Consider the standard action of $G=\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{C}^2$, the resulting quotient $\mathbb{C}^2/G$ is a singular variety $Spec(\mathbb{C}[x^2,xy,y^2])$. It is known that there is a commutative diagram,
\begin{array} &Z&\xrightarrow{p} &\mathbb{C}^2&\\ \downarrow{f}&&\downarrow{\pi}\\ Y&\xrightarrow{\tau} &\mathbb{C}^2/G& \end{array} Here Z is the blow up of $\mathbb{C}^2$ at (0,0) and Y is the blow up of $\mathbb{C}^2/G$ at $(0,0)$.
One can check by hand that $\mathbb{C}[x,y]\otimes\rho_0$,$\mathbb{C}[x,y]\otimes\rho_1$ form a set of generators of the triangulated category $D^b_G(\mathbb{C}^2)$.(the $\rho$'s are irreducible representations of $G$.) By results of Bridgeland, King and Reid, we know that $Rp_*f^*:D^b(Y)\rightarrow D^b_G(\mathbb{C}^2)$ is an equivalence.
My question is: can one explicitly construct a object of $D^b(Y)$ whose image under the above functor is $\mathbb{C}[x,y]\otimes\rho_1$.(one can find similar thing for $\mathbb{C}[x,y]\otimes\rho_0$ easily, namely $\mathcal{O}_Y$, but the $\rho_1$ part is really causing problems in our case.)
Thanks for the help.