show that if G is a (possibly disconnected) unipotent group with a Heisenberg idempotent e,then there exists a connected unipotent group U 0 and a minimal idempotent f 0 ∈ DU 0 (U 0 ) such that
MG,e ∼= MU 0 ,f 0 . Finally, we complete the proof of Theorem 1.4(ii) for the case C ∈ C+
p in §3.2 andthe case C ∈ C−
in
§3.3
p

Acknowledgments
I would like to thank V. Drinfeld for introducing me to this subject, for the many useful discussions
and for suggesting corrections and improvements. This work was partially supported by World
Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

2

Proof of Theorem 1.4(i)In this section, we prove the first part of the main theorem. We begin by reducing to the case of
Heisenberg idempotents and further to the case of special Heisenberg idempotents.

3

2.1

Reduction to special Heisenberg idempotents

Let us recall the notions of admissible pairs and Heisenberg idempotents defined in [BD08]. A
multiplicative local system on a connected unipotent group G is a nonzero Ql -local system L on G
equipped with an isomorphism µ∗ L ∼= L  L. It follows that L must be of rank 1, and that it has
a natural G-equivariant structure and we can consider it as an object of DG (G). According to [Bo,Lem. 7.3], we can equivalently think of a multiplicative local system on G as a central extensionof G by the abstract group Qp /Zp if we choose an identification of Qp /Zp with µp∞ , the group of∗
p-power-th roots of unity in Ql .
Remark 2.1. In particular, any central extension 0 → C → G̃ → G → 0 of G by an abelian group
C along with a character C → µp∞ determines a multiplicative local system on G.
It will often be necessary to work with perfect schemes. For a scheme X over k, we denote by Xperf
its perfectization. We have X(k) = Xperf (k). For a perfect connected commutative unipotent
group U over k, we let U ∗ denote its Serre dual which parametrizes the multiplicative Ql -local
systems on U . U ∗ is also a perfect connected commutative unipotent group. (See [Bo, Appendix
A.5], [Bé, §1] for details.)
Definition 2.2. ([BD08, Defn. 1.30]) Let G be a unipotent group. Let (N, L) be a pair consistingof a connected subgroup N ⊂ G and a multiplicative local system L on N . We say that the pair
(N, L) is admissible if the following conditions hold:(i) Let G0 = NG (N, L), the normalizer of the pair (N, L) in G (in particular N / G0 and L is G0 equivariant, see also [BD08, Defn. 1.29]) and let G0◦ denote its neutral connected component.
Then G0◦ /N is commutative.
(ii) The group morphism φL : (G0◦ /N )perf → (G0◦ /N )∗perf that is defined in this situation (seeRemark 2.5 below) is an isogeny.
(iii) (Geometric Mackey condition) For every g ∈ G(k) − G0 (k), we have
L|(N ∩N g )◦  Lg |(N ∩N g )◦ ,where N g = g −1 N g and Lg is the multiplicative local system on N g obtained from L bytransport of structure.
Definition 2.3. Let (N, L) be an admissible pair as above. If we further have that dim(G0 /N ) ≤ 1,
we say that (N, L) is a special admissible pair.
Remark 2.4. For an admissible pair (N, L), let e0 := L[2 dim N ](dim N ) ∈ DG0 (G0 ). Then e0 ∈DG0 (G0 ) is in fact a minimal idempotent on the subgroup G0 ⊂ G (see [Bo, Prop. 8.2], [De]).
Remark 2.5. In view of statement (i) of the definition, we have the commutator map G0◦ ×G0◦ → N .
The pullback of L via the commutator induces a skew-symmetric biextension of (G0◦ /N )perf ×(G0◦ /N )perf . (See [Bo, Appendix A.6] for more on biextensions.) This defines the map φL of(ii). More explicitly, for g ∈ G0◦ , let cg : G0◦ → N be the commutator with g map: G0◦ 3 h 7→
hgh−1 g −1 . Then we can check that c∗g L is a multiplicative local system on G0◦ whose restriction
to the normal subgroup N is trivial, whence we can think of it as a multiplicative local system on
G0◦ /N . Explicitly, the map φL : (G0◦ /N )perf → (G0◦ /N )∗perf is defined by g 7→ c∗g L. Often, we
say that the map φL itself is skew-symmetric. Also the conjugation action of G0 on G0◦ induces anaction of π0 (G0 ) on G0◦ /N . Since L is G0 -equivariant, it follows that φL is π0 (G0 )-equivariant. See[BD08, §3.3] for more.
4

Definition 2.6. If (N, L) is an admissible pair for G such that G0 = G, we say that (N, L) is a
Heisenberg admissible pair1 for G. In this case the minimal idempotent e := L[2 dim N ](dim N ) ∈DG (G) is said to be a Heisenberg idempotent. Further, if we have dim(G/N ) ≤ 1 we say that theidempotent is a special Heisenberg idempotent.
Remark 2.7. It is easy to see that if e ∈ DG (G) is a Heisenberg idempotent as above, then ne =dim(N ). This is because L[dim(N )] is a perverse sheaf. Hence the functional dimension de =1◦
2 dim(G /N ).
Example 2.8. Let us consider an instructive example of the fake Heisenberg group H. As avariety, H = Ga × Ga and we define multiplication by (x, a) · (y, b) = (x + y, a + b + xy p ). Note that(x, a)−1 = (−x, xp+1 − a). The center of H is N := 0 × Ga and H/N ∼= Ga is commutative. Let
L be the multiplicative local system on N corresponding to the Artin-Schreier central extension∗
x7→xp −x
0 → Fp → Ga −−−−−→ Ga → 0 (and the choice of a primitive p-th root of unity ζ ∈ Ql ). Since N
is central, it is clear that NH (N, L) = H. The commutator with (y, b) map c(y,b) : H → N ∼= Gappdefined in Remark 2.5 is given by c(y,b) (x, a) = xy − x y =: cy (x) (since in this example thecommutator map is independent of a, b). Let us pass to the perfectization, Ga,perf of Ga . We canidentify Ga,perf with its Serre dual G∗a,perf . This identification ψ can be described as follows (see[Bé, §1.1] for details):
For t ∈ Ga,perf , we have the homothety t : Ga,perf → Ga,perf . Then we define ψ(t) = t∗ L, thepullback (considered as an element of G∗a,perf ) of the Artin-Schreier multiplicative local system by
the homothety t.
We see that in this example, the map φL : Ga,perf → G∗a,perf is defined by y 7→ c∗y L. If we identifyG∗a,perf with Ga,perf as above, we can check that this map is given by y 7→ y p − y 1/p . This map
is an isogeny with kernel Fp2 . Hence (N, L) is a Heisenberg admissible pair on H. The object
e := L[2](1) is the corresponding Heisenberg idempotent and de = 21 .
0
In [BD08, §1.12], the induction (with compact support) functor indG
G0 : DG0 (G ) → DG (G) is defined0
for closed subgroups G ⊂ G. In the following theorem, Boyarchenko and Drinfeld prove that every
minimal idempotent in DG (G) comes from an admissible pair and in particular from a Heisenberg
idempotent on a subgroup by induction.
Theorem 2.9. [BD08] (i) Let (N, L) be an admissible pair for a unipotent group G and let e0 ∈0
DG0 (G0 ) be the corresponding Heisenberg idempotent on G0 as defined above. Then e := indGG0 e ∈0
DG (G) is a minimal idempotent. The functional dimensions of e and e are related as follows:de = de0 + dim(G/G0 ).
(ii) In the situation of (i), the induction functor induces an equivalence of modular categories
MG0 ,e0 ∼= MG,e .
(iii) Every minimal idempotent e ∈ DG (G) comes from an admissible pair by the procedure describedin (i). Hence every minimal idempotent e ∈ DG (G) comes from induction from a Heisenberg
idempotent e0 on some subgroup G0 .
Let us now strengthen this last statement and prove that every minimal idempotent can in fact be
induced from a special Heisenberg idempotent. Since we know that every minimal idempotent canbe induced from a Heisenberg idempotent, it suffices to prove that every Heisenberg idempotent can
1Note that in this situation, condition (iii) of the definition is vacuous.

5

be induced from a special Heisenberg idempotent. For ease of notation, henceforth in this section
let us work in the setting of perfect unipotent groups, namely all groups considered are assumed tobe perfect unipotent. We will use results proved in Appendix A.
Let us suppose we have a Heisenberg admissible pair (N, L) on a perfect unipotent group G. Let
e = L[2 dim N ](dim N ) ∈ DG (G) be the corresponding Heisenberg idempotent. Let H = G◦ andlet Γ = G/H = π0 (G). Since G is unipotent, Γ is a p-group. Note that we have the Γ-equivariant
skew-symmetric isogeny φL : H/N → (H/N )∗ .
Proposition 2.10. Let W be a connected subgroup of H containing N such that W/N is a Γinvariant isotropic subgroup of H/N . Then we can extend L to a multiplicative local system L0 on
W . The pair (W, L0 ) is admissible for G.
Proof. Since W/N is isotropic, the pullback of L by the commutator map cW : W × W → N istrivial. Hence by [Bo, Prop. 7.7] we can extend L to a multiplicative local system L0 on W .
The Γ-invariance of W/N implies that W / G. Let W 0 be the connected subgroup of H such thatW 0 /N = ((W/N )⊥ )◦ . Then we see that NG (W, L0 )◦ = W 0 and W 0 /W is commutative. Note thatthe commutator map cW 0 : W 0 × W 0 → W factors as cW 0 : W 0 × W 0 → N ⊂ W . It follows that the
skew-symmetric biextension φL0 : W 0 /W → (W 0 /W )∗ defined using the pair (W, L0 ) is the sameas the skew-symmetric biextension of W 0 /W = ((W/N )⊥ )◦ /(W/N ) induced from the isotropic
subspace W/N for skew-symmetric biextension φL : H/N → (H/N )∗perf . It follows that φL0 is an
isogeny. Since W / G, the geometric Mackey condition is immediate. Hence (W, L0 ) is an admissible
pair for G.
Corollary 2.11. Let G be a unipotent group. Every minimal idempotent e ∈ DG (G) comes froma special admissible pair.
Proof. Let (N, L) be an admissible pair for a minimal idempotent e. We assume without lossof generality that the normalizer of the pair (N, L) equals G. (Else we can replace G by thenormalizer of the pair.) We will now continue to use previous notations. Let us now choose aconnected subgroup W ⊂ H containing N such that W/N is a Γ-invariant maximal isotropic
subgroup of H/N . The existence of W is guaranteed by Corollary A.11. Then we can extend L toa multiplicative local system L0 on W . In this case NG (W, L0 )◦ = W 0 as defined above. Since W/N
is maximal isotropic, dim(W 0 /W ) ≤ 1 by Corollary A.11 and hence (W, L0 ) is a special admissible
pair for the minimal idempotent e.

