The Chern character for classical matrix groups

We give explicit formulas for representations of classical matrix groups whose Chern characters have lowest order terms equal to standard characteristic classes. For SO(2r), the Euler class e does not arise in this way, but 2r-le does arise in this way. A complex representation p of a compact Lie group G gives rise to a complex vector bundle (p over the classifying space BG of G. In this way one obtains a ring homomorphism R(G) -* K(BG) from the complex representation ring of G to the complex K-theory of BG. Composing with the Chern character yields a ring homomorphism, R(G) -* K(BG) -h H**(BG;Q) Our interest here is in determining which cohomology classes on BG occur as the lowest order terms of some Chern character ch((p). We will consider the classical matrix groups U(n), SU(n), Sp(n), and SO(n), and we will produce explicit representations p whose Chern characters ch(Qp) have lowest order terms equaling standard characteristic classes. In all cases considered here, the lowest order terms of the Chern characters ch((p) turn out to be integral classes. For U(n), SU(n), Sp(n), and SO(2r + 1), every homogeneous integral cohomology class modulo torsion on BG arises as the lowest order term of some Chern character ch((p); for SO(2r), the Euler class does not so arise, but 2r-1 times the Euler class does. In the cases considered here where BG has integral cohomology which is free of torsion, namely, U(n), SU(n), and Sp(n), general results of Atiyah and Hirzebruch ([3], ?2.5 and ?4.8) imply that the lowest order terms of any Chern character ch((p) are integral classes and that any homogeneous integral class arises in this way. Thus our only contribution in these cases is the explicit form of certain representations which give rise to standard characteristic classes. For SO(n), whose classifying space has 2-torsion, the general results of Atiyah and Hirzebruch do not apply directly. It may be possible to obtain some general results in the presence of torsion by using the Atiyah-Hirzebruch spectral sequence together with integrality results such as those of Adams [1] and Maunder [7]; we have Received by the editors October 1, 1996. 1991 Mathematics Subject Classification. Primary 55R40. The author was partially supported by NSA grants MDA904-94-H-2025 and MDA904-96-10067, and by Purdue University Calumet Scholarly Research Awards. ?1998 American Mathematical Society 1237 This content downloaded from 207.46.13.172 on Fri, 07 Oct 2016 06:27:24 UTC All use subject to http://about.jstor.org/terms