Envelopes and classifying spaces

Abstract

For a split semisimple algebraic group 𝐻 with its split maximal torus 𝑆, let 𝑓 ∶ CH(𝐻) → CH(𝑆) 𝑊 be the restriction homomorphism of the Chow rings CH of the classifying spaces  of 𝐻 and 𝑆, where 𝑊 is the Weyl group. A constraint on the image of 𝑓, given by the Steenrod operations, has been applied to the spin groups in a previous paper. Here, we describe and apply to the spin groups another constraint, which is given by the reductive envelopes of 𝐻. We also recover in this way some older results on orthogonal groups. K E Y W O R D Saffine algebraic groups, chow groups, classifying spaces, orthogonal groups, spin groups CONSTRAINTLet 𝐻 be a split semisimple algebraic group (over an arbitrary field) with a split maximal torus 𝑆 and the Weyl group 𝑊. The image of the restriction homomorphism CH(𝐻) → CH(𝑆), where CH(𝐻) is the Chow ring of the classifying space 𝐻 of 𝐻, defined in [14], consists of 𝑊-invariant elements. The resulting homomorphism of graded ringsis rationally an isomorphism: Its kernel and cokernel are killed by the torsion index of 𝐻, see [15, Theorem 1.3(1)]. Note that CH(𝑆) is canonically identified with the symmetric ring ( Ŝ) of the character group Ŝ of 𝑆, [3, section 3.2] (see also [7, section 3]). In particular, the group CH(𝑆) is free of torsion so that the kernel of 𝑓 actually coincides with the ideal Tors CH(𝐻) of torsion elements in CH(𝐻). Therefore, determination of the image Im 𝑓 of 𝑓 is equivalent to determination of the quotient ring CH(𝐻)∕ Tors CH(𝐻). This quotient ring is relevant for some applications (e.g., [2] and [10]), where the torsion in CH(𝐻) is irrelevant.In [9], a (given by the Steenrod operations) constraint on Im 𝑓 has been described. There is another constraint on Im 𝑓, which is given by any envelope of 𝐻-a split reductive group 𝐺 such that 𝐻 is its semisimple part. Namely, 𝐺 contains 𝐻 as a normal subgroup, has a split maximal torus 𝑇 with 𝑇 ∩ 𝐻 = 𝑆, the Weyl group of 𝐺 coincides with the Weyl group 𝑊 of 𝐻, and the square (1.1) This work has been accomplished during author's stay at the Max Planck Institute for Mathematics in Bonn.