2.2

Analysis of Heisenberg idempotents

Let G be a unipotent group. Let (N, L) be a Heisenberg admissible pair for G and let e =L[2 dim N ](dim N ) ∈ DG (G) be the corresponding Heisenberg idempotent. Let H = G◦ and let)
Γ = G/H. We see that in this case ne = dim(N ) and hence de = dim(H/N. We have the Γ2equivariant skew-symmetric isogeny φL : (H/N )perf → (H/N )∗perf . Let KL = kerφL . As defined∗in [Bo, Appendix A.10] or [Da, §2.3] we have a quadratic form (see Definition A.12) θ : KL → Ql
associated to the skew-symmetric isogeny φL and we have a metric group (KL , θ). We also have an
action of Γ on this metric group. According to Appendix B, we have the pointed positive spherical

6

modular category M(KL , θ) associated to the metric group M(KL , θ). We see in Theorem 2.12
below that MH,e ∼= M(KL , θ).
Consider the full subcategory eDH (G) ⊂ DH (G), the Ql -linear triangulated monoidal category ofH-equivariant complexes. According to [De, §2.4], we have a gradingM
eDH (H γ̃),eDH (G) =H γ̃∈G/Ha monoidal action of Γ on eDH (G) and a crossed braiding which give eDH (G) the structure of abraided Γ-crossed category (see [DGNO, §4.4.3] for definition, see also Appendix B.2) with identity
component eDH (H) and unit object e. The equivariantization (see [DGNO, §4.4.4]) eDH (G)Γ is a
braided monoidal category equivalent to the Hecke subcategory eDG (G) ⊂ DG (G).
fG,e denote the full subcategory of eDH (G) consisting of perverse sheaves shifted by dim N .
Let M
fG,e is the full subcategory consisting of shifted (by dim(N ))Note that by definition MH,e ⊂ Mperverse sheaves supported on H ⊂ G. The following theorem is proved in [De].
fG,e is closed under convolution and we have a faithfully graded braided
Theorem 2.12. ([De]) (i) MfΓ-crossed structure on MG,e with trivial component MH,e .
(ii) MH,e is a pointed modular category equivalent to the modular category M(KL , θ) associated tothe metric group (KL , θ).
fΓ is a modular category and MG,e ∼fΓ .
(iii) The equivariantization M
=M
G,e
G,e
Remark 2.13. By statements (i) and (ii) above, we have a braided action of Γ on M(KL , θ) coming
fG,e . This induces an action of Γ on the metric groupfrom the braided Γ-crossed structure of M
(KL , θ) which is the same as the action which we had defined before.
Remark 2.14. We note that whenever we have a minimal idempotent e on a unipotent group G, wehave the modular category MG,e . Moreover, if e is a Heisenberg idempotent as described above,fG,e whose identity component is the pointed modular
we also have a braided Γ-crossed category M
fG,e )Γ is equivalent to MG,e .
category MH,e ∼= M(KL , θ) and whose equivariantization (MExample 2.15. Let H be the fake Heisenberg group with the Heisenberg admissible pair (N, L)
and corresponding Heisenberg idempotent e defined in Example 2.8. We saw that in this example,
the skew-symmetric isogeny
φL : Ga,perf → G∗a,perf ∼= Ga,perf
is given by y 7→ y p − y 1/p and that the kernel KL = Fp2 . In this case, (see [Da, Appendix A]) the
metric group associated to the skew-symmetric isogeny is the anisotropic metric group (Fp2 , ζ N m(·) )where N m : Fp2 → Fp is the norm map. Hence in this example, the modular category MH,e isequivalent to Manis:= M(Fp2 , ζ N m(·) ).
p
fG,e is a rigid monoidal category. According to [De, §2.3] the duality
Remark 2.16. The category M
fG,e 3 X 7→ X ∨ = Dι∗ X[2 dim N ](dim N ) where D denotes Verdier duality
functor is given by M
and ι : G → G is the inversion map. We have natural isomorphisms X ∼= (X ∨ )∨ which give us afG,e . Hence M
fΓ ⊂ eDG (G) also has a sphericalcanonical pivotal (in fact spherical) structure on M
G,estructure which is the same as the one obtained using the twist θ on DG (G).
In the remainder of this subsection, we study certain invariants of the modular category MG,e incase of Heisenberg and special Heisenberg idempotents. We refer to [DGNO, §2.4.1] for the notion

7

of Frobenius-Perron dimension and [DGNO, §6] for the notions of Gauss sums and multiplicative
central charge.
We prove the following proposition for all Heisenberg idempotents, and later restrict to the classof special Heisenberg idempotents. Recall that for a Heisenberg idempotent e ∈ DG (G), de =1
2 dim(H/N ). We continue to use the assumptions and notations introduced at the beginning of
this subsection.
Proposition 2.17. Let e ∈ DG (G) be a Heisenberg idempotent.
(i) The Gauss sums τ ± (MG,e ) are equal to (−1)2de pk · |Γ| for some integer k ≥ 0. Hence themultiplicative central charge of MG,e equals (−1)2de .
(ii) The Frobenius-Perron dimension of MG,e equals |Γ|2 · |KL |.
Proof. Since MH,e is equivalent to the modular category corresponding to the metric group (KL , θ),by [Da, Prop. 5] we have τ ± (MH,e, ) = τ ± (KL , θ) = (−1)2de pk for some integer k ≥ 0. Suppose(M, φ) ∈ MΓH,e ⊂ MG,e , where M ∈ MH,e and φ is the equivariant structure. The twist θ(M,φ) in
MG,e is the same as the twist θM in MH,e as an automorphism of M . Let E be the full subcategoryof MΓH,e consisting of objects (M, φ) such that M is isomorphic to a direct sum of copies of theunit object e. Hence it follows that the twist when restricted to E is trivial, or in other words, E is
an isotropic subcategory of MG,e . Moreover, it is clear that E ∼= Rep(Γ). The braided Γ-crossedfcategory MG,e is precisely the corresponding de-equivariantization and the identity component
MH,e is the corresponding fiber category. (See [DGNO, §4.4.7-8].) Hence by [DGNO, Thm. 6.16],
τ ± (MG,e ) = τ ± (MH,e ) · |Γ|. Statement (i) now follows, since τ ± (MH,e ) = (−1)2de pk . To prove
fG,e ) = |Γ| · FPdim(MH,e ) = |Γ| · |KL | by [DGNO, Prop. 2.21] and that(ii), note that FPdim(MΓfG,e ) by [DGNO, 4.26].
f ) = |Γ| · FPdim(MFPdim(MG,eRemark 2.18. Since (KL , θ) is a metric group coming from a skew-symmetric biextension, it follows from [Da] that |KL | is an even power of p. Moreover, Γ is a p-group. Hence we see that
FPdim(MG,e ) = p2k for some k ∈ Z+ . We also see that the multiplicative central charge of MG,eis 1 if dim(H/N ) is even and −1 if dim(H/N ) is odd.
Remark 2.19. Suppose the Heisenberg admissible pair (N, L) is special, i.e. dim(H/N ) ≤ 1. Hence
H/N = 0 or Ga . Hence the induced action of the p-group Γ on H/N is trivial.
Now suppose that we have any admissible pair such that the action of Γ on H/N is trivial. (By theabove remark, every special admissible pair is of this type.) The action of Γ on the metric group(KL , θ) is clearly trivial too. Since Γ acts trivially on H/N , the commutator map from H × G
in fact maps to N . We will now use some of the tools developed in [De]. For g ∈ G we havethe commutator map cg : H → N defined h 7→ hgh−1 g −1 . We recall from [De, §4.1] that in thise := {g ∈ G|c∗ L ∼situation, the objects of eDH (G) are supported on K
= Ql } ⊃ N . The objects ofgeeDH (H) are supported on K := K ∩ H. By Remark 2.5, we see that K/N = KL = ker(φL ). We
can readily modify the proofs of Proposition 4.5, 4.6 from [De] and prove the following:
Proposition 2.20. Let e ∈ DG (G) be a Heisenberg idempotent as above such that the action of Γon H/N is trivial. Then we have the following:fG,e is a central (see Definition B.1) braided Γ-crossed category with identity component(i) M
M(KL , θ).
e ⊂ G. Let ek denote the right translate of e by k. Then ek ∈ M
fG,e ⊂ eDH (G). The(ii) Let k ∈ Kkkee are all the simple
isomorphism class of e only depends on the coset N k ∈ K/N
and the e , k ∈ K
8

