  In this paper we construct equivalences of monoidal categories relating three
geometric or representation-theoretic categorical incarnations of the affine
Hecke algebra of a connected reductive algebraic group $G$ over a field of
positive characteristic: a category of Harish-Chandra bimodules for the Lie
algebra of $G$; the derived category of equivariant coherent sheaves on (a
completed version of) the Steinberg variety of the Frobenius twist $G^{(1)}$ of
$G$; a derived category of constructible sheaves on the affine flag variety of
reductive group which is Langlands dual to $G^{(1)}$. These constructions build
on the localization theory developed by the first author with Mirkovi\'c and
Rumynin and previous work of ours (partly joint with L. Rider), and provide an
analogue for positive-characteristic coefficients of a construction of the
first author. As an application, we prove a conjecture by Finkelberg-Mirkovi\'c
giving a geometric realization of the principal block of algebraic
representations of $G$.
