Modular categories associated with unipotent groups

Abstract

Let $G$ be a unipotent algebraic group over an algebraically closed field $\mathtt{k }$ of characteristic $p>0$ and let $l\ne p$ be another prime. Let $e$ be a minimal idempotent in $\mathcal{D }_G(G)$, the $\overline{\mathbb{Q }}_l$-linear triangulated braided monoidal category of $G$-equivariant (for the conjugation action) $\overline{\mathbb{Q }}_l$-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 $\mathcal{M }_{G,e}$. In this paper, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class $\mathfrak{C }_p^{\pm }$.

Appendices

Appendix: Skew-symmetric biextensions

In this appendix, we prove some general results about connected commutative (perfect) unipotent groups and skew-symmetric biextensions (see [2, Appendix A.6-11],[5, §2] for more on skew-symmetric biextensions). Since it is convenient to talk about skew-symmetric biextensions in the setting of perfect schemes, we will work in the category of perfect unipotent groups over $\mathtt{k }$, even though many of the results and arguments also work in the category of unipotent groups. Let $\mathfrak{cpu }$ denote the category of commutative perfect unipotent groups over $\mathtt{k }$ and let $\mathfrak{cpu }^\circ$ be the full subcategory of connected commutative perfect unipotent groups. The category $\mathfrak{cpu }$ is abelian and $\mathfrak{cpu }^\circ$ is an exact subcategory. Serre duality defines an involution $\mathfrak{cpu }^\circ \ni V\mapsto V^*\in \mathfrak{cpu }^\circ$ which is an exact anti-autoequivalence of $\mathfrak{cpu }^\circ$. For $V\in \mathfrak{cpu }^\circ$, we think of $V^*\in \mathfrak{cpu }^\circ$ as the space parametrizing the multiplicative local systems on $V$ (after fixing an embedding $\mathbb{Q }_p/\mathbb Z _p\hookrightarrow \overline{\mathbb{Q }}_l^*$). We refer to [2, Appendix A] for more on the notions of Serre duality and biextensions.

1.1 Group actions on a connected commutative unipotent group

Let $\Gamma$ be a group. Let $V\in \mathfrak{cpu }$ (resp. $\in \mathfrak{cpu }^\circ$) be equipped with an action of $\Gamma$. Let $\Gamma -\mathfrak{cpu }$ (resp. $\Gamma -\mathfrak{cpu }^\circ$) denote the category whose objects are $V$ (resp. connected $V$) equipped with a $\Gamma$-action and whose morphisms are $\Gamma$-equivariant maps of commutative perfect unipotent groups. Then, $\Gamma -\mathfrak{cpu }$ is an abelian category and $\Gamma -\mathfrak{cpu }^\circ$ 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 $\mathtt{k }$ and $\Gamma$ acts by linear transformations.

Definition 4.1

An object $V\in \Gamma -\mathfrak{cpu }^\circ$ is called simple if it is nonzero and has no proper connected $\Gamma$-invariant subgroups.

It is easy to see that:

Lemma 4.2

Every $V\in \Gamma -\mathfrak{cpu }^\circ$ has a filtration $\{0\}=V_0\subset V_1\subset \cdots \subset V_{n-1}\subset V_n=V$ by connected $\Gamma$-invariant subgroups such that each successive quotient $V_{i+1}/V_i$ is simple.

We see that a morphism in $\Gamma$-$\mathfrak{cpu }^\circ$ to a simple object is either zero or surjective and that a morphism from a simple object is either zero or has finite kernel. Hence, we have the following:

Lemma 4.3

Let $V,W$ be simple objects in $\Gamma -\mathfrak{cpu }^\circ$ and $f:V\rightarrow W$ a morphism in $\Gamma -\mathfrak{cpu }^\circ$. Then, $f$ is either zero or an isogeny.

Lemma 4.4

Let $V\in \Gamma -\mathfrak{cpu }^\circ$ be simple. Then, $p\cdot V=0$, i.e. as a connected commutative perfect unipotent group, $V\cong \mathbb{G }_{a,perf}^r$ for some positive integer $r$.