fG,e .
objects of the braided Γ-crossed category M
fG,e is pointed with thee(iii) For k1 , k2 ∈ K we have an isomorphism ek1 ∗ ek2 ∼= ek1 k2 . Hence M
egroup of isomorphism classes of simple objects being K/Nwhich fits into a central extensione
0 → KL = K/N ⊂ K/N→ Γ → 0.
Proof. Since the action of Γ on the metric group (KL , θ) obtained from the braided Γ-crossed
fG,e is trivial (see also Remark 2.13), M
fG,e is a central braided crossed category by
category M
Definition B.1.
For g ∈ G, let δg denote the skyscraper sheaf supported at g ∈ G whose stalk at g is Ql . By
definition, the right translate eg = e ∗ δg . Note that since e ∈ DG (G), we have the braiding
e c∗ L ∼isomorphism δg ∗ e ∼= e ∗ δg between the left and right translates of e by g ∈ G. For k ∈ K,
k = Qlkk
and hence we can define an H-equivariant structure on the translate e = L [2 dim N ](dim N ).
fG,e ⊂ eDH (G). For n ∈ N , a trivialization of the stalk Ln of L at n defines anHence ek ∈ M
e Hence the isomorphism classisomorphism e → en and also isomorphisms ek → enk for each k ∈ K.
kof e only depends on the coset N k. It is clear that we have an isomorphism ek1 ∗ ek2 ∼= ek1 k2 . InkfG,e is an invertible simple object. From this, or using [De, Prop. 3.10, 3.11]
particular each e ∈ Mke are all the simple objects of M
fG,e .
we see that the e for k ∈ K
fG,e are 1, or in other
Proposition 2.21. The categorical dimensions of all the simple objects of M
fwords, the spherical structure on the pointed category MG,e is positive (and of course integral).
fΓ are positive integers.
Hence the categorical dimensions of all simple objects of MG,e ∼=M
G,ee Note that ek is
Proof. We must prove that the categorical dimension d(ek ) = 1 for all k ∈ K.
supported on N k and that its dual (ek )∨ = Dι∗ (ek )[2 dim N ](dim N ) (see [De, §2.3] and Remark 2.16
above) is supported on k −1 N . Here D denotes Verdier duality and ι : G → G is the inversion map.
Let us identify N k and k −1 N with N by the evident maps. Under this identification ι : N k → k −1 Ngets identified with ι : N → N , µ : N k × k −1 N → N gets identified with the multiplication for
N (hence this identification is compatible with convolutions), ek gets identified with e and Verdier
duality is also compatible. By definition, the categorical dimension d(ek ) equals the compositione → ek ∗ (ek )∨ → (ek )∨∨ ∗ (ek )∨ → e. Once we make the identifications as above, we see thatfG,e . Hence categorical dimensions of all simple objects ofd(ek ) = d(e) and e is the unit object of M
fG,e are 1. Moreover, the categorical dimensions of all simple objects of MG,e ∼fΓ are positive
M
=M
G,e
f
integral multiples of those of MG,e (see [DGNO, Prop. 4.26]). Hence the categorical dimensions of
all simple objects of MG,e must also be positive integers.

2.3

Completion of the proof of Theorem 1.4(i)Let e ∈ DG (G) be a minimal idempotent. Then there exists a special Heisenberg idempotent
e0 (coming from a special admissible pair (N, L) on G) on a subgroup G0 ⊂ G satisfying the0
geometric Mackey condition such that e ∼= indGG0 e . By Theorems 2.9(ii) and 2.12(iii), we have
0)
π
(G
0
f 0 0 . Since e0 is special Heisenberg idempotent on G0 , we can apply
MG,e ∼= MG0 ,e0 ∼
= M
G ,e

9

fG0 ,e0 is a central braided π0 (G0 )-crossed catePropositions 2.20 and 2.21 and conclude that M
fG0 ,e0 is positive and the identity component is equivalent togory, the spherical structure on M
M(KL , θ), where (KL , θ) is the metric group coming from the admissible pair (N, L). We recall
that KL = ker(φL ) ⊂ (G0◦ /N )perf . Since G0 is unipotent, π0 (G0 ) is a p-group. By Theorem 2.9(i),0◦de = dim(G /N2)(=0 or 1) + dim G/G0 . If de ∈ Z, G0◦ /N = 0 and then M(KL , θ) ∼= Vec. Hence in
10◦ /N ∼ G whence the metric
this case MG,e ∈ C+
byDefinition
C.1(i).
If
d
∈
+
Z,
then
G
=
e
a
p
2
group (KL , θ) is Witt-equivalent to the anisotropic metric group (see Appendix A.3) (Fp2 , ζ N m(·) )by Theorem A.13. Hence in this case MG,e ∈ C−
p by Definition C.1(ii). This completes the proofof Theorem 1.4(i).

3Proof of Theorem 1.4(ii)Let C ∈ C±
p . In order to prove Theorem 1.4(ii), we first show that it is enough to find a possibly
disconnected unipotent group G with a Heisenberg idempotent e such that C ∼= MG,e .

3.1

Passing from a disconnected group to a connected group

Let G be a possibly disconnected unipotent group with a Heisenberg admissible pair (N, L). As
before, let H = G0 . Let e ∈ DG (G) denote the corresponding Heisenberg idempotent. We will
prove that there exists a connected unipotent group U 0 and a minimal idempotent f 0 ∈ DU 0 (U 0 )such that the modular categories MG,e and MU 0 ,f 0 are equivalent.
Proposition 3.1. Let V be a unipotent group with an action of G by automorphisms. Let K be aG-equivariant multiplicative local system on V . Then we have the following:(a) (V o N, K  L) is a Heisenberg admissible pair for V o G.
(b) We have an equivalence eDG (G) ∼= (f  e)D(V oG) (V o G), where f = K[2 dim V ](dim V ),given by M 7→ f M . Here f e is the Heisenberg idempotent corresponding to the admissible
pair above.
Proof. Here K  L = p∗1 K ⊗ p∗2 L is the external tensor product where p1 , p2 are the two projections
from V o N to V and N respectively. Since K is G-equivariant multiplicative local system (andhence N -equivariant) on V , it is easy to check that K  L is a multiplicative local system on V o N
and that in fact it is also V o G-equivariant. We also check that the corresponding morphism((V o H)/(V o N ))perf → ((V o H)/(V o N ))∗perf can be identified with the map (H/N )perf →(H/N )∗perf corresponding to the admissible pair (N, L). For part (b), we use the fact that f isG-equivariant.
Let us now proceed to construct the connected unipotent group U 0 . First, we embed G into a
connected unipotent group U . Recall the well known theorem (due to Chevalley) that given an
algebraic group along with a closed subgroup, we can find a representation of the group on a vector
space and a line in the vector space such that the subgroup is precisely the stabilizer of the line.
(See for example [H, §11.2].) In case the subgroup is unipotent, it can be characterized as thestabilizer of a point in the vector space (any nonzero point on the chosen line). Hence we can find a10

vector space V with a U -action and a multiplicative local system K on V (in other words K ∈ V ∗ ,which can be thought of as the vector space we start with) such that G is precisely the stabilizer
of K ∈ V ∗ under the U -action. Now let U 0 = V o U . Let N 0 = V o N and L0 = K  L. We will
now prove that the pair (N 0 , L0 ) is admissible for U 0 , and that its normalizer is G0 := V o G.
Proposition 3.2. (a) For u ∈ U 0 − G0 we haveL0 |(N 0 ∩u N 0 )0  u L0 |(N 0 ∩u N 0 )0 .
(b) G0 is the normalizer of the pair (N 0 , L0 ) and it is an admissible pair for U 0 .
(c) Let f 0 ∈ DU 0 (U 0 ) be the corresponding minimal idempotent. Then we have an equivalenceeDG (G) ∼= f 0 DU 0 (U 0 ).
Proof. (a) Note that since V is a connected normal subgroup of U 0 , we have V ⊂ (N 0 ∩ u N 0 )0 . We
have L0 |V ∼= u K. We have chosen K such that the stabilizer of K ∈ V ∗ under= K and that u L0 |V ∼the action of U is precisely G. Hence we have that u K  K for each u ∈ U 0 − G0 . Hence (a) follows.
(b) It is clear that G0 is contained in the normalizer of (N 0 , L0 ). But from (a) we see that itis precisely the normalizer. This combined with (a) and Prop. 3.1 implies that (N 0 , L0 ) is anadmissible pair for U 0 .
0(c) Note that we have f 0 = indU
G0 (f  e), where f is an is Prop. 3.1. Hence (c) follows from Prop.
3.1(b) and Theorem 2.9(ii).

3.2

The case C ∈ C+pω∼
Let C ∈ C+
p . By Proposition C.2, we have a realization C = Z(VecΓ ) equipped with positive∗
spherical structure for some finite p-group Γ and an ω ∈ H 3 (Γ, Ql ). Let us choose a representative
×
3-cocycle ω : Γ3 → Ql . Since Γ is a finite p-group, we may assume that ω takes values in a finite
×
subgroup µpn ⊂ Ql of pn -th roots of unity for some n. Let W denote the ring scheme of Witt
vectors of length n. With its additive structure, W is a connected commutative unipotent groupsuch that pn · W = 0. We have the Artin-Schreier sequence 0 → C := Z/pn Z → W → W → 0. We
fix an identification µpn ∼= C.

