Some Tits systems with affine Weyl groups in Chevalley groups over Dedekind domains
Let A be a Dedekind domain with P a nonzero principal prime ideal (p) such that the canonical homomorphism \(A^*\to (A/P)^*\) of multiplicative groups of units is surjective. Let R be the subring A[1/p] of the quotient field of A. Let \(\Phi\) be a reduced irreducible root system and \(\Phi_ a=\Phi \times Z\) the corresponding affine root system. The authors study the structure of the elementary subgroup E(\(\Phi\),R) of the Chevalley group G(\(\Phi\),R) over R. They construct a Tits system with affine Weyl group \(W_ a\)- the subgroup of \(S_{\Phi a}\) generated by all affine Weyl reflections \(\sigma_{\alpha}\) given by \(\sigma_{\alpha}(\beta)=(\sigma_{\alpha}(b),m-<b,a>n)\), where \(\alpha\) varies over \(\Phi_ a.\) In the case of a field with a nontrivial discrete valuation, A its ring of integers and P the prime ideal of A, their result specializes to a result of \textit{N. Iwahori} and \textit{H. Matsumoto} [Publ. Math., Inst. Hautes Etud. Sci. 25, 5-48 (1965; Zbl 0228.20015)]. The construction also generalizes a result of the second author [Tsukuba J. Math. 3, No.2, 41- 51 (1979; Zbl 0475.17005)] for Kac-Moody groups. The proof proceeds like those of \textit{R. Steinberg} for classical Chevalley algebras [Yale Univ. Lect. Notes (1968; Zbl 0307.22001)] and \textit{D. Peterson} and \textit{V. Kac} [Proc. