Notation for $G$-sheaf

Question

Let $G$ be a finite group, regarded as an algebraic group, and $X$ be a $G$-variety (as in $X$ is a variety with an action by $G$). Then according to Bridgeland, King, and Reid in The McKay correspondence as an equivalence of derived categories, a $G$-sheaf is a quasi-coherent sheaf $E$ of $\mathcal{O}_X$-modules, together with a collection of isomorphisms $\lambda_g^E:E \to g^*E$ satisfying the appropriate conditions (I am somewhat familiar with these already).

However later in the section they state the following (with some paraphrasing):

If $G$ acts trivially on $X$, then any $G$-sheaf decomposes as a direct sum $$E = \bigoplus E_i \otimes \rho_i,$$ where $\{\rho_0,...,\rho_k\}$ are the irreducible representations of $G$. The sheaves $E_i$ are just ordinary sheaves on $X$.

My question is what does the notation $E_i \otimes \rho_i$ mean exactly? I know that if $G$ acts trivially then an equivariant structure is essentially a choice of representation $G \to \operatorname{Aut}_{\mathcal{O}_X} (E)$, so I believe that $G$-sheaves would decompose in this way, but the notation makes no sense to me, as $\rho_i$ are (I think?) representations of $G$ on vector spaces.

As a follow-up (and this may be trivial after my first question), how is the fact that $[-]^G$ (taking invariants of a $G$-sheaf $E$) and $-\otimes \rho_0$ (letting $G$ act trivially on $E$) adjoint functors?

Thanks in advance.

Answer 1

Just a guess. A representation $\rho$ of $G$ is a $G$-equivariant sheaf on a point. Then we can pull $\rho$ back along $a:X \to pt$ to get a $G$-equivariant sheaf on $X$. This sheaf won't necessarily be quasicoherent, but tensoring with a quasicoherent sheaf $E$ should make it so, by extension of scalars from $\mathbb{k}_X$ to $\mathcal{O}_X$. Here $\mathbb{k}$ is the (I assume) algebraically closed ground field, and $\mathbb{k}_X$ is the constant sheaf $a^*(\mathbb{k})$.

For your second question, the functors $- \otimes \rho_0$ and $[-]^G$ provide adjunctions between the category $QC_G(X)$ of $G$-sheaves on $X$ and $QC(X)$, the category of quasicoherent sheaves on $X$. Note that $QC_G(X)$ is just the category of sheaves over the sheaf of rings that locally looks like $\mathcal{O}_X(U)[G]$, the group ring of $G$. The adjunction then becomes the adjunction corresponding to the extension of scalars $\mathcal{O}_X \to \mathcal{O}_X[G]$.

Answer 2

Just to make the answer of @leibnewtz explicit --- since the action of $G$ is trivial, $g^*E = E$ (canonical isomorphism) for any quasicoherent sheaf on $X$, hence the compositions $$ E_i \otimes \rho_i \to E_i \otimes \rho_i = g^*(E_i \otimes \rho_i) $$ (where the first morphism is $1_{E_i}$ tensored with the $g$-action $\rho_i \to \rho_i$ and the second arrow is the canonical isomorphism) provide the required equivariant structure on $E_i \otimes \rho_i$.