Proof

Consider the morphism $p\cdot \text{ Id }_V: V\rightarrow V$ in $\Gamma -\mathfrak{cpu }^\circ .$ Since some power of $p$ annihilates $V$, the map $p\cdot \text{ Id }_V$ is not an isogeny. Hence, it must be zero. $\square$

Lemma 4.5

Let $V\in \Gamma -\mathfrak{cpu }^\circ$ be simple. Let ${\gamma }\in \Gamma$ be a central element whose order is a power of $p$. Then, ${\gamma }$ acts trivially on $V$.

Proof

Suppose ${\gamma }^{p^n}=1$. Since ${\gamma }\in \Gamma$ is central, ${\gamma }:V\rightarrow V$ is a morphism in $\Gamma -\mathfrak{cpu }^\circ$ and so is $({\gamma }-\text{ Id })$. Since V is simple, $p\cdot V=0$. Hence, $({\gamma }-\text{ Id })^{p^n}={\gamma }^{p^n}-\text{ Id }=0.$ Hence, ${\gamma }-\text{ Id }$ cannot be an isogeny. Hence, ${\gamma }=\text{ Id }$ since $V$ is simple. $\square$

Remark 4.6

If $\Gamma$ is a (possibly infinite) $p$-group and $V\in \Gamma -\mathfrak{cpu }^\circ$ is simple, then the center $Z(\Gamma )$ acts trivially on $V$.

Proposition 4.7

If $\Gamma$ is a (possibly infinite) nilpotent $p$-group, then there is a unique simple object in $\Gamma -\mathfrak{cpu }^\circ$, namely $\mathbb{G }_{a,perf}$ equipped with the trivial action of $\Gamma$.

Proof

We prove this by induction on the length of the ascending central series of $\Gamma$. By the remark, the proposition holds for commutative $p$-groups $\Gamma$. ($Z(\Gamma )=\Gamma$ must act trivially on a simple $V$, and consequently $V$ must be one dimensional.) Now suppose $\Gamma$ is such that the proposition holds true for all $p$-groups of lower nilpotence class. Let $V\in \Gamma -\mathfrak{cpu }^\circ$ be simple. By the remark above, $Z(\Gamma )$ acts trivially on $V$. Hence, the $\Gamma$-action on $V$ comes from an action of the quotient $\Gamma /Z(\Gamma )$. $V$ is simple for this action. Hence, by our inductive hypothesis, $V\cong \mathbb{G }_{a,perf}$ with trivial action of $\Gamma /Z(\Gamma )$. Hence, the action of $\Gamma$ is trivial as well. $\square$

Corollary 4.8

Let $\Gamma$ be a nilpotent $p$-group. Let $V\in \Gamma -\mathfrak{cpu }^\circ$ with $\dim (V)=n$. Then, there is a filtration $\{0\}=V_0\subset V_1\subset \cdots \subset V_{n-1}\subset V_n=V$ by connected $\Gamma$-invariant subgroups such that each $V_{i+1}/V_i\cong \mathbb{G }_a$ with $\Gamma$ acting trivially.

1.2 Invariant isotropic subgroups

As before, let $\Gamma$ be a group and let $V\in \Gamma -\mathfrak{cpu }^\circ$. Let $\phi :V\rightarrow V^*$ be a $\Gamma$-equivariant skew-symmetric biextension (see [2, Appendix A.10]). We can think of such a skew-symmetric biextension as a bimultiplicative local system on $V\times V$ whose restriction to the diagonal is trivial. Let $W\subset V$ be a connected subgroup. Let $W^\perp (\in \mathfrak{cpu })$ be the kernel of the composition $V\stackrel{\phi }{\rightarrow } V^*\twoheadrightarrow W^*$. We say that $W$ is isotropic if $W\subset W^{\perp }$. In this section, we prove that if $\dim (V)\ge 2$ and if $\Gamma$ is a nilpotent $p$-group, then there exists a non-trivial $\Gamma$-invariant isotropic subgroup.

Lemma 4.9

Let $n$ be the smallest integer such that $p^n\cdot V=0$. Let $r$ be an integer such that $\frac{n}{2}\le r \le n-1.$ Then, $p^r\cdot V$ is a proper $\Gamma$-invariant isotropic subgroup.

