Airy sheaves for reductive groups

We construct a class of $\ell$-adic local systems on $\mathbb{A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation $y''(z)=zy(z)$. We employ the geometric Langlands correspondence to construct the sought-after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ng\^o and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For $\mathrm{GL}_n$, we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behaviour of the local systems at $\infty$. These conjectures in particular imply cohomological rigidity of Airy sheaves.