Stably Cayley groups in characteristic zero. (Q2929551)

Let \(G\) be a connected affine algebraic group over a field \(k\) and let \(\mathfrak g\) be the Lie algebra of \(G\). Consider the action of \(G\) on itself by conjugation and on \(\mathfrak g\) via the adjoint representation. If there is a \(G\)-equivariant birational isomorphism \(G\dashrightarrow\mathfrak g\) defined over \(k\), then \(G\) is called a Cayley group over \(k\). If \(G\times\mathbb G_m^r\) is Cayley over \(k\) for some \(r\geqslant 0\), then \(G\) is called stably Cayley over \(k\). These notions were introduced by \textit{N. Lemire, V. L. Popov} and \textit{Z. Reichstein} [in J. Am. Math. Soc. 19, No. 4, 921-967 (2006; Zbl 1103.14026)], where, in particular, Cayley and stably Cayley simple groups have been classified in the case where \(k\) is an algebraically closed field of characteristic zero.NEWLINENEWLINE In the paper under review, the authors study reductive Cayley groups over an arbitrary field \(k\). They give a criterion for a reductive group \(G\) to be stably Cayley, formulated in terms of its character lattice, and obtain the following classifications:NEWLINENEWLINE Theorem 1. Let \(k\) be a field of characteristic \(0\) and let \(G\) be an absolutely simple \(k\)-group. Then the following conditions are equivalent: (a) \(G\) is stably Cayley over \(k\); (b) \(G\) is an arbitrary \(k\)-form of one of the following groups: \(\text{SL}_3\), \(\text{PGL}_n\) (\(n=2\) or \(n\geqslant 3\) odd), \(\text{SO}_n\) (\(n\geqslant 5\)), \(\text{Sp}_{2n}\) (\(n\geqslant 1\)), \(\text G_2\), or an inner \(k\)-form of \(\text{PGL}_n\) (\(n\geqslant 4\) even).NEWLINENEWLINE Theorem 2. Let \(k\) be a field of characteristic \(0\) and let \(G\) be a simple (but not necessarily absolutely simple) \(k\)-group. Then the following conditions are equivalent: (a) \(G\) is stably Cayley over \(k\); (b) \(G\) is isomorphic to \(R_{l/k}(G_1)\), where \(l/k\) is a finite field extension and \(G_1\) is either a stably Cayley absolutely simple group over \(l\) (i.e., one of the groups listed in Theorem 1) or an outer \(l\)-form of \(\text{SO}_4\). -- Here \(R_{l/k}\) is the Weil functor of restriction of scalars.