Central elements in affine mod <i>p</i> Hecke algebras via perverse -sheaves

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