Generic character sheaves on parahoric subgroups

We study parabolic induction producing $\ell$-adic sheaves on a parahoric subgroup scheme in the loop group of a reductive group. Under a genericity assumption on the input data, we prove that it produces conjugation equivariant perverse sheaves on the parahoric subgroup; this is upgraded to a $t$-exact equivalence of categories of $\ell$-adic sheaves. An iterative version of the construction produces such a perverse sheaf starting from a geometric analogue of the data considered by J.-K. Yu and J. Kim. We prove, under a mild condition on $q$, that generic parabolic induction from a parahoric torus realizes the character of the representation arising from the associated parahoric Deligne--Lusztig induction, which is known to parametrize the Fintzen--Kaletha--Spice twist of types. In the simplest interesting setting, our construction produces a simple perverse sheaf associated to a sufficiently nontrivial multiplicative local system on a torus, resolving a conjecture of Lusztig.