We will now construct a unipotent group G along with a Heisenberg idempotent e on it suchfG,e = Vecω as braided crossed categories (see proof of Proposition
that (see Remark 2.14) MΓC.2). Hence as required we will have MG,e ∼= C. Consider the group ring
= Z(VecωΓ ) ∼= (VecωΓ )Γ ∼e := W(Γ) = M aps(Γ, W). This is a connected commutative unipotent group (isomorphic toH∆e via the diagonal. Note
W|Γ| ) with an action of Γ by left translations. We embed C ,→ W ,→ Heethat under this embedding Γ acts trivially on C ⊂ H. Let H := H/C as a Γ-module. Hence wehave an exact sequencee →H→0
0→C→H(1)∗of Γ-modules. It is a Γ-equivariant central extension of H by C ⊂ Ql . Let L denote the corree is the group algebra,
sponding Γ-equivariant multiplicative local system on H. Note that since Hie
H (Γ, H) = 0 for i > 0 by Shapiro’s lemma. Hence from the long exact sequence of cohomology groups associated to the above short exact sequence, we get an isomorphism δ : H 2 (Γ, H) →11

H 3 (Γ, C). Hence there exists an f ∈ H 2 (Γ, H) thats maps to ω (considered now as an element of
H 3 (Γ, C)) and we get a corresponding extension
0 → H → G → Γ → 0.

(2)

Note that (H, L) is a special Heisenberg admissible pair for G. Let e be the corresponding idempotent.
For γ ∈ Γ let us choose lifts γ̃ ∈ G such that γ˜1 γ˜2 = f (γ1 , γ2 )γg
1 γ2 where we view f as a 2fG,e (upcocycle. We see that by Proposition 2.20, the translates eγ̃ are all the simple objects of M
to isomorphism). For a, b ∈ Γ, let us fix a trivialization of the stalk Lf (a,b) . This determines anisomorphism ηa,b : e → ef (a,b) . For a, b, c ∈ Γ we have a f (b, c) + f (a, bc) = f (a, b) + f (ab, c) =: h.
From the Γ-equivariant multiplicative local system structure of L and the chosen trivializations η·,·we get two trivializations of Lh . These two trivializations differ by δf = ω by definition of thefG,e convolve asboundary map. The simple objects of M∼=−1e ηa,beã ∗ eb̃ −→ eãb̃ = ef (a,b)ab ∼= ef (a,b) ∗ eab −→ eab .
eef
For a, b, c ∈ Γ, we have identifications eãb̃c̃ ∼= (eã ∗ eb̃ ) ∗ ec̃ ∼= ef (a,b) ∗ ef (ab,c) ∗ eabc and eãb̃c̃ ∼= eã ∗ (eb̃ ∗aff
ec̃ ) ∼
= e f (b,c) ∗ ef (a,bc) ∗ eabc . Each of the previous two compositions are further identified with eabc
using the Γ-equivariant structure of e and the chosen isomorphisms η·,· . These two identifications
fG,e corresponds to ω ∈ H 3 (Γ, Q∗l ) i.e. M
fG,e ∼
differ by ω. Hence the associativity constraint of M=ωfVecΓ and by Proposition 2.21, the spherical structure on MG,e is positive. Hence by Theorem
2.12(iii) after equivariantization, MG,e ∼= (VecωΓ )Γ ∼=2 Z(VecωΓ ) ∼= C as modular categories.

This (along with §3.1) proves Theorem 1.4(ii) in the case when C ∈ C+p.

3.3

The case C ∈ C−p

3
Let C ∈ C−
p . By Proposition C.2, there exists a faithfully graded central (Definition B.1) braided Γcrossed category D (equipped with positive spherical structure) with identity component Manis
:=
p
N
m(·)
Γ
) such that C ∼M(Fp2 , ζ
= D as modular categories where Γ is a finite p-group.

We refer to Appendix B.2 for an introduction to L
central braided crossed categories. A (faithfullyCγ with identity component M(A, θ) (where
graded) central braided Γ-crossed category D =γ∈Γ(A, θ) is some metric group) is pointed by Remark B.4. Hence, the set of isomorphism classes ofsimple objects in D is a group which is a central extension of Γ by A, the group of isomorphismclasses of simple objects of C1 = M(A, θ). The class of the central extension gives us an elementf ∈ H 2 (Γ, A). Now, we can start from a central extension of Γ by A corresponding to an f ∈H 2 (Γ, A) and ask whether there exist central braided Γ-crossed categories D giving rise to thiscentral extension. According to [ENO09, §8], such central braided crossed categories exist when a∗certain obstruction O4 (f ) ∈ H 4 (Γ, Ql ) vanishes, in which case they are classified up to equivalence2

See proof of Proposition C.2.
All braided crossed categories that we consider abstractly, as well as those that arise from other structures inthis paper are faithfully graded even though sometimes we may not state this explicitly.
3

12

∗

by a certain H 3 (Γ, Ql )-torsor. We refer to [ENO09] for more on classification of braided crossed
categories and in particular to op. cit. §4 for the definition of various higher categorical groups(for example Pic(·), EqBr(·)) associated to braided fusion categories that play an important role inthe classification. (See also Appendix B.2.)
Remark 3.3. Since M(A, θ) is a non-degenerate braided fusion category, we have an equivalenceΘ : Pic(M(A, θ)) ∼= EqBr(M(A, θ)) (see [ENO09, §5.4] for details or Appendix B.2). According to[ENO09, §8], f ∈ H 2 (Γ, A) classify morphisms of 1-groups Γ → Pic(M(A, θ)) ∼= EqBr(M(A, θ))lifting the trivial group homomorphism Γ → Pic(M(A, θ)) ∼= EqBr(M(A, θ)), or equivalently
braided actions of Γ on M(A, θ) lifting the trivial action of Γ on (A, θ). Such a morphism
Γ → Pic(M(A, θ)) corresponding to f ∈ H 2 (Γ, A) can be lifted to a morphism of 2-groups
Γ → Pic(M(A, θ)) if and only if the obstruction O4 (f ) vanishes. In this case, the lifts are classified∗
by a certain H 3 (Γ, Ql )-torsor. According to [ENO09] (see Theorem B.2), the lifts Γ → Pic(M(A, θ))as above classify the central braided crossed categories corresponding to the f ∈ H 2 (Γ, A) up toequivalence.
f be a central braided Γ-crossed category with identity component Manis
Let M
corresponding top
an f ∈ H 2 (Γ, Fp2 ) (this implies that the corresponding obstruction O4 (f ) vanishes). Then we wish
fG,e ∼fto construct a unipotent group G with a Heisenberg idempotent e such that M= M.
Let us now show that if we can realize one such central braided Γ-crossed category corresponding
to f from a unipotent group, then we can realize all.
f M
f0 be two central braided Γ-crossed categories corresponding to the class
Proposition 3.4. Let M,2f ∈ H (Γ, Fp2 ). Suppose there exists a unipotent group G with a Heisenberg idempotent e such thatfG,e ∼
f Then there exists a unipotent group U with a Heisenberg idempotent e0 such thatM
= M.
fU,e0 ∼
f0 .
M
=M
×

f0 differs from M
f in the associativity constraint by an element ω ∈ H 3 (Γ, Q ). By
Proof. Indeed M
lassumption, we have an extension 0 → H → G → Γ → 0 with a Heisenberg idempotent e ∈ DG (G)f∼fG,e . By §3.2, we also
corresponding to a Heisenberg admissible pair (N, L) for G such that M=Mhave a connected commutative unipotent group V equipped with a Γ-action and a Γ-equivariant
multiplicative local system K on V and an extension 0 → V → W → Γ → 0 corresponding to thefW,K[2 dim V ](dim V ) ∼3-cocycle ω (namely such that M= VecωΓ ). Let us consider the product of thegroup extensions:
0 → H × V → U → Γ → 0.
Now we can check that (N × V, L  K) is a Heisenberg admissible pair for U and that the corref0 .
sponding braided Γ-crossed category is M
Hence in order to complete the proof of Theorem 1.4(ii), given f ∈ H 2 (Γ, Fp2 ), or equivalently acentral extension 0 → Fp2 → KΓ → Γ → 0 (such that the obstruction O4 (f ) vanishes) it is enoughto construct a unipotent group U with π0 (U ) = Γ along with a Heisenberg idempotent such thatthe corresponding braided Γ-crossed category is one corresponding to the given central extension.
(See §3.3.4 for details.)

13

3.3.1

The central extension and obstruction in two step nilpotent case

Let H be a two step nilpotent unipotent group such that its center N is connected and H/N iscommutative. Let Γ be a finite p-group with an outer action on H given by a group homomorphism
Γ → Out(H). This induces a braided action of Γ on the braided monoidal category DH (H). The
outer action also induces an honest action of Γ on the center N as well as on H/N = Inn(H) since
we have assumed that Inn(H) is commutative. Let us assume that the induced action of Γ on H/Nis trivial.
Let L be a Γ-equivariant multiplicative local system on N , such that (N, L) is a Heisenberg admissible pair on H. This means that the induced (Γ-equivariant) skew-symmetric biextension map
φL : (H/N )perf → (H/N )∗perf is an isogeny whose kernel is denoted KL as before. We have thecorresponding non-degenerate quadratic form θ on KL associated to this skew-symmetric isogeny.
The admissible pair (N, L) gives us the Heisenberg idempotent e = L[2 dim N ](dim N ) on H.
As before, let K ⊂ H be such that K/N = KL . For h ∈ H, let ch : H → N be the “commutatorwith h” map: H 3 g 7→ ghg −1 h−1 ∈ N . Recall that by Remark 2.5 (see also [De, Prop. 4.5]),
K = {k ∈ H|c∗k L ∼= Ql }.

