Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different.

## Premodular categories

In braided spherical fusion (=premodular) categories $\mathcal{C}$ with braiding $c_{-,-}$, we can define the double braiding: $$\gamma_{A,X}\colon A \otimes X \to A \otimes X$$ $$\gamma_{A,X} := c_{X,A} \circ c_{A,X}$$

Now we can define *encirclings*, where we braid an object around another and close the loop on the right hand side:

$$\Delta_{A,X}\colon A \to A$$ $$\Delta_{A,X} = \operatorname{tr}_X(\gamma_{A,X})$$

(The trace is defined via the pivotal structure.)

The **"killing" lemma** is quite well known.
Assume that $A$ is simple. Then:

$$\Omega := \bigoplus_{X \text{ simple}} d(X) X$$
$$\Delta_{A,\Omega} = \left\{ \begin{matrix} d(\Omega) \cdot 1_A && \qquad A \text{ is in the symmetric centre} \\ 0 && \qquad \text{otherwise} \end{matrix}\right.$$
$A$ is in the *symmetric centre* if braids trivially with every object. One says that $A$ is killed if it's not in the symmetric centre, hence the name of the lemma.

## $G$-crossed braided fusion categories

Now consider $G$-crossed braided fusion categories. If you think of fusion categories as some sort of categorified group (algebra), then a $G$-crossed braided fusion category is a categorified crossed module. It is, amongst other things, a $G$-graded fusion category:

$$\mathcal{C} = \bigoplus_{g \in G} \mathcal{C}_g$$

It also has a monoidal $G$-action:

$${}^g(-)\colon \mathcal{C}_h \to \mathcal{C}_{ghg^{-1}} \quad \forall g, h \in G$$

And it has a crossed braiding:

$$ c_{A,X}\colon A \otimes X \to {}^gX \otimes A\quad \forall A \in \mathcal{C}_g$$

This means that the double braiding is now *not an endomorphism anymore*:

$$ \gamma_{A,X}\colon A \otimes X \to {}^{ghg^{-1}}A \otimes {}^g X \quad\forall A \in \mathcal{C}_g, X \in \mathcal{C}_h$$

You can only take the trace on the right hand side, and thus define encircling, if $X \cong {}^gX$. The only canonical way I can think of is to demand that the grade of $A$ is trivial, i.e. $g = e$, since there is a coherence isomorphism $\epsilon_X\colon {}^eX \to X$. Then we can define:

$$\Omega_h := \bigoplus_{X \text{ simple } \in \mathcal{C}_h} d(X) X$$ $$\Delta_{A, \Omega_h} = \operatorname{tr}_{\Omega_h}\left((1_A \otimes \epsilon_{\Omega_h}) \circ \gamma_{A,\Omega_h} \right) \qquad \forall A \in \mathcal{C}_e$$

But now source and target aren't equal: $$\Delta_{A,\Omega_h}\colon A \to {}^hA$$

In some cases we have $A \cong {}^hA$, and then one can ask whether the endomorphism is 0 or not, which leads me to my question:

**Is there a similar lemma to the standard killing lemma in the $G$-crossed case?**