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Let G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs 
$(M,{\mathcal O})$ where M is the identity component of the centralizer of a semisimple element in G and 
${\mathcal O}$ is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.
In this paper we construct an action of the affine Hecke category (in its ‘Soergel bimodules’ incarnation) on the principal block of representations of a simply connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical 
$p$-Kazhdan–Lusztig polynomials.
