I am reading Achar's book about perverse sheaves. Now I am trying to solve the exercise 1.10.5 in this book (all varieties are assumed over $\mathbb{C}$ and sheaves are over a field $k$):
Define $$ \mathcal{N} =\left\{ \left[ \begin{matrix} x& y\\ z& -x\\ \end{matrix} \right] \mid x^2+yz=0 \right\} ,\widetilde{\mathcal{N} }=\left\{ \left( \left[ \begin{matrix} x& y\\ z& -x\\ \end{matrix} \right] ,\left[ u:v \right] \right) \in \mathcal{N} \times \mathbb{P} ^1\mid \left[ \begin{matrix} x& y\\ z& -x\\ \end{matrix} \right] \left[ \begin{array}{l} u\\ v\\ \end{array} \right] =0 \right\} , $$ $\mu:\widetilde{\mathcal{N} } \to \mathcal{N}$ is the natural projection and $i:0 \to \mathcal{N}$ is the closed inclusion. Compute $i^*\mu_*\underline{k}_{\widetilde{\mathcal{N} }}$ and $i^!\mu_*\underline{k}_{\widetilde{\mathcal{N} }}$, where all functors are considered in the derived category (hence they are the derived functors of the usual ones).