Proof

It is clear that $p^r\cdot V$ is a proper $\Gamma$-invariant subgroup for each positive integer $r\le n-1$. For $r\ge \frac{n}{2}$, $p^r\cdot (p^r\cdot V)=0$, hence $p^r\cdot (p^r\cdot V)^*=0$. Hence, we see that the composition

$$\begin{aligned} p^r\cdot V\subset V\rightarrow V^*\twoheadrightarrow (p^r\cdot V)^* \end{aligned}$$

is zero. ($p^r\cdot x\mapsto p^r\cdot \phi (x)\mapsto p^r\cdot \phi (x)|_{p^r\cdot V}=0$ in $(p^r\cdot V)^*$ which is annihilated by $p^r$). $\square$

Proposition 4.10

Let $\Gamma$ be a nilpotent $p$-group, $V\in \mathfrak{cpu }^\circ$ and let $\phi :V\rightarrow V^*$ be a $\Gamma$-equivariant skew-symmetric biextension. If $\dim (V)\ge 2$, then there exists a nonzero $\Gamma$-invariant isotropic subgroup in $V$.

Proof

If $p\cdot V\ne 0$, then the previous lemma gives us a proper $\Gamma$-invariant isotropic subgroup in $V$. Hence, let us assume $p\cdot V=0$, in which case the previous lemma does not tell us anything. Let us use Corollary 4.8. Since $\dim (V)\ge 2$, there exists a two-dimensional $\Gamma$-invariant subgroup $V_2\subset V$. Since $p\cdot V_2 =0, V_2\cong \mathbb{G }_{a,perf}^2.$ Consider the induced $\Gamma$-equivariant skew-symmetric biextension of $V_2$. Let $V_1\subset V_2$ be a one-dimensional $\Gamma$-fixed subgroup guaranteed by Corollary 4.8. If $V_1$ is isotropic in $V_2$, then it is also isotropic in $V$ and we are done. If not, then its perpendicular $V_1^{\perp _{V_2}}$ in $V_2$ must be one-dimensional and $V_1+(V_1^{\perp _{V_2}})^\circ =V_2.$ Since $V_1\subset V_2$ is $\Gamma$-stable, $(V_1^{\perp _{V_2}})^\circ$ must also be $\Gamma$-invariant and since it is one dimensional, the action of $\Gamma$ on $(V_1^{\perp _{V_2}})^\circ$ must be trivial and consequently action of $\Gamma$ on $V_2$ must be trivial. Now, by [2, Lemma A.29], $V_2$ has a one-dimensional isotropic subgroup $W$. Since the action of $\Gamma$ on $V_2$ is trivial, $W$ is $\Gamma$-stable. $\square$

Corollary 4.11

Let $\Gamma$ be a nilpotent $p$-group and let $V\in \Gamma -\mathfrak{cpu }^\circ$ be equipped with a $\Gamma$-equivariant skew-symmetric biextension $V\stackrel{\phi }{\rightarrow }V^*$. Then, there exists a $\Gamma$-invariant maximal isotropic subgroup $W\subset V$ and for any such $W$, $\dim (W^{\perp })\le \dim (W)+1$ and $W$ is also maximal among all (not necessarily $\Gamma$-invariant) isotropic subgroups of $V$.

Proof

Clearly maximal $\Gamma$-invariant isotropic subgroups exist. Let $W$ be any $\Gamma$-invariant isotropic subgroup of $V$. Then, $(W^\perp )^\circ$ is also $\Gamma$-stable and we have the induced $\Gamma$-equivariant skew-symmetric biextension of $(W^{\perp })^\circ /W$. $\Gamma$-invariant isotropic subgroups of $V$ containing $W$ are in a natural bijective correspondence with $\Gamma$-invariant isotropic subgroups of $(W^{\perp })^\circ /W$. If $W$ is a maximal $\Gamma$-invariant isotropic subgroup, then $(W^{\perp })^\circ /W$ has no nonzero $\Gamma$-invariant isotropic subgroups. Hence, this space has dimension less than or equal to 1. $\square$

1.3 Metric groups associated with skew-symmetric biextensions

Definition 4.12

Let $A$ be a finite abelian group and let $\theta : A\rightarrow \overline{\mathbb{Q }}_l^*$ be a function. We say that $\theta$ is a quadratic form if $\theta (a)=\theta (-a)$ and if the symmetric function $b(a,b):=\frac{\theta (a+b)}{\theta (a)\theta (b)}$ is a bicharacter $b:A\times A\rightarrow \overline{\mathbb{Q }}_l^*$. We say that $\theta$ is non-degenerate if the bicharacter $b$ is non-degenerate. A pre-metric group is a pair $(A,\theta )$ where $A$ is a finite abelian group and $\theta :A\rightarrow \overline{\mathbb{Q }}_l^*$ is a quadratic form. If the quadratic form is non-degenerate, we say that $(A,\theta )$ is a metric group.

