The talk is based on a joint work with A. Neshitov, N. Semenov,
V. Petrov (arXiv:1505.07083).
Following the motivic Galois group approach we relate a subcategory
generated by the oriented motive of a versal flag variety
with the category of integer/modular representations of certain Hecke-type
algebras. As an application, we describe
irreducible blocks of the affine nil-Hecke algebra (over a finite field of
characteristic dividing the order of the Weyl group)
using recent results on motives of twisted flag varieties.