On the Algebraic K-Theory of Some Homogeneous Varieties

The $K$-theory of inner twisted forms of homogeneous varieties $G/H$ with connected reductive algebraic groups $H\subset G$ of the same rank is computed. We provide an explicit isomorphism with the $K$-theory of certain central simple algebras associated to the considered variety, as a consequence one has that $K_0(G/H)$ is a free abelian group of rank $[W(G):W(H)]$. The result is used for the computation of the $K$-theory of some affine homogeneous varieties including an octonionic projective plane and quaternionic projective spaces.

2010 Mathematics Subject Classification: 19E08, 14M17

Keywords and Phrases: affine homogeneous varieties, K-theory, central simple algebras

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