Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Abstract

Let G be a split semisimple linear algebraic group over a field k 0 . Let E be a G-torsor over a field extension k of k 0 . Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel. Consider a twisted form E/B of the variety of Borel subgroups G/B over k.Following the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an isomorphism between the Grothendieck groups of the h-motivic subcategory generated by E/B and the category of finitely generated projective modules of certain Hecke-type algebra H which depends on the root datum of G, on the torsor E and on the formal group law of the theory h.In particular, taking h to be the Chow groups with finite coefficients Fp and E to be a generic G-torsor we prove that all indecomposable submodules of an affine nil-Hecke algebra H of G with coefficients in Fp are isomorphic to each other and correspond to the (non-graded) generalized Rost-Voevodsky motive for (G, p).

Oriented cohomology sheaves on double moment graphs

“…We have âu (1 Yu ) = [ X u ] and by the same arguments as in [NPSZ,Example 2.3] we conclude that ⊕ u âu is an isomorphism, hence, it gives a direct sum decomposition of h(pt)-modules…”

“…Note that for Chow theory (even for Chow motives) this was done in [CM06,Thm. 16]; for an arbitrary h G and P = Q = B this is [NPSZ,Example 3.6]. We also show that this direct sum decomposition is compatible with the forgetful map (Corollary 3.4) and with the geometric action of W (Corollary 3.6).…”

Rost motives, affine varieties, and classifying spaces

“…The projectors from Theorem defining a ‐equivariant motivic decomposition of of Theorem immediately give a decomposition of considered as a nongraded module over itself into a direct sum of its pairwise isomorphic (as nongraded modules) indecomposable one‐sided ideals. Thus, the following proposition holds (see also ). Proposition In the above notation there is a decomposition of nongraded ‐modules, where .…”

“…Besides this, we remark that our equivariant motivic decompositions from Section 5 automatically provide decompositions of affine nil‐Hecke algebras modulo a prime and give new information about its modular representations (see Section 9 and ).…”

Hopf-theoretic approach to motives of twisted flag varieties

“…Due to nilpotency results [CNZ21, § 5] (cf. [PS17, NPSZ18, Theorem 5.5]) one can lift motivic decompositions of the -motives of twisted flag varieties to a motivic decomposition of the -equivariant -motive of .…”