Let G be a reductive algebraic group over a field k of positive characteristic, and denote by Rep(G) the category of finite dimensional algebraic representations of G. Over the last decades, the theory of perverse sheaves has become ubiquitous in the study of representations of algebraic groups. For instance, the geometric Satake equivalence asserts that there is an equivalence of categories between Rep(G) and the category of perverse sheaves (with coefficients in k) on the affine Grassmannian associated with the Langlands dual group of G over the complex numbers. This result has recently been used by Riche-Williamson to obtain character formulas for simple modules. Building on the geometric Satake equivalence and earlier work of Arkhipov-Bezrukavnikov-Braverman-Gaitsgory-Mirkovi\'c, I will explain the construction of a category of perverse sheaves which should be equivalent to the category of G_1T-modules, whose study is often useful for the understanding of Rep(G).