Given a metric group $(A,\theta )$, we have the associated non-degenerate bicharacter $b$. A subgroup $B\subset A$ is said to be isotropic if $\theta |_B=1$. In this case, $B\subset B^\perp$. For an isotropic subgroup $B$, the subquotient $(B^\perp /B, \theta ^{\prime })$ is also a metric group, where $\theta ^{\prime }$ is the quadratic form on $B^\perp /B$ induced by $\theta$. A metric group is said to be anisotropic if it has no non-trivial isotropic subgroups. If $L$ is a maximal isotropic subgroup of a metric group $(A,\theta )$, then the subquotient $(L^\perp /L,\theta ^{\prime })$ is an anisotropic metric group whose isomorphism class is independent of the choice of maximal isotropic subgroup $L$. We say that the metric group $(A,\theta )$ is Witt-equivalent to the anisotropic metric group $(L^\perp /L,\theta ^{\prime })$.

Let us now give two examples of anisotropic metric $p$-groups which are of particular interest in the theory of character sheaves on unipotent groups (see Theorem 4.13 below):

• The trivial metric group $(0, 1)$.

• $(\mathbb F _{p^2}, \zeta ^{{Nm}(\cdot )})$ where $Nm: \mathbb F _{p^2}\rightarrow \mathbb{F }_p$ is the norm map and $\zeta \in \overline{\mathbb{Q }}_l^*$ is a primitive $p$th root of unity.

Now, let $V$ be a connected commutative perfect unipotent group and let $\phi :V\rightarrow V^*$ be a skew-symmetric isogeny. According to [2, Appendix A.10], [5, §2.3], to such a skew-symmetric isogeny, we can associate a metric group $(A,\theta )$. Here, $A$ is the kernel of $\phi$. Then, we can define the associated non-degenerate quadratic form $\theta :A\rightarrow \overline{\mathbb{Q }}_l^*$ (see [2, Appendix A.10], [5, §2.3]). The following theorem is proved in [5]:

Theorem 4.13

([5]) The metric group $(A,\theta )$ associated with the skew-symmetric isogeny $\phi :V\rightarrow V^*$ is Witt-equivalent to the trivial anisotropic metric group if $\dim (V)$ is even and is Witt-equivalent to the anisotropic metric group $(\mathbb F _{p^2}, \zeta ^{Nm(\cdot )})$ if $\dim (V)$ is odd.

Appendix: The modular category associated with a metric group and braided crossed categories

1.1 The modular category associated with a metric group

Given a pre-metric group $(A,\theta )$ (see Definition 4.12), there exists a corresponding pointed braided category $\mathcal{M }(A,\theta )$ whose group of isomorphism classes of simple objects is isomorphic to $A$ and the braiding on the square of a simple object is given by $\theta$. This determines the braided monoidal category up to equivalence. We equip this braided pointed category with the positive spherical structure. With this structure, the twist (an automorphism of the identity functor on the pointed braided category) is given by $\theta$. We denote this pre-modular category by $\mathcal{M }(A,\theta )$ (see [7, §2.11.4-5 and Appendix D] for more details). If $(A,\theta )$ is a metric group, $\mathcal{M }(A,\theta )$ is a modular category.

The modular category defined by the metric group $(\mathbb F _{p^2},\zeta ^{Nm(\cdot )})$ is of particular interest and we denote it by $\mathcal{M }_p^{anis}$.

1.2 Braided crossed categories and equivariantization

Let $(A,\theta )$ be a metric group and let $\Gamma$ be a finite group. Let $\mathcal{D }$ be a faithfully graded braided $\Gamma$-crossed category with identity component $\mathcal{M }(A,\theta )$. This means that we have the following structure on $\mathcal{D }$:

• A grading $\mathcal{D }=\bigoplus \limits _{{\gamma }\in \Gamma }\mathcal{C }_{\gamma }$ where each $\mathcal{C }_{\gamma }$ is nonzero and $\mathcal{C }_1\cong \mathcal{M }(A,\theta )$.

• A monoidal action of $\Gamma$ on $\mathcal{D }$ such that ${\gamma _1}(\mathcal{C }_{\gamma _2})\subset \mathcal{C }_{{\gamma _1}{\gamma _2}{\gamma _1}^{-1}}$ for each ${\gamma _1},{\gamma _2}\in \Gamma$.

• For ${\gamma }\in \Gamma , X\in \mathcal{C }_{\gamma }$ and $Y\in \mathcal{D }$ isomorphisms

  $$\begin{aligned} c_{X,Y}:X\otimes Y\stackrel{\cong }{\rightarrow } {\gamma }(Y)\otimes X \end{aligned}$$

  functorial in $X,Y$ and satisfying certain compatibility conditions.

For a braided $\Gamma$-crossed category $\mathcal{D }$, the equivariantization $\mathcal{D }^\Gamma$ has the structure of a braided monoidal category. We refer to [7, §4.4.3] for a precise definition and properties of braided crossed categories and related concepts.

In particular, the monoidal action of $\Gamma$ on $\mathcal{D }$ restricts to an action of $\Gamma$ on the braided monoidal category $\mathcal{M }(A,\theta )$ by braided autoequivalences which in turn determines an action of $\Gamma$ on the metric group $(A,\theta )$.

Definition 5.1

We say that such a braided $\Gamma$-crossed category as above is central if the induced action of $\Gamma$ on the metric group $(A,\theta )$ is trivial.

Let $\mathcal{D }=\bigoplus \limits _{{\gamma }\in \Gamma }\mathcal{C }_{\gamma }$ be a braided $\Gamma$-crossed category. Each $\mathcal{C }_{\gamma }$ is a $\mathcal{C }_1$-module category. In fact, by [11, Thm. 6.1], it is an invertible (under tensor product of module categories, see op. cit. §3) $\mathcal{C }_1$-module category. For a braided fusion category $\mathcal{C }_1$, let $\underline{\underline{\text{ Pic }}}(\mathcal{C }_1)$ denote the categorical 2-group whose objects are invertible $\mathcal{C }_1$-module categories under tensor product, 1-morphisms are equivalences of $\mathcal{C }_1$-module categories and 2-morphisms are isomorphisms of such equivalences. As proved in op. cit., the assignment $\gamma \mapsto \mathcal{C }_{\gamma }$ along with the braided $\Gamma$-crossed structure of $\mathcal{D }$ defines a morphism of 2-groups $\Gamma \rightarrow \underline{\underline{\text{ Pic }}}(\mathcal{C }_1)$. In fact, the following result is proved in op. cit. §7.8

Theorem 5.2

([11, Thm. 7.12]) Equivalence classes of faithfully graded braided $\Gamma$-crossed categories with trivial component $\mathcal{C }_1$ are in bijection with morphisms of categorical 2-groups $\Gamma \rightarrow \underline{\underline{\text{ Pic }}}(\mathcal{C }_1)$.

We can truncate (by forgetting the highest level morphisms and identifying isomorphic lower ones) the 2-group $\underline{\underline{\text{ Pic }}}(\mathcal{C }_1)$ to a 1-group $\underline{\text{ Pic }}(\mathcal{C }_1)$ and further to an ordinary group $\text{ Pic }(\mathcal{C }_1)$ of equivalence classes of invertible $\mathcal{C }_1$-module categories. Let ${\underline{\text{ EqBr }}}(\mathcal{C }_1)$ denote the 1-group whose objects are braided autoequivalences of $\mathcal{C }_1$ and whose morphisms are isomorphisms of such braided autoequivalences. We can truncate this to get an ordinary group ${\text{ EqBr }}(\mathcal{C }_1)$. According to [11, §5.4], there is a monoidal functor $\Theta :\underline{\text{ Pic }}(\mathcal{C }_1)\rightarrow {\underline{\text{ EqBr }}}(\mathcal{C }_1)$ which is an equivalence in case the braided fusion category $\mathcal{C }_1$ is non-degenerate.

Remark 5.3

A morphism of 1-groups $\Gamma \rightarrow {\underline{\text{ EqBr }}}(\mathcal{C }_1)$ is equivalent to a braided action of $\Gamma$ on $\mathcal{C }_1$. Note that a morphism $\Gamma \rightarrow \underline{\underline{\text{ Pic }}}(\mathcal{C }_1)$ gives us a morphism $\Gamma \rightarrow {\underline{\text{ EqBr }}}(\mathcal{C }_1)$ using the monoidal functor $\Theta$. In other words, we get a braided action of $\Gamma$ on $\mathcal{C }_1$. This is precisely the braided action coming from the braided crossed category structure defined by the morphism $\Gamma \rightarrow \underline{\underline{\text{ Pic }}}(\mathcal{C }_1)$.

Now, let us consider a faithfully graded central braided $\Gamma$-crossed category $\mathcal{D }$ with trivial component $\mathcal{M }(A,\theta )$ (where $(A,\theta )$ is a metric group), corresponding to a morphism $\Gamma \rightarrow \underline{\underline{\text{ Pic }}}(\mathcal{M }(A,\theta ))$. Centrality means that the induced map $\Gamma \rightarrow O(A,\theta )\cong {\text{ EqBr }}(\mathcal{M }(A,\theta ))\cong \text{ Pic }(\mathcal{M }(A,\theta ))$ is trivial. Here, $O(A,\theta )$ is the group of automorphisms of $A$ preserving the quadratic form $\theta$. Hence, in this case, all the graded components $\mathcal{C }_{\gamma }$ are equivalent to the trivial module category $\mathcal{M }(A,\theta )$ as $\mathcal{M }(A,\theta )$-module categories.

Remark 5.4

In particular, this implies that $\mathcal{D }$ is pointed.

Let us equip $\mathcal{D }$ with the positive spherical structure (see [7, Ex. 2.26]). Since the identity component $\mathcal{M }(A,\theta )$ is a non-degenerate braided category and the grading on $\mathcal{D }$ is faithful, the equivariantization $\mathcal{D }^\Gamma$ is also a non-degenerate braided category (by [7, Proposition 4.56(ii)]). Since $\mathcal{D }$ is equipped with a (positive) spherical structure, $\mathcal{D }^{\Gamma }$ also has the structure of a modular category.

Proposition 5.5

Let $\mathcal{D }$ be faithfully graded braided $\Gamma$-crossed category with identity component $\mathcal{M }(A,\theta )$, where $\Gamma$ is a finite group and $(A,\theta )$ is a metric group. Let $\mathcal{C }$ be the braided monoidal category $\mathcal{D }^\Gamma$. Let $L\subset A$ be a $\Gamma$-invariant isotropic subgroup and form the subquotient metric group $(L^{\perp }/L,\theta ^{\prime })$. Then, there exists a (faithfully graded) braided crossed category whose identity component is equivalent to $\mathcal{M }(L^{\perp }/L,\theta ^{\prime })$ and whose equivariantization is equivalent to the braided monoidal category $\mathcal{C }$.

Proof

The full subcategory of $\mathcal{M }(A,\theta )$ generated by the simple objects corresponding to $L\subset A$ with the restriction of associativity and the braiding is the symmetric monoidal category $\mathcal{M }(L,1=\theta |_L)\cong \text{ Vec }_L$ and is equipped with a fiber functor $\text{ Vec }_L\rightarrow \text{ Vec }$. The braided action of $\Gamma$ on $\mathcal{M }(A,\theta )$ induces one on $\text{ Vec }_L$ and after equivariantization, we get a symmetric monoidal category $\mathcal E :=\text{ Vec }_L^{\Gamma }\subset \mathcal{D }^\Gamma = \mathcal{C }$ equipped with the fiber functor $\mathcal{E }=\text{ Vec }_L^{\Gamma }\rightarrow \text{ Vec }_L\rightarrow \text{ Vec }$. The centralizer of $\mathcal{E }$ in $\mathcal{C }$ is the subcategory $\mathcal{E }^{\prime }=\mathcal{M }(L^{\perp },\theta |_{L^{\perp }})^{\Gamma }$ of $\mathcal{M }(A,\theta )^\Gamma$. The de-equivariantization (see [7, §4.4.7,8]) $\mathcal{C }\boxtimes _{\mathcal{E }}\text{ Vec }$ is a faithfully graded braided crossed category with identity component $\mathcal{E }^{\prime }\boxtimes _{\mathcal{E }}\text{ Vec }\cong \mathcal{M }(L^{\perp }/L,\theta )$ and whose equivariantization is equivalent to $\mathcal{C }$ (see [7, Prop. 4.56]). $\square$

