On G-linebundles and \(K_ G^{\bullet}(X)\)
Let G be a finite reductive group, acting on an algebraic variety X. The author continues the study of the group \(K_ G(X)\) of G-linearized bundles over X [see \textit{G. Ellingsrud} and the author, Math. Ann. 251, 253-261 (1980; Zbl 0425.14012)]. The main results are the descriptions of \(K_ G(X)\) for the case of dim \(X\leq 1\). If X is a smooth curve and \(B\subset X\) is the ramification locus for the map \(X\to X/G,\) then in \(Pic_ G(X)\) there exists a subgroup \(Pic^ B_ G(X)\) generated by G- invariant divisors with support outside B. It is proved the existence of an equivariant first Chern class from \(Pic^ B_ G(X)\) to \(K_ G(X)\) and equivariant Riemann-Roch formula for line-bundles in \(Pic^ B_ G(X)\). The paper contains interesting examples and discussions.
scientific article; zbMATH DE number 3889717