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Let $G$ be a split connected reductive group over a finite field ofcharacteristic $p > 2$ such that $G_\text{der}$ is absolutely almost simple. Wegive a geometric construction of perverse $\mathbb{F}_p$-sheaves on the Iwahoriaffine flag variety of $G$ which are central with respect to the convolutionproduct. We deduce an explicit formula for an isomorphism from the sphericalmod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. Wealso give a formula for the central integral Bernstein elements in the Iwahorimod $p$ Hecke algebra. To accomplish these goals we construct a nearby cyclesfunctor for perverse $\mathbb{F}_p$-sheaves and we use Frobenius splittingtechniques to prove some properties of this functor. We also prove that certainequal characteristic analogues of local models of Shimura varieties arestrongly $F$-regular, and hence they are $F$-rational and have pseudo-rationalsingularities.

Central elements in affine mod $p$ Hecke algebras via perverse $\mathbb{F}_p$-sheaves