If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then $$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$ where $(\star)$ is true when $i$ is in addition regular. Here $k$ denotes the unit D module/$\ell$-adic sheaf.

Now consider replacing $k_X$ by $\text{IC}_X$ (and assume $i$ plays nicely with the Whitney stratifications). Is there anything that can be said about $i^*\text{IC}_X$ and $i^!\text{IC}_X$?

I'm even interested in the simple case where $Z$ and $U=X\setminus Z$ are smooth, giving a two-term stratification of $X$.

If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then $$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$ where $(\star)$ is true when $i$ is in addition regular. Here $k$ denotes the unit D module/$\ell$-adic sheaf.

Now consider replacing $k_X$ by $\text{IC}_X$ (and assume $i$ plays nicely with the Whitney stratifications). Is there anything that can be said about $i^*\text{IC}_X$ and $i^!\text{IC}_X$, apart from them being perverse sheaves?

I'm even interested in the simple case where $Z$ and $U=X\setminus Z$ are smooth, giving a two-term stratification of $X$.

If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then $$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$ where $(\star)$ is true when $i$ is in addition regular. Here $k$ denotes the unit D module/$\ell$-adic sheaf.

Now consider replacing $k_X$ by $\text{IC}_X$ (and assume $i$ plays nicely with the Whitney stratifications). Is there anything that can be said about $i^*\text{IC}_X$ and $i^!\text{IC}_X$, beyond that they are perverse sheaves?

I'm even interested in the simple case where $Z$ and $U=X\setminus Z$ are smooth, giving a two-term stratification of $X$.

If $i:Z\hookrightarrow X$ is a closed embedding of codimension $c$, then $$i^*k_X\ =\ k_Z , \ \ \ i^!k_X\ \stackrel{(\star)}{=}\ k_Z[2c]$$ where $(\star)$ is true when $i$ is in addition regular. Here $k$ denotes the unit D module/$\ell$-adic sheaf.

Now consider replacing $k_X$ by $\text{IC}_X$ (and assume $i$ plays nicely with the Whitney stratifications). Is there anything that can be said about $i^*\text{IC}_X$ and $i^!\text{IC}_X$?

I'm even interested in the simple case where $Z$ and $U=X\setminus Z$ are smooth, giving a two-term stratification of $X$.