Remark 5.6

Note that in the situation above, $\text{ Vec }_L\cong \text{ Rep }(L^*)$, where $L^*=\text{ Hom }(L,\overline{\mathbb{Q }}_l^*)$ is the Pontryagin dual. The automorphism group $\Gamma ^{\prime }$ of the fiber functor $\mathcal{E }\rightarrow \text{ Vec }$ is an extension of groups $0\rightarrow L^*\rightarrow \Gamma ^{\prime }\rightarrow \Gamma \rightarrow 0$. In particular, in case both $\Gamma$, $A$ are $p$-groups, it follows that $\Gamma ^{\prime }$ is one too.

Appendix: The class $\mathfrak{C }_p^{\pm }$ of modular categories

In this appendix, we define and study some properties of the class $\mathfrak{C }_p^{\pm }$ of modular categories. We will follow [8], 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 4.3 above. We also use the notions of central braided crossed categories and equivariantization, as well as that of the modular category $\mathcal{M }(A,\theta )$ associated with metric group $(A,\theta )$ defined in Appendix 5 above.

Definition 6.1

• (i) We say that a modular category $\mathcal{C }$ belongs to the class $\mathfrak{C }_p^{+}$ if $\mathcal{C }\cong \mathcal{D }^\Gamma$ as a modular category, where $\mathcal{D }$ is a (faithfully graded and positive spherical) central braided $\Gamma$-crossed category with identity component $\mathcal{M }(A,\theta )$ such that $\Gamma$ is a finite $p$-group and $(A,\theta )$ is a metric $p$-group Witt-equivalent to the trivial metric group.

• (ii) We say that a modular category $\mathcal{C }$ belongs to the class $\mathfrak{C }_p^{-}$ if $\mathcal{C }\cong \mathcal{D }^\Gamma$ as a modular category, where $\mathcal{D }$ is a (faithfully graded and positive spherical) central braided $\Gamma$-crossed category with identity component $\mathcal{M }(A,\theta )$ such that $\Gamma$ is a finite $p$-group and $(A,\theta )$ is a metric $p$-group Witt-equivalent to the anisotropic metric group $(\mathbb F _{p^2},\zeta ^{Nm(\cdot )})$.

• (iii) The class $\mathfrak{C }_p^{\pm }$ is defined to be the union $\mathfrak{C }_p^{+}\cup \mathfrak{C }_p^{-}$.

By Proposition 5.5, we have the following:

Proposition 6.2

• (i) A modular category $\mathcal{C }\in \mathfrak{C }_p^{+}$ if and only if it can be realized as the center of a pointed fusion category (equipped with the positive spherical structure) whose group of isomorphism classes of simple objects is a finite $p$-group.

• (ii) A modular category $\mathcal{C }\in \mathfrak{C }_p^{-}$ if and only if it can be realized as $\mathcal{D }^{\Gamma }$ where $\Gamma$ is a $p$-group and $\mathcal{D }$ is a faithfully graded central braided $\Gamma$-crossed category (equipped with positive spherical structure) with identity component $\mathcal{M }_p^{anis}=\mathcal{M }(\mathbb F _{p^2},\zeta ^{Nm(\cdot )})$.

Proof

A pointed fusion category (with positive spherical structure) can be equivalently thought of as a faithfully graded (positive spherical) central braided crossed category with identity component $\text{ Vec }$ and that the equivariantization of such a category is equivalent to its Drinfeld center (see [7, §4.4.10]). Both the statements now follow from Proposition 5.5 and Remark 5.6. $\square$

We now state another equivalent characterization of the class $\mathfrak{C }_p^{\pm }$ of modular categories proved in [8]. We refer to [7, §2.4.1-3 and §6] for the notions of Frobenius–Perron dimension, categorical dimension, Gauss sums and multiplicative central charge.

