KO-Rings of Full Flag Varieties

TL;DR

This paper computes the KO-groups of full flag varieties using a novel approach that relates the Witt ring to Tate cohomology, revealing an exterior algebra structure generated by G's representations.

Contribution

It introduces a type-independent method to compute KO-groups of flag varieties by linking the Witt ring with Tate cohomology, providing explicit algebraic descriptions.

Findings

Witt ring of flag varieties is an exterior algebra

Generators of the Witt ring are determined by G's representations

Method applies uniformly across different types of Lie groups

Abstract

We present type-independent computations of the KO-groups of full flag varieties, i.e. of quotient spaces G/T of compact Lie groups by their maximal tori. Our main tool is the identification of the Witt ring, a quotient of the KO-ring, of these varieties with the Tate cohomology of their complex K-ring. The computations show that the Witt ring is an exterior algebra whose generators are determined by representations of G.