DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 483-492
Bob Oliver
Title: Vector Bundles over Classifying Spaces
Let $\kk(X)$ denote the Grothendieck group of the monoid of (complex) vector
bundles over any given space $X$. This is not in general the same as the
$K$-theory group $K(X)$. When $X=BG$, the
classifying space of a compact Lie group $G$, then $K(BG)$ has already been
described by Atiyah and Segal as a certain completion of the representation
ring $R(G)$. The main result described here is
that the Grothendieck group $\kk(BG)$ also can be described explicitly, in
terms of the representation rings of certain subgroups of $G$, and compared
with both $R(G)$ and $K(BG)$.