For a split semisimple algebraic group ๐ป with its split maximal torus ๐, let ๐ โถ CH(๎ฎ๐ป) โ CH(๎ฎ๐) ๐ be the restriction homomorphism of the Chow rings CH of the classifying spaces ๎ฎ of ๐ป and ๐, where ๐ is the Weyl group. A constraint on the image of ๐, 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 ๐ป. 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 ๐ป be a split semisimple algebraic group (over an arbitrary field) with a split maximal torus ๐ and the Weyl group ๐. The image of the restriction homomorphism CH(๎ฎ๐ป) โ CH(๎ฎ๐), where CH(๎ฎ๐ป) is the Chow ring of the classifying space ๎ฎ๐ป of ๐ป, defined in [14], consists of ๐-invariant elements. The resulting homomorphism of graded ringsis rationally an isomorphism: Its kernel and cokernel are killed by the torsion index of ๐ป, see [15, Theorem 1.3(1)]. Note that CH(๎ฎ๐) is canonically identified with the symmetric ring ๎ฟ( ล) of the character group ล of ๐, [3, section 3.2] (see also [7, section 3]). In particular, the group CH(๎ฎ๐) is free of torsion so that the kernel of ๐ actually coincides with the ideal Tors CH(๎ฎ๐ป) of torsion elements in CH(๎ฎ๐ป). Therefore, determination of the image Im ๐ of ๐ is equivalent to determination of the quotient ring CH(๎ฎ๐ป)โ Tors CH(๎ฎ๐ป). This quotient ring is relevant for some applications (e.g., [2] and [10]), where the torsion in CH(๎ฎ๐ป) is irrelevant.In [9], a (given by the Steenrod operations) constraint on Im ๐ has been described. There is another constraint on Im ๐, which is given by any envelope of ๐ป-a split reductive group ๐บ such that ๐ป is its semisimple part. Namely, ๐บ contains ๐ป as a normal subgroup, has a split maximal torus ๐ with ๐ โฉ ๐ป = ๐, the Weyl group of ๐บ coincides with the Weyl group ๐ of ๐ป, and the square (1.1) This work has been accomplished during author's stay at the Max Planck Institute for Mathematics in Bonn.