(3)

Since L is Γ-equivariant, we get a braided action of Γ on eDH (H) and hence on MH,e ∼= M(KL , θ).
Since the action of Γ on H/N is trivial, we see that the braided action of Γ on MH,e induces thetrivial action on KL . According to [ENO09] this braided action corresponds to a certain central
extension of Γ by KL as described in the previous section. Let us construct this braided action andcentral extension explicitly.
Let us pullback the exact sequence 0 → Inn(H) → Aut(H) → Out(H) → 0 along the outer action
map and obtain the central (since induced action of Γ on Inn(H) is trivial) extension
0 → Inn(H) → AutΓ (H) → Γ → 0.
For γ ∈ Γ, let Autγ (H) ⊂ Aut(H) be the Inn(H)-coset that describes the outer action of γ on
H. In particular, Aut1 (H) = Inn(H) = H/N . For γ ∈ Γ and γ̃ ∈ Autγ (H), let cγ̃ : H → N
be the map cγ̃ (h) = hγ̃(h−1 ) ∈ N . (cγ̃ (h) ∈ N since Γ acts trivially on H/N .) In particular, forN h ∈ Aut1 (H) = H/N , cN h : H → N is the commutator with h map ch defined previously. For
each γ̃ ∈ Autγ (H), cγ̃ is a group homomorphism. Hence c∗γ̃ L is a multiplicative local system on H
that is trivial when restricted to N since L is Γ-equivariant. For γ ∈ Γ, let
Kγ = {γ̃ ∈ Autγ (H)|c∗γ̃ L ∼= Ql }.

(4)

By (3), we see that K1 = K/N = KL . We have the relationcγ˜1 γ˜2 (h) = cγ˜1 (h)γ1 (cγ˜2 (h)).
Let KΓ be the disjoint union of all the Kγ for γ ∈ Γ. Then we can check that in fact KΓ is a groupand we have a central extension
0 → K1 = KL → KΓ → Γ → 0.
14

KΓ has an honest action on H and it induces the given outer action of Γ on H. We also havea braided action of KΓ on eDH (H) and MH,e and the action of KL ⊂ KΓ corresponds to the
conjugation action by the simple objects of MH,e . Hence this induces the braided action of thequotient Γ on MH,e .
Thus from the outer action of Γ on H and the Γ-equivariant multiplicative local system L on thecenter N , we get a two cocycle f : Γ × Γ → KL ⊂ H/N = Inn(H) corresponding to the central
extension constructed above by choosing lifts γ̃ ∈ Kγ ⊂ Autγ (H) ⊂ Aut(H) for each γ ∈ Γ suchthatγ˜1 γ˜2 = f (γ1 , γ2 )γg
1 γ2 .
Let F : Γ × Γ → K ⊂ H be a lift of f to K ⊂ H. Hence we get a 3-cocycle ω : Γ3 → N such that
we haveã(F (b, c))F (a, bc) = ω(a, b, c)F (a, b)F (ab, c), for a, b, c ∈ Γ.
(5)
Remark 3.5. Thus starting from an outer action of Γ on H we get an ω ∈ H 3 (Γ, N ). This is theobstruction to the existence of an extension of Γ by H corresponding to given outer action (see[EM, §7]).
Remark 3.6. Furthermore, if we have a Γ-equivariant Heisenberg admissible pair (N, L) as above,
then we get a braided action of Γ on MH,e and the corresponding central extension 0 → KL →KΓ → Γ → 0.
∗This data gives rise to an obstruction β ∈ H 4 (Γ, Ql ) to the existence of a central braided Γcrossed category with trivial component MH,e . On the other hand we have the boundary map∗
δ : H 3 (Γ, N ) → H 4 (Γ, Ql ) coming from the Γ-equivariant multiplicative local system L on N∗
e → N → 0). Next
(which can be thought of as a short exact sequence of Γ-modules 0 → Ql → N
we show that δω = β.
3.3.2

The cohomological obstructions

We continue to use the notations of the previous section in which we described a 3-cocycle ω ∈H 3 (Γ, N ) which is the obstruction to the existence of an extension of Γ by H corresponding to
the given outer action. We have a Γ-equivariant multiplicative local system L on N which gives∗
rise to a boundary map δ : H 3 (Γ, N ) → H 4 (Γ, Ql ). We note that the outer action respected theHeisenberg admissible pair (N, L) on H and that the induced action of Γ on H/N (and hence on
KL ⊂ H/N ) was trivial. The outer action defines a braided action of Γ on MH,e and we constructed
the corresponding central extension
0 → KL → KΓ → Γ → 0.
∗

From this data, we get a β ∈ H 4 (Γ, Ql ) which is the obstruction to the existence a central braided
Γ-crossed category corresponding to the braided action of Γ on MH,e . In this situation:
Proposition 3.7. We have β = δω.
Proof. Let us first describe the boundary map δ explicitly. Let ω ∈ H 3 (Γ, N ) be represented by a
3-cocycle ω : Γ × Γ × Γ → N . For a, b, c ∈ Γ, let us choose a trivialization of the stalk Lω(a,b,c) .
We have ω(ab, c, d)−1 · ω(a, b, cd)−1 · ã ω(b, c, d) · ω(a, bc, d) · ω(a, b, c) = 1 for a, b, c, d ∈ Γ. Hence15

using our chosen trivializations of Lω(·,·,·) and the Γ-equivariant multiplicative structure of the local
system L, we get an automorphism of the stalk L1 = Ql . This automorphism is δω(a, b, c, d).
∗

We now describe the obstruction β ∈ H 4 (Γ, Ql ) explicitly. By Proposition 2.20 (see also [De, §4.3]),
the simple objects of MH,e can be described as the translates ek of e by k ∈ K ⊂ H and moreover,for k1 , k2 ∈ K we can identify ek1 ∗ ek2 with ek1 k2 . The isomorphism class of ek only depends onthe coset N k. For n ∈ N and any k ∈ K, a choice of trivialization of the stalk Ln gives us an
isomorphism ek → enk . For a, b, c ∈ Γ, we have a chosen trivialization of the stalk Lω(a,b,c) as in
the previous paragraph.
Note that for a ∈ Γ, we have chosen lifts ã ∈ Ka ⊂ Auta (H) (see §3.3.1). We define the corresponding braided automorphism (also denoted by ã) of MH,e as pullback via ã−1 : H → H.
Let MH,e,ã denote the corresponding braided MH,e -bitorsor which is equal to MH,e as a lefttorsor and where the right action is twisted by the braided automorphism ã. For X ∈ MH,e and−1
ea, b ∈ Γ, ãb̃(X) = eF (a,b) ∗ ab(X)∗ eF (a,b) . We will define a system of products (see [ENO09, §8.6])
F (a,b) where we have used theMH,e,ã MH,e MH,e,b̃ → MH,e,ab
e mapping (X, Y ) 7→ X ∗ ã(Y ) ∗ e
identifications of all bitorsors with MH,e as left-torsors. Now for a, b, c ∈ Γ, we have two functors
MH,e,ã × MH,e,b̃ × MH,e,c̃ → MH,e,abc
f corresponding to the two bracketings. They are given byã

(X, Y, Z) 7→ X ∗ ã(Y ) ∗ ãb̃(Z) ∗ e F (b,c)F (a,bc) ,(X, Y, Z) 7→ X ∗ ã(Y ) ∗ ãb̃(Z) ∗ eF (a,b)F (ab,c) .
Hence by (5), the trivialization of the stalk Lω(a,b,c) induces a natural isomorphism ζa,b,c between
these two functors. Now for a, b, c, d ∈ Γ we have five functors MH,e,ã ×MH,e,b̃ ×MH,e,c̃ ×MH,e,d˜ →MH,e,abcd
g corresponding to the five bracketings and a pentagon formed by the natural isomorphisms∗

between these functors defined using the ζ·,·,· defined above. Let β(a, b, c, d) ∈ Ql denote thenatural automorphism of one of the five functors obtained from the above pentagon (namely βmeasures the failure of the commutativity of the pentagon). It follows that β is a 4-cocycle andthat the associativity constraint ζ can be modified to satisfy the pentagon axiom if and only if β iscohomologically trivial, or in other words, β is the obstruction to the existence of the central braided
crossed category. Comparing with the explicit computation of δω above, we see that β = δω.

3.3.3

Constructing an outer action from a central extension

Let Γ be a finite p-group and let
0 → Fp2 → KΓ → Γ → 0
be a central extension. We will now reverse the construction of §3.3.1. Namely, we will construct a
two step nilpotent group H with connected center N and an embedding Fp2 ⊂ H/N with an actionof KΓ such that Fp2 ⊂ H/N = Inn(H) acts by its natural conjugation action on H (this induces
an outer action of Γ on H) and a Γ-equivariant Heisenberg admissible pair (N, L) as described in§3.3.1, such that the corresponding metric group is the anisotropic metric group (Fp2 , θ = ζ N m(·) )and such that the corresponding extension of Γ by KL = Fp2 obtained using the construction of§3.3.1 coincides with KΓ .
16

