  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.
