Non-abelian Rees construction and pure motives

The classical Rees construction (of common use in commutative algebra and Hodge theory) interpolates between filtrations, viewed as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔾</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> {{G}ₘ} -equivariant vector bundles on the affine line, and their associated gradings. Various non-abelian versions have been proposed, where the multiplicative group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔾</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> {{G}ₘ} is replaced by an arbitrary reductive group. Building on a construction due to P. O’Sullivan, we present a Galois correspondence between quasi-homogeneous spaces and certain monoidal categories, and apply it to monoidal categories of motives with concrete applications to algebraic cycles. In particular, we give a new proof and generalization of the Clozel–Deligne theorem about numerical equivalence on abelian varieties over finite fields.