Let us fix a positive integer n and let W be the ring scheme of Witt vectors of length n. Let
e = M aps(Γ, W), or the group algebra of Γ over the ring W. Then N
e is a connected commuN
tative unipotent group with a left Γ-action by right translations of functions. Note that we havee ) = 0 for i > 0 by Shapiro’s lemma.
H i (Γ, N
e of constant maps which is a Γ-equivariant map withWe have the embedding C := Z/pn Z ⊂ W,→N
e /C which is a connected commutative
Γ acting trivially on W. Let us form the quotient N := N
unipotent group equipped with the induced action of Γ.
Lemma 3.8. We have the following structure:(i) A Γ-equivariant multiplicative local system L on N .
(ii) Embeddings Ga ,→ W ⊂ N such that the induced Γ-action is trivial on W and L restricts to
the Artin-Schreier local systems on W and Ga .
Proof. Consider the Γ-equivariant central extension
e → N → 0.
0→C→NThis gives us (i), the Γ-equivariant multiplicative local system L on N (after choosing an identifi∗
e , we getcation C ∼= µpn ⊂ Ql ). Restricting the above central extension to the constants W ⊂ N
the central extension 0 → C → W → W/C → 0, which can be identified with the Artin-Schreier
w7→F (w)−w

central extension 0 → C → W−−−−−−−−→W → 0, where F is the relative Frobenius map. Thise /C = N and we see thatgives us an identification W ∼= W/C and an embedding W ∼= W/C ⊂ N
under this embedding L|W is the Artin-Schreier multiplicative local system. The action of Γ on
W is trivial under this embedding. Also, we have Ga ,→ W under which the Artin-Schreier local
system on W restricts to the Artin-Schreier local system on Ga .
An immediate consequence is the following:
Lemma 3.9. Let B : Ga × Ga → Ga ⊂ N be the skew-symmetric biadditive map given byB(λ1 , λ2 ) = λ1 λ2 p − λ1 p λ2 ∈ Ga ⊂ N . The pullback B ∗ L induces a skew-symmetric biextension
Ga,perf → (Ga,perf )∗ which is an isogeny with kernel Fp2 and the induced metric is the anisotropic
metric.
Proof. This follows from Example 2.8 since L|Ga is the Artin-Schreier multiplicative local system
and B identifies with the commutator map of Example 2.8.
eλ := {φ ∈ M aps(KΓ , W)|φ(kx) = φ(x) + k p λ for each k ∈ Fp2 , x ∈ KΓ }, theFor λ ∈ Ga , let N
e0 is the set of Fp2 -invariant maps and hence can bespace of Fp2 -quasi-invariant maps. Note that N
e = M aps(Γ, W). We have the natural action of KΓ on N
eλ given by yφ(x) = φ(xy).
identified with Neeλ is aFor k ∈ Fp2 ⊂ KΓ , its action on Nλ is given by kφ(x) = φ(xk = kx) = φ(x) + k p λ. Each N
e -bitorsor and addition induces Γ-equivariant maps N
eλ × Neλ → Neλ +λ for λi ∈ Ga .
N
1
2
1
2
eλ /C. In particular N0 = N . For φ ∈ N
eλ , let φ denote its image in Nλ .
For λ ∈ Ga , define Nλ := NeNote that for a constant function w ∈ W ⊂ N , w = F (w) − w considered as an element of the W
embedded in N (see Lemma 3.8), where F : W → W is the relative Frobenius. We have the induced17

action of KΓ on each Nλ and the induced Γ-equivariant addition maps Nλ1 × Nλ2 → Nλ1 +λ2 forλi ∈ Ga . For k ∈ Fp2 ⊂ KΓ , its action on Nλ is given bykφ = φ + k p λ = φ + F (k p λ) − k p λ
2= φ + k p λp − k p λ = φ + kλp − k p λ= φ + B(k, λ).
Let us now define the unipotent group H. The underlying variety H = {(λ, f )|λ ∈ Ga , f ∈ Nλ }.
Define a product on H by setting(λ1 , f1 ) · (λ2 , f2 ) = (λ1 + λ2 , f1 + f2 + λ1 λ2 p )).
(Here λ1 λ2 p ∈ Ga ⊂ N .) Note that (λ, f )−1 = (−λ, λp+1 − f ). The center of H is N andInn(H) = H/N = Ga with the inner action of µ ∈ Ga given by µ (λ, f ) = (λ, f + B(µ, λ)) where
B is as defined in Lemma 3.9. The commutator map [·, ·] : H × H → N ⊂ H is given by[(λ1 , f1 ), (λ2 , f2 )] = B(λ1 , λ2 ). By this, Remark 2.5 and Lemma 3.9 we have:
Lemma 3.10. The pair (N, L) is a Heisenberg admissible pair on H and the corresponding metricgroup is the anisotropic metric group (Fp2 , θ).
The action of KΓ on each Nλ induces an action of KΓ on the group H defined by x(λ, f ) = (λ, xf ).
The induced action of Γ on N is the original action. The induced action on H/N is trivial. As
before, the action of k ∈ Fp2 ⊂ KΓ is given by k(λ, f ) = (λ, f + B(k, λ)). We see that this identifies
with the inner action of k ∈ Fp2 ⊂ Ga = H/N.
We can now prove:
Proposition 3.11. The central extension obtained from the data as above (using the constructionof §3.3.1) is the central extension 0 → Fp2 → KΓ → Γ → 0 that we started out with.
Proof. It is clear (for example by Lemma 3.10) that in this case KL = Fp2 . For y ∈ KΓ , letcy : H → N be defined by h = (λ, φ) 7→ h · y(h−1 ) = φ − yφ. This is a group homomorphism
e by (λ, φ) 7→ φ − yφ. One canwhich can in fact be lifted to a group homomorphism Cy : H → N
readily verify that this is indeed a well defined homomorphism. This means that c∗y L is the trivial
multiplicative local system on H. Hence using (4) we see that KΓ is indeed the central extension
obtained by the construction of §3.3.1.
3.3.4

Completion of the proof

∼ f0Γ
f0
Let C ∈ C−
p . By Proposition C.2, C = M as modular categories, where M is a (positive spherical)
anis
central braided Γ-crossed category with trivial component Mp
and Γ is a finite p-group. The
f0 gives
group of isomorphism classes of simple objects in the central braided crossed category M∗
2
4
us an f ∈ H (Γ, Fp2 ) such that the corresponding obstruction β ∈ H (Γ, Ql ) vanishes. Note thatβ can be defined as an element of H 4 (Γ, µp ). Hence β must vanish in some H 4 (Γ, µpn ). Let us
fix one such n. Let H be the two-step nilpotent unipotent group equipped with the outer actionof Γ and the Γ-equivariant admissible pair (N, L) corresponding to f ∈ H 2 (Γ, Fp2 ) constructedin the previous section (for the chosen integer n). Let ω ∈ H 3 (Γ, N ) be the obstruction to the18existence of a group extension corresponding to the outer action of Γ on H constructed above. The
multiplicative local system L was defined using the Γ-equivariant central extension 0 → µpn ∼=C→e → N → 0. Since H i (Γ, Ne ) = 0 for positive integers i, the boundary map defines an isomorphismNδ : H 3 (Γ, N ) ∼= H 4 (Γ, µpn ). Since β = δω = 0, we conclude that ω vanishes and hence there exists
an extension 0 → H → G → Γ → 0. It follows that (N, L) is a special Heisenberg admissible pairfG,e has identity component
for G and that the corresponding central braided Γ-crossed category M
anis
Mp
and group of isomorphism classes of simple objects is the central extension corresponding
to f .
f0 and M
fG,e both give rise to the same central extension f . By Proposition 3.4, there
Hence Mf0 ∼fU,e0 as sphericalexists a unipotent group U with a Heisenberg idempotent e0 such that M=MΓ∼∼
f 0 = C as modular categories.
central braided crossed categories. Hence MU,e0 = MU,eBy §3.1, we can pass to a connected unipotent group. Hence the proof of Theorem 1.4 is now
complete.

A

Appendix: Skew-symmetric biextensions

In this appendix, we prove some general results about connected commutative (perfect) unipotent
groups and skew-symmetric biextensions. (See [Bo, Appendix A.6-11],[Da, §2] for more on skewsymmetric biextensions.) Since it is convenient to talk about skew-symmetric biextensions in thesetting of perfect schemes, we will work in the category of perfect unipotent groups over k, eventhough many of the results and arguments also work in the category of unipotent groups. Let
cpu denote the category of commutative perfect unipotent groups over k and let cpu◦ be the fullsubcategory of connected commutative perfect unipotent groups. The category cpu is abelian andcpu◦ is an exact subcategory. Serre duality defines an involution cpu◦ 3 V 7→ V ∗ ∈ cpu◦ which is anexact anti-autoequivalence of cpu◦ . For V ∈ cpu◦ , we think of V ∗ ∈ cpu◦ as the space parametrizing∗
the multiplicative local systems on V (after fixing an embedding Qp /Zp ,→ Ql ). We refer to [Bo,Appendix A] for more on the notions of Serre duality and biextensions.

A.1

Group actions on a connected commutative unipotent groupLet Γ be a group. Let V ∈ cpu (resp. ∈ cpu◦ ) be equipped with an action of Γ. Let Γ-cpu (resp.
Γ-cpu◦ ) denote the category whose objects are V (resp. connected V ) equipped with a Γ-action andwhose morphisms are Γ-equivariant maps of commutative perfect unipotent groups. Then Γ-cpu isan abelian category and Γ-cpu◦ is an exact subcategory. We prove some results in this situation
which reduce to basic well-known results in the case when V is a vector space over k and Γ acts bylinear transformations.
Definition A.1. An object V ∈ Γ-cpu◦ is called simple if it is nonzero and has no proper connectedΓ-invariant subgroups.
It is easy to see that:
Lemma A.2. Every V ∈ Γ-cpu◦ has a filtration {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vn−1 ⊂ Vn = V byconnected Γ-invariant subgroups such that each successive quotient Vi+1 /Vi is simple.