Proposition 6.3

([8]) A modular category $\mathcal{C }$ belongs to $\mathfrak{C }_p^{\pm }$ if and only if the following three conditions are satisfied:

• (i) The Frobenius–Perron dimension of $\mathcal{C }$ equals $p^{2k}$ for some $k\in \mathbb Z ^+$.

• (ii) The categorical dimensions of all simple objects of $\mathcal{C }$ are positive integers.

• (iii) The multiplicative central charge of $\mathcal{C }$ is $\pm 1$.

If $\mathcal{C }$ satisfies the conditions above then the multiplicative central charge of $\mathcal{C }$ is $1$ iff $\mathcal{C }\in \mathfrak{C }_p^{+}$ and the multiplicative central charge of $\mathcal{C }$ is $-1$ iff $\mathcal{C }\in \mathfrak{C }_p^{-}$.

Proof

Suppose that $\mathcal{C }\in \mathfrak{C }_p^{\pm }$. Hence, by Proposition 6.2, we have $\mathcal{C }\cong \mathcal{D }^\Gamma$ where $\mathcal{D }$ is some faithfully graded positive spherical central braided $\Gamma$-crossed category with trivial component $\mathcal{M }(A,\theta )$ where the metric group $(A,\theta )$ is either $(0,1)$ (in case $\mathcal{C }\in \mathfrak{C }_p^{+}$) or $(\mathbb F _{p^2},\zeta ^{Nm(\cdot )})$ (in case $\mathcal{C }\in \mathfrak{C }_p^{-}$) and $\Gamma$ is a finite $p$-group. The proof is very similar to the proofs of Propositions 2.17 and 2.21. We have that $\text{ FPdim }(\mathcal{C })=\text{ FPdim }(\mathcal{D }^\Gamma )=|\Gamma |\cdot \text{ FPdim }(\mathcal{D })=|\Gamma |^2\cdot \text{ FPdim }(\mathcal{M }(A,\theta ))=|\Gamma |^2\cdot |A|$ which is an even power of $p$ since $|A|$ is either 1 or $p^2$ and $\Gamma$ is a finite $p$-group. The categorical dimensions of all simple objects of $\mathcal{D }$ are 1 since $\mathcal{D }$ is pointed and equipped with the positive spherical structure. Hence, it follows that the categorical dimensions of all objects of $\mathcal{D }^\Gamma \cong \mathcal{C }$ must be positive integers. Finally, by [7, Thm. 6.16], the Gauss sums of $\mathcal{C }$ are given by $\tau ^{\pm }(\mathcal{C })=\tau ^{\pm }(\mathcal{D }^\Gamma )=|\Gamma |\cdot \tau ^{\pm }(\mathcal{M }(A,\theta ))$

$$\begin{aligned} =|\Gamma |\cdot \tau ^{\pm }(A,\theta )= \left\{ \begin{array}{ll} |\Gamma | &{} \text{ if } \mathcal{C }\in \mathfrak{C }_p^{+}\\ -p\cdot |\Gamma |&{} \text{ if } \mathcal{C }\in \mathfrak{C }_p^{-}. \end{array}\right. \end{aligned}$$

Hence, we see that the multiplicative central charge of $\mathcal{C }$ is 1 if $\mathcal{C }\in \mathfrak{C }_p^{+}$ and $-1$ if $\mathcal{C }\in \mathfrak{C }_p^{-}$. $\square$

Remark 6.4

The last statement implies that the classes $\mathfrak{C }_p^{+}$ and $\mathfrak{C }_p^{-}$ are disjoint.

We have completed the proof of the “only if” part of the proposition. We refer to [8] for the “if” part. $\square$

The class $\mathfrak{C }_p^{-}$ may also be characterized as follows:

Proposition 6.5

([8]) A modular category $\mathcal{C }\in \mathfrak{C }_p^{-}$ if and only if the modular category $\mathcal{C }\boxtimes \mathcal{M }_p^{anis} \in \mathfrak{C }_p^{+}$.