representation of a group and its center (I asked the following question at StackExchange but received no answer.)
Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \otimes V^*$ for each (representative of isomorphism class of) simple object of $C$.
I want to show that $D=\mathrm{Rep}(G/Z(G))$.
It seems that this problem is claimed in Tensor Categories By Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik without a complete proof.
I appreciate any help.
 A: I'll work over $\mathbb{C}$ below for simplicity, although it can be replaced by an algebraically closed field of characteristic $0$. 
Assuming that "fusion subcategory generated by" means what I think it means (tensor products, direct sums, direct summands), here is a general observation.

Lemma: The fusion subcategory of $\text{Rep}(G)$ generated by some reps $V_i$ is $\text{Rep}(G/N)$ where $N$ is the intersection of the kernels of $G$ acting on each $V_i$. 

This follows in turn from the standard fact that $\text{Rep}(H)$ is generated as a fusion category (in the above sense) by any faithful representation of $H$; see, for example, this MO question. From here it suffices to show that the intersection of the kernels of $G$ acting on $V \otimes V^{\ast}$ for each irrep $V$ is $Z(G)$.
Recall that the regular representation $\mathbb{C}[G]$ decomposes, as a representation of $G \times G$ (acting on the left and right), as $\bigoplus_V V \boxtimes V^{\ast}$ where $\boxtimes$ denotes the external tensor product. If we restrict to the diagonal copy of $G$ in $G \times G$, we get that the permutation representation coming from $G$ acting on itself by conjugation decomposes as $\bigoplus_V V \otimes V^{\ast}$ where $\otimes$ is now the ordinary tensor product of representations. The kernel of this permutation representation is clearly $Z(G)$, as desired. 
