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We geometrize the mod $p$ Satake isomorphism of Herzig andHenniart-Vign\'eras using Witt vector affine flag varieties for reductivegroups in mixed characteristic. We deduce this as a special case of a formula,stated in terms of the geometry of generalized Mirkovi\'c-Vilonen cycles, forthe Satake transform of an arbitrary pararhoric mod $p$ Hecke algebra withrespect to an arbitrary Levi subgroup. Moreover, we prove an explicit formulafor the convolution product in an arbitrary parahoric mod $p$ Hecke algebra.Our methods involve the constant term functors inspired from the geometricLanglands program, and we also treat the case of reductive groups in equalcharacteristic. We expect this to be a first step towards a geometrization of amod $p$ Local Langlands Correspondence.

Geometrization of the Satake transform for mod $p$ Hecke algebras