Algebraic K-theory of varieties $SL_{2n}/Sp_{2n}$, $E_6/F_4$ and their twisted forms

TL;DR

This paper computes the algebraic K-theory of certain affine homogeneous varieties and their twisted forms, providing explicit generators and extending results to new classes of algebraic varieties.

Contribution

It offers the first explicit calculations of K-theory for varieties like $SL_{2n}/Sp_{2n}$ and $E_6/F_4$, including their twisted forms, with detailed generators.

Findings

K-theory of $SL_{2n}/Sp_{2n}$ and $E_6/F_4$ computed

Explicit generators for their K-theory algebras provided

K-theory for some twisted forms also determined

Abstract

Let $SL_{2n}$, $Sp_{2n}$, $E_6 = G^{sc}(E_6)$, $F_4 = G(F_4)$ be simply connected split algebraic groups over an arbitrary field $F$. Algebraic K-theory of affine homogeneous varieties $SL_{2n}/Sp_{2n}$ and $E_6/F_4$ is computed. Moreover, explicit elements that generate $K_*(SL_{2n}/Sp_{2n})$ and $K_*(E_6/F_4)$ as $K_*(F)$-algebras are provided. For some twisted forms of these varieties K-theory is also computed.