IC complex on $\Bbb C = \Bbb C^* \sqcup \{0\}$ Let $X = \Bbb C$ stratified as $\Bbb C^* \sqcup \{0\}$. Let $\mathscr L$ be a local system on $\Bbb C^*$. 

How to describe $IC(U, \mathscr L)$ ? 

By describe I mean : compute its stalk at $0$, maybe find a short exact sequences of complexes involving $IC(U, \mathscr L)$, ...
My thoughts : I don't think we can have $IC(U, \mathscr L) \cong j_* \mathscr L[1]$ or $j_!\mathscr L[1]$. Indeed, they are "dual" (up to changing $\mathscr L$ to $\mathscr L^{\vee}$) to each other, and $\Bbb D_X(IC(U, \mathscr L)) \cong IC(U, \mathscr L^{\vee})$). But in general, $0$ is in the support of $j_* \mathscr L[1]$ but is not in the support of $j_! \mathscr L$. However I don't see other candidates. 
The definition is the image in the category of perverse sheaves of $^pj_! \mathscr L[1] \to \ ^pj_* \mathscr L[1]$ but I don't know how to compute it. 
Another related question is : 

If $V$ is a skyscraper sheaf at $0$, how to describe extensions between $\mathscr L[1]$ and $V$, in the category of perverse sheaves ?

I know everything can be phrased in term of quivers but if possible I would like to use the basic definitions only.
 A: I'm not sure if this description is necessarily better than the definition you've already given, but here goes.
Let $i:0\to X$ be the inclusion. One way to characterize the IC complex is by the property that it doesn't have any copy of $V = i_*\mathbb C$ as a quotient or a subobject. For example, $j_{!*}\mathcal L[1]\subseteq {}^pj_{!}\mathcal L[1]$ in the category of perverse sheaves
$$
\operatorname{Hom}_{\operatorname{Perv}(X)}(V, j_{!*}\mathcal L[1]) \subseteq \operatorname{Hom}_{\operatorname{Perv}(X)}(V, {}^pj_{*}\mathcal L[1]) =0.
$$
And dually we can see that all maps from $j_{!*}\mathcal L[1]$ to $V$ are $0$.
Now, $j_{!*}\mathcal L[1]$ is a subobject of ${}^pj_*\mathcal L[1]$, precisely the largest subobject that doesn't have $V$ as a quotient: if $W$ is such that $j_{!*}\mathcal L[1]\subseteq W\subseteq {}^pj_*\mathcal L[1]$, then taking $j^*$ we have that $\mathcal L[1]\subseteq j^* W \subseteq \mathcal L[1]$, so the quotient $W/j_{!*}\mathcal L[1]$ is supported at $0$. If $W$ has no quotients supported at $0$, then it must equal $j_{!*}\mathcal L[1]$.
Let $\mathcal F = {}^pj_*\mathcal L[1]$. The biggest quotient of $\mathcal F$ supported on $0$ is $$i_*\operatorname{Hom}_{\operatorname{Perv}(X)}(\mathcal F, V) = i_*\operatorname{Hom}_{\operatorname{Perv}(X)}(\mathcal F, i_*\mathbb C)\cong  i_*\operatorname{Hom}_{\operatorname{Perv}(X)}(i^*\mathcal F, \mathbb C)  = i_* \left(i^*\mathcal F\right)^\vee. $$
So we can see that whenever $i^*\mathcal F = 0$, then the IC extension coincides with the pushforward. In general, we have that
$$
i^*\mathcal F = H^0(\mathbb C^*, \mathcal L) = \operatorname{ker}(\rho - 1).
$$
So we know that if $r$ is the rank of $\mathcal L$ and $m$ is the multiplicity of $1$ as an eigenvalue of its monodromy, the IC extension is the quotient of ${}^pj_*\mathcal L[1]$ by a skyscraper sheaf of dimension $m$. I'm not really sure if this constitutes a good description of this sheaf.

To answer the second question, this computation is useful, using the fact that $V=i_*\mathbb C = i_!\mathbb C$ and the adjunction relations:
$$
\operatorname{Ext}^\bullet(\mathcal L[1], i_*\mathbb C) = 
\operatorname{Hom}_{D(X)}(\mathcal L[1], i_*\mathbb C) = 
\operatorname{Hom}_{D(X)}(i^*\mathcal L[1], \mathbb C).
$$
So
$$
\operatorname{Ext}^1(\mathcal L[1], V) \cong (i^*\mathcal L)^\vee.
$$
Dually,
$$
\operatorname{Ext}^\bullet(i_!\mathbb C, \mathcal L[1]) = 
\operatorname{Hom}_{D(X)}(i_!\mathbb C, \mathcal L[1]) = 
\operatorname{Hom}_{D(X)}(\mathbb C, i^!\mathcal L[1]).
$$

A more practical approach to both questions is using the description of perverse sheaves as representations of a quiver, which is sort of outsourcing the computations to the proof of the theorem.

The category of perverse sheaves on $X$ relative to the stratification $\mathbb C^* \sqcup 0$ is equivalent to the category of pairs of vector spaces $(\Psi,\Phi)$ together with two maps:
  $$
\Psi \overset{\operatorname{can}}{\underset{\operatorname{var}}{\rightleftarrows}} \Phi.
$$
  The maps are subject to the condition that $\operatorname{var}\circ \operatorname{can} + 1$ is invertible.

Under this equivalence, the usual functors are given as follows:


*

*$j^*$ assigns to $(\Psi,\Phi)$ the local system with stalk $\Psi$ and monodromy $\operatorname{var}\circ \operatorname{can} + 1$.

*${}^pj_*$ assigns the local system with stalk $W$ and monodromy $\rho$ the diagram $ W \underset{1}{\overset{\rho - 1}{\rightleftarrows}} W$. This can be seen from the adjunction $j^*\vdash j_*$.

*$i_*$ maps the vector space $W$ to $(0,W)$.

*By adjunction, $i^*$ assigns to $(\Psi,\Phi)$ the vector space $\operatorname{coker} (\operatorname{can})$, and $i^!(\Psi,\Phi) = \ker (\operatorname{var})$.


I've seen this proved in Malgrange's book Equations différentielles à coefficients polynomiaux, and I think it is the topic of the paper How to Glue Perverse Sheaves by Beilinson.