19

We see that a morphism in Γ-cpu◦ to a simple object is either zero or surjective and that a morphismfrom a simple object is either zero or has finite kernel. Hence we have the following:
Lemma A.3. Let V, W be simple objects in Γ-cpu◦ and f : V → W a morphism in Γ-cpu◦ . Thenf is either zero or an isogeny.
Lemma A.4. Let V ∈ Γ-cpu◦ be simple. Then p · V = 0, i.e. as a connected commutative perfectunipotent group, V ∼= Gra,perf for some positive integer r.
Proof. Consider the morphism p · IdV : V → V in Γ-cpu◦ . Since some power of p annihilates V , themap p · IdV is not an isogeny. Hence it must be zero.
Lemma A.5. Let V ∈ Γ-cpu◦ be simple. Let γ ∈ Γ be a central element whose order is a power ofp. Then γ acts trivially on V .
nProof. Suppose γ p = 1. Since γ ∈ Γ is central, γ : V → V is a morphism in Γ-cpu◦ and so isnn(γ − Id). Since V is simple p · V = 0. Hence (γ − Id)p = γ p − Id = 0. Hence γ − Id cannot be an
isogeny. Hence γ = Id since V is simple.
Remark A.6. If Γ is a (possibly infinite) p-group and V ∈ Γ-cpu◦ is simple, then the center Z(Γ)
acts trivially on V .
Proposition A.7. If Γ is a (possibly infinite) nilpotent p-group, then there is a unique simple
object in Γ-cpu◦ , namely Ga,perf equipped with the trivial action of Γ.
Proof. We prove this by induction on the length of the ascending central series of Γ. By the remark,the proposition holds for commutative p-groups Γ. (Z(Γ) = Γ must act trivially on a simple V ,and consequently V must be one dimensional.) Now suppose Γ is such that the proposition holds
true for all p-groups of lower nilpotence class. Let V ∈ Γ-cpu◦ be simple. By the remark above,Z(Γ) acts trivially on V . Hence the Γ-action on V comes from an action of the quotient Γ/Z(Γ).
V is simple for this action. Hence by our inductive hypothesis, V ∼= Ga,perf with trivial action of
Γ/Z(Γ). Hence the action of Γ is trivial as well.
Corollary A.8. Let Γ be a nilpotent p-group. Let V ∈ Γ-cpu◦ with dim(V ) = n. Then there is afiltration {0} = V0 ⊂ V1 ⊂ · · · ⊂ Vn−1 ⊂ Vn = V by connected Γ-invariant subgroups such that each
Vi+1 /Vi ∼= Ga with Γ acting trivially.

A.2

Invariant isotropic subgroups

As before, let Γ be a group and let V ∈ Γ-cpu◦ . Let φ : V → V ∗ be a Γ-equivariant skew-symmetricbiextension. (See [Bo, Appendix A.10].) We can think of such a skew-symmetric biextension as abimultiplicative local system on V × V whose restriction to the diagonal is trivial. Let W ⊂ V beφa connected subgroup. Let W ⊥ (∈ cpu) be the kernel of the composition V → V ∗  W ∗ . We saythat W is isotropic if W ⊂ W ⊥ . In this section, we prove that if dim(V ) ≥ 2 and if Γ is a nilpotent
p-group, then there exists a non-trivial Γ-invariant isotropic subgroup.
Lemma A.9. Let n be the smallest integer such that pn · V = 0. Let r be an integer such thatnr
2 ≤ r ≤ n − 1. Then p · V is a proper Γ-invariant isotropic subgroup.

20

Proof. It is clear that pr · V is a proper Γ-invariant subgroup for each positive integer r ≤ n − 1.
For r ≥ n2 , pr · (pr · V ) = 0, hence pr · (pr · V )∗ = 0. Hence we see that the composition
pr · V ⊂ V → V ∗  (pr · V )∗is zero. (pr · x 7→ pr · φ(x) 7→ pr · φ(x)|pr ·V = 0 in (pr · V )∗ which is annihilated by pr .)Proposition A.10. Let Γ be a nilpotent p-group, V ∈ cpu◦ and let φ : V → V ∗ be a Γ-equivariant
skew-symmetric biextension. If dim(V ) ≥ 2, then there exists a nonzero Γ-invariant isotropic
subgroup in V .
Proof. If p · V 6= 0, then the previous lemma gives us a proper Γ-invariant isotropic subgroup in V .
Hence let us assume p · V = 0, in which case the previous lemma does not tell us anything. Let us
use Corollary A.8. Since dim(V ) ≥ 2, there exists a two-dimensional Γ-invariant subgroup V2 ⊂ V .
Since p · V2 = 0, V2 ∼= G2a,perf . Consider the induced Γ-equivariant skew-symmetric biextension of
V2 . Let V1 ⊂ V2 be a one-dimensional Γ-fixed subgroup guaranteed by Corollary A.8. If V1 is⊥Visotropic in V2 , then it is also isotropic in V and we are done. If not, then its perpendicular V1 2
⊥V2 ◦⊥V
) = V2 . Since V1 ⊂ V2 is Γ-stable, (V1 2 )◦ must⊥V
also be Γ-invariant and since it is one dimensional, the action of Γ on (V1 2 )◦ must be trivial andin V2 must be one-dimensional and V1 + (V1consequently action of Γ on V2 must be trivial. Now by [Bo, Lemma A.29], V2 has a one-dimensional
isotropic subgroup W . Since the action of Γ on V2 is trivial, W is Γ-stable.
Corollary A.11. Let Γ be a nilpotent p-group and let V ∈ Γ-cpu◦ be equipped with a Γ-equivariant
φskew-symmetric biextension V → V ∗ . Then there exists a Γ-invariant maximal isotropic subgroup
W ⊂ V and for any such W , dim(W ⊥ ) ≤ dim(W ) + 1 and W is also maximal among all (notnecessarily Γ-invariant) isotropic subgroups of V .
Proof. Clearly maximal Γ-invariant isotropic subgroups exist. Let W be any Γ-invariant isotropic
subgroup of V . Then (W ⊥ )◦ is also Γ-stable and we have the induced Γ-equivariant skew-symmetricbiextension of (W ⊥ )◦ /W . Γ-invariant isotropic subgroups of V containing W are in a naturalbijective correspondence with Γ-invariant isotropic subgroups of (W ⊥ )◦ /W . If W is a maximal
Γ-invariant isotropic subgroup, then (W ⊥ )◦ /W has no nonzero Γ-invariant isotropic subgroups.
Hence this space has dimension less than or equal to 1.

A.3

Metric groups associated to skew-symmetric biextensions∗

Definition A.12. Let A be a finite abelian group and let θ : A → Ql be a function. We sayθ(a+b)
that θ is a quadratic form if θ(a) = θ(−a) and if the symmetric function b(a, b) := θ(a)θ(b)is a∗bicharacter b : A × A → Ql . We say that θ is non-degenerate if the bicharacter b is non-degenerate.
∗
A pre-metric group is a pair (A, θ) where A is a finite abelian group and θ : A → Ql is a quadratic
form. If the quadratic form is non-degenerate, we say that (A, θ) is a metric group.
Given a metric group (A, θ), we have the associated non-degenerate bicharacter b. A subgroup
B ⊂ A is said to be isotropic if θ|B = 1. In this case, B ⊂ B ⊥ . For an isotropic subgroup B, thesubquotient (B ⊥ /B, θ0 ) is also a metric group, where θ0 is the quadratic form on B ⊥ /B inducedby θ. A metric group is said to be anisotropic if it has no non-trivial isotropic subgroups. If L
21

is a maximal isotropic subgroup of a metric group (A, θ), then the subquotient (L⊥ /L, θ0 ) is ananisotropic metric group whose isomorphism class is independent of the choice of maximal isotropic
subgroup L. We say that the metric group (A, θ) is Witt-equivalent to the anisotropic metric group(L⊥ /L, θ0 ).
Let us now give two examples of anisotropic metric p-groups which are of particular interest in thetheory of character sheaves on unipotent groups (see Theorem A.13 below):• The trivial metric group (0, 1).
∗• (Fp2 , ζ N m(·) ) where N m : Fp2 → Fp is the norm map and ζ ∈ Ql is a primitive p-th root ofunity.
Now let V be a connected commutative perfect unipotent group and let φ : V → V ∗ be a skewsymmetric isogeny. According to [Bo, Appendix A.10], [Da, §2.3], to such a skew-symmetric isogeny,
we can associate a metric group (A, θ). Here A is the kernel of φ. Then we can define the associated∗non-degenerate quadratic form θ : A → Ql (see [Bo, Appendix A.10], [Da, §2.3]). The following
theorem is proved in [Da]:Theorem A.13. ([Da]) The metric group (A, θ) associated to the skew-symmetric isogeny φ : V →V ∗ is Witt-equivalent to the trivial anisotropic metric group if dim(V ) is even and is Witt-equivalent
to the anisotropic metric group (Fp2 , ζ N m(·) ) if dim(V ) is odd.

B

Appendix: The modular category associated to a metric groupand braided crossed categories

B.1

The modular category associated to a metric group

Given a pre-metric group (A, θ) (see Definition A.12), there exists a corresponding pointed braided
category M(A, θ) whose group of isomorphism classes of simple objects is isomorphic to A andthe braiding on the square of a simple object is given by θ. This determines the braided monoidal
category up to equivalence. We equip this braided pointed category with the positive sphericalstructure. With this structure, the twist (an automorphism of the identity functor on the pointed
braided category) is given by θ. We denote this pre-modular category by M(A, θ). (See [DGNO,§2.11.4-5 and Appendix D] for more details.) If (A, θ) is a metric group, M(A, θ) is a modular
category.
The modular category defined by the metric group (Fp2 , ζ N m(·) ) is of particular interest and wedenote it by Manis
.
p

