Central elements in affine mod p Hecke algebras via perverse Fp-sheaves

Let G be a split connected reductive group over a finite field of characteristic p > 2 such that G(der) is absolutely almost simple. We give a geometric construction of perverse F-p-sheaves on the Iwahori affine flag variety of G which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod p Hecke algebra to the center of the Iwahori mod p Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod p Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse Fp-sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly F-regular, and hence they are F-rational and have pseudo-rational singularities.