Generic character sheaves on parahoric subgroups

We study parabolic induction producing $\mathrm{\ell }$-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 $\mathrm{\ell }$-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.