B.2

Braided crossed categories and equivariantization

Let (A, θ) be a metric group and let Γ be a finite group. Let D be a faithfully graded braided
Γ-crossed category with identity component M(A, θ). This means that we have the following
structure on D:L• A grading D =Cγ where each Cγ is non-zero and C1 ∼= M(A, θ).
γ∈Γ22• A monoidal action of Γ on D such that γ1 (Cγ2 ) ⊂ Cγ1 γ2 γ1 −1 for each γ1 , γ2 ∈ Γ.
• For γ ∈ Γ, X ∈ Cγ and Y ∈ D isomorphisms∼=

cX,Y : X ⊗ Y → γ(Y ) ⊗ X
functorial in X, Y and satisfying certain compatibility conditions.
For a braided Γ-crossed category D, the equivariantization DΓ has the structure of a braided
monoidal category. We refer to [DGNO, §4.4.3] for a precise definition and properties of braided
crossed categories and related concepts.
In particular, the monoidal action of Γ on D restricts to an action of Γ on the braided monoidal
category M(A, θ) by braided autoequivalences which in turn determines an action of Γ on themetric group (A, θ).
Definition B.1. We say that such a braided Γ-crossed category as above is central if the induced
action of Γ on the metric group (A, θ) is trivial.
L
Let D =Cγ be a braided Γ-crossed category. Each Cγ is a C1 -module category. In fact, byγ∈Γ[ENO09, Thm. 6.1], it is an invertible (under tensor product of module categories, see op. cit.
§3) C1 -module category. For a braided fusion category C1 , let Pic(C1 ) denote the categorical 2group whose objects are invertible C1 -module categories under tensor product, 1-morphisms areequivalences of C1 -module categories and 2-morphisms are isomorphisms of such equivalences. As
proved in op. cit., the assignment γ 7→ Cγ along with the braided Γ-crossed structure of D defines
a morphism of 2-groups Γ → Pic(C1 ). In fact, the following result is proved in op. cit. §7.8
Theorem B.2. ([ENO09, Thm. 7.12]) Equivalence classes of faithfully graded braided Γ-crossed
categories with trivial component C1 are in bijection with morphisms of categorical 2-groups Γ →Pic(C1 ).
We can truncate (by forgetting the highest level morphisms and identifying isomorphic lower ones)
the 2-group Pic(C1 ) to a 1-group Pic(C1 ) and further to an ordinary group Pic(C1 ) of equivalenceclasses of invertible C1 -module categories. Let EqBr(C1 ) denote the 1-group whose objects are
braided autoequivalences of C1 and whose morphisms are isomorphisms of such braided autoequivalences. We can truncate this to get an ordinary group EqBr(C1 ). According to [ENO09, §5.4],
there is a monoidal functor Θ : Pic(C1 ) → EqBr(C1 ) which is an equivalence in case the braided
fusion category C1 is non-degenerate.
Remark B.3. A morphism of 1-groups Γ → EqBr(C1 ) is equivalent to a braided action of Γ on C1 .
Note that a morphism Γ → Pic(C1 ) gives us a morphism Γ → EqBr(C1 ) using the monoidal functor
Θ. In other words we get a braided action of Γ on C1 . This is precisely the braided action comingfrom the braided crossed category structure defined by the morphism Γ → Pic(C1 ).
Now let us consider a faithfully graded central braided Γ-crossed category D with trivial component M(A, θ) (where (A, θ) is a metric group), corresponding to a morphism Γ → Pic(M(A, θ)).
Centrality means that the induced map Γ → O(A, θ) ∼= EqBr(M(A, θ)) ∼= Pic(M(A, θ)) is trivial. Here O(A, θ) is the group of automorphisms of A preserving the quadratic form θ. Hence inthis case, all the graded components Cγ are equivalent to the trivial module category M(A, θ) as
M(A, θ)-module categories.
Remark B.4. In particular, this implies that D is pointed.
23

Let us equip D with the positive spherical structure (see [DGNO, Ex. 2.26]). Since the identity
component M(A, θ) is a non-degenerate braided category and the grading on D is faithful, theequivariantization DΓ is also a non-degenerate braided category (by [DGNO, Proposition 4.56(ii)]).
Since D is equipped with a (positive) spherical structure, DΓ also has the structure of a modular
category.
Proposition B.5. Let D be faithfully graded braided Γ-crossed category with identity component
M(A, θ), where Γ is a finite group and (A, θ) is a metric group. Let C be the braided monoidal
category DΓ . Let L ⊂ A be a Γ-invariant isotropic subgroup and form the subquotient metricgroup (L⊥ /L, θ0 ). Then there exists a (faithfully graded) braided crossed category whose identity
component is equivalent to M(L⊥ /L, θ0 ) and whose equivariantization is equivalent to the braided
monoidal category C.
Proof. The full subcategory of M(A, θ) generated by the simple objects corresponding to L ⊂ Awith the restriction of associativity and the braiding is the symmetric monoidal category M(L, 1 =θ|L ) ∼= VecL and is equipped with a fiber functor VecL → Vec. The braided action of Γ on
M(A, θ) induces one on VecL and after equivariantization, we get a symmetric monoidal category
E := VecΓL ⊂ DΓ = C equipped with the fiber functor E = VecΓL → VecL → Vec. The centralizerof E in C is the subcategory E 0 = M(L⊥ , θ|L⊥ )Γ of M(A, θ)Γ . The de-equivariantization (see[DGNO, §4.4.7,8]) C E Vec is a faithfully graded braided crossed category with identity component
E 0 E Vec ∼= M(L⊥ /L, θ) and whose equivariantization is equivalent to C. (See [DGNO, Prop.
4.56].)∗
Remark B.6. Note that in the situation above, VecL ∼= Rep(L∗ ), where L∗ = Hom(L, Ql ) is the
Pontryagin dual. The automorphism group Γ0 of the fiber functor E → Vec is an extension ofgroups 0 → L∗ → Γ0 → Γ → 0. In particular, in case both Γ, A are p-groups, it follows that Γ0 is
one too.

C

Appendix: The class C±
p of modular categories

In this appendix, we define and study some properties of the class C±
p of modular categories. We
will follow [DGNO2], but we give an alternative equivalent definition. In the following definition, we
use the notions of metric groups and Witt-equivalence that we discussed in Appendix A.3 above.
We also use the notions of central braided crossed categories and equivariantization, as well asthat of the modular category M(A, θ) associated to metric group (A, θ) defined in Appendix B
above.
∼ Γ
Definition C.1. (i) We say that a modular category C belongs to the class C+
p if C = D asa modular category, where D is a (faithfully graded and positive spherical) central braided
Γ-crossed category with identity component M(A, θ) such that Γ is a finite p-group and (A, θ)
is a metric p-group Witt-equivalent to the trivial metric group.
∼ Γ(ii) We say that a modular category C belongs to the class C−
p if C = D as a modular category,where D is a (faithfully graded and positive spherical) central braided Γ-crossed category withidentity component M(A, θ) such that Γ is a finite p-group and (A, θ) is a metric p-group
Witt-equivalent to the anisotropic metric group (Fp2 , ζ N m(·) ).
+−(iii) The class C±
p is defined to be the union Cp ∪ Cp .

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By Proposition B.5 we have the following:
Proposition C.2. (i) A modular category C ∈ C+
p if and only if it can be realized as the center of apointed fusion category (equipped with the positive spherical structure) whose group of isomorphismclasses of simple objects is a finite p-group.
Γ(ii) A modular category C ∈ C−
p if and only if it can be realized as D where Γ is a p-group and D
is a faithfully graded central braided Γ-crossed category (equipped with positive spherical structure)with identity component Manis= M(Fp2 , ζ N m(·) ).
p
Proof. A pointed fusion category (with positive spherical structure) can be equivalently thought ofas a faithfully graded (positive spherical) central braided crossed category with identity component
Vec and that the equivariantization of such a category is equivalent to its Drinfeld center. (See[DGNO, §4.4.10].) Both the statements now follow from Proposition B.5 and Remark B.6.
We now state another equivalent characterization of the class C±
p of modular categories proved in[DGNO2]. We refer to [DGNO, §2.4.1-3 and §6] for the notions of Frobenius-Perron dimension,categorical dimension, Gauss sums and multiplicative central charge.
Proposition C.3. ([DGNO2]) A modular category C belongs to C±
p if and only if the following
three conditions are satisfied:(i) The Frobenius-Perron dimension of C equals p2k for some k ∈ Z+ .
(ii) The categorical dimensions of all simple objects of C are positive integers.
(iii) The multiplicative central charge of C is ±1.
If C satisfies the conditions above then the multiplicative central charge of C is 1 iff C ∈ C+
p and themultiplicative central charge of C is −1 iff C ∈ C−.
p
∼ Γ
Proof. Suppose that C ∈ C±
p . Hence by Proposition C.2, we have C = D where D is some faithfully
graded positive spherical central braided Γ-crossed category with trivial component M(A, θ) whereN m(·) ) (in case C ∈ C− ) and Γ is a
the metric group (A, θ) is either (0, 1) (in case C ∈ C+pp ) or (Fp2 , ζfinite p-group. The proof is very similar to the proofs of Propositions 2.17 and 2.21. We have that
FPdim(C) = FPdim(DΓ ) = |Γ| · FPdim(D) = |Γ|2 · FPdim(M(A, θ)) = |Γ|2 · |A| which is an evenpower of p since |A| is either 1 or p2 and Γ is a finite p-group. The categorical dimensions of all simple
objects of D are 1 since D is pointed and equipped with the positive spherical structure. Hence it
follows that the categorical dimensions of all objects of DΓ ∼= C must be positive integers. Finally,by [DGNO, Thm. 6.16] the Gauss sums of C are given by τ ± (C) = τ ± (DΓ ) = |Γ| · τ ± (M(A, θ))(|Γ|= |Γ| · τ (A, θ) =−p · |Γ|±if C ∈ C+
p
if C ∈ C−
p.

−
Hence we see that the multiplicative central charge of C is 1 if C ∈ C+
p and -1 if C ∈ Cp .
−Remark C.4. The last statement implies that the classes C+
p and Cp are disjoint.

We have completed the proof of the “only if ” part of the proposition. We refer to [DGNO2] for the“if ” part.

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The class C−
p may also be characterized as follows:
Proposition C.5. ([DGNO2]) A modular category C ∈ C−
p if and only if the modular categoryanis+
C  Mp ∈ Cp .

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