Computation of nearby cycles, monodromy action and action of $sl_{2}$ on $gr(\Psi)$ for Picard-Lefshetz family. Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{1}$ be a map that sends $(x,y)$ to $xy$. Let $U \hookrightarrow \mathbb{A}^{2}$ be the preimage $f^{-1}(\mathbb{A}^{2}\setminus \{ 0\})$ and $X:=f^{-1}(0)$. Consider the shifted constant sheaf $\mathbb{C}_{U}[2]$ on $U$. Let $\Psi(\mathbb{C}_{U}[2])$ denote the nearby cycles functor applied to $\mathbb{C}_{U}[2]$. It gives us a perverse sheaf on $X$.
My question is: how to compute this perverse sheaf on $X$ and the action of monodromy on it? Also if the action is unipotent then how to calculate the action of "Lefshetz" $sl_{2}$ on $\operatorname{gr}(\Psi(\mathbb{C}_{U}[2]))$?
Thanks!
 A: Sorry I am not an expert on perverse sheaves hence I am not sure I will completely answer the question.
Let $X_1$ and $X_2$ the two branches of $X$ ($x=0$ and $y=0$) and $O$ the point of intersection. In the following, everything will be on $X$, so when I write $\mathbb{C}_Z$ with $Z\subset X$ a closed subset, I mean $i_*\mathbb{C}$ the pushforward of the constant sheaf along the inclusion.
Finally, let $\omega_X$ be the dualizing sheaf on $X$ and $\Psi=\Psi(\mathbb{C}[2])$. (With the shifted nearby cycles functor).
We can resolve the constant sheaf on $X$ using the closed covering $X_1$ and $X_2$. We get a distinguished triangle in the derived category of $X$ :
$$\mathbb{C}_X\longrightarrow\mathbb{C}_{X_1}\oplus\mathbb{C}_{X_2}\longrightarrow\mathbb{C}_O\overset{+1}\longrightarrow$$
Dually, we get a distinguised triangle
$$\mathbb{C}_O\longrightarrow\mathbb{C}_{X_1}(1)[2]\oplus\mathbb{C}_{X_2}(1)[2]\longrightarrow\omega_X\overset{+1}\longrightarrow$$
In the above triangles, every map is a (sum or difference of) the canonical restriction/inclusion.
You have the following diagram with distinguished rows an column :
$$\require{AMScd}
\begin{CD}
{}@.{}@.\mathbb{C}_{X_1}[1]\oplus\mathbb{C}_{X_2}[1]\\
@.@.@VVV\\
\mathbb{C}_O@>>>\Psi@>>>\omega_X(-1)[-1]@>{+1}>>\\
@VVV@|@VVV\\
\mathbb{C}_X[1]@>>>\Psi@>>>\mathbb{C}_O(-1)@>{+1}>>\\
@VVV@.@VV{+1}V\\
\mathbb{C}_{X_1}[1]\oplus\mathbb{C}_{X_2}[1]\\
@VV{+1}V
\end{CD}
$$
It is self dual.
The monodromy is unipotent, in fact $(T-1)^2=0$. The morphism $T-1$ is the composition $\Psi\rightarrow\mathbb{C}_O\rightarrow\Psi$. ($T-1$ happen to be shift by $(-1)$ but this is very exceptional)
The gr is given by $\operatorname{gr}\psi=\mathbb{C}_O \oplus \left(\mathbb{C}_{X_1}[1]\oplus\mathbb{C}_{X_2}[1]\right)\oplus\mathbb{C}_O(-1)$ and the monodromy induce an isomorphism between the weight 2 and the weight 0 part.
