Rationality of almost simple algebraic groups
The author establishes rationality properties for a class of almost simple adjoint algebraic groups. Due to results of \textit{A. S. Merkur'ev} [K-Theory 7, No. 1, 1-3 (1993; Zbl 0811.16011) and Publ. Math., Inst. Hautes Etud. Sci. 84, 189-213 (1996; Zbl 0884.20029)] one cannot hope for stable rationality in general when the group involves division algebras of degree divisible by 4. Therefore the following main result is in some sense sharp. Let \(k\) be a field and let \(G\) be an almost simple adjoint \(k\)-group. Let \(m(G)\) be the maximal length of the black segments of its Tits index that are defined over \(k\). That is, the almost simple group with root system spanned by the segment must be defined over \(k\). If \(G\) is of classical type \(X^{(d)}_{n,r}\) with \(n-rd\leq 2\), or \(G\) is of exceptional type with \(m(G)\leq 3\) then \(G\) is either rational or stably rational over \(k\). Here, following Tits, \(d\) denotes the rank of the division algebra associated with \(G\), \(n\) is the rank and \(r\) is the \(k\)-rank. The proof is case by case, using Tits diagrams and many earlier results.
scientific article; zbMATH DE number 1374927