On the algebraic K-theory of some homogeneous varieties
It is known see, e.g. [\textit{I. A. Panin}, K-Theory 8, No. 6, 541--585 (1994; Zbl 0854.19002)] that the \(K\)-theory of homogeneous projective varieties can be expressed in terms of the \(K\)-theory of certain separable algebras. In the present paper, the author provides a unified approach to the \(K\)-theory of homogeneous varieties and computes it for inner forms of \(G/H,\) where \(H \subset G\) is a pair of connected split reductive groups of the same rank (all algebraic groups are supposed to be smooth algebraic varieties over an arbitrary field \(k).\) The first main theorem asserts that, for \(r = [W(G):W(H)],\) \(W(\ldotp)\) denoting the Weyl group, there exist \(r\) vector bundles over \(G/H,\) \(V_1, \ldots, V_r,\) such that \(K_\ast (G/H) = \displaystyle\mathop\oplus^r_{i = 1} K_\ast (k) [V_i].\) Suppose moreover that \(G\) is simply connected semisimple (this is not actually a restriction) and put \(\overline G = G/Z(G)\); fix a 1-cocycle \(\gamma:Gal (k^{\mathrm{sep}/k}) \to \overline G (k^{\mathrm{sep}})\) and twist the variety \(X = G/H\) to get \(X_\gamma = (G/H)_\gamma\).NEWLINENEWLINENEWLINEThe second main theorem gives an isomorphism \(\displaystyle\mathop\oplus^r_{i = 1} K_\ast (A(\lambda_i)_\gamma) {\displaystyle\buildrel\sim \over \rightarrow} K_\ast (X_\gamma),\) where the separable algebras \(A(\lambda_i)_\gamma\) are obtained as follows : let \(R(G) = K_0(\mathrm{Rep}_k(G))\) be the representation ring of \(G\); the author shows that \(R(H)\) is a free \(R(G)\)-module of rank \(r,\) with a basis consisting of the classes of irreducible representations of \(H\); the proof of this property introduces \(r\) minimal and indecomposable (in some sense) elements \(\lambda_i \in w_i \wedge^+_G,\) one for each chamber \(w_i {\mathcal C} (G),\) \(\wedge^+_G\) denoting a cone of dominant weights; a theorem of \textit{J.-P. Serre} [Publ. Math., Inst. Hautes Étud. Sci. 34, 37--52 (1968; Zbl 0195.50802)] associates to such a \(\lambda_i\) an irreducible \(G\)-module \(V_G(\lambda_i)\); then \(A(\lambda_i) = \mathrm{End}_k(V_G (\lambda_i))\). \smallskip\noindent As an application, the author recovers known results concerning homogeneous varieties such that twisted flag varieties, even-dimensional affine quadrics, quaternionic projective spaces, octonionic projective planes.
scientific article; zbMATH DE number 6039008