Our understanding of algebraic representations of reductive algebraic groups in positive characteristic has seen big advances in the last years and has been largely transformed into the geometric theory of studying parity sheaves on affine Grassmannians and affine flag varieties or, equivalently and more combinatorially, the diagrammatic Hecke category. This has led, among other things, to a geometric proof of the linkage principle and a greatly simplified proof of Lusztig's character formula for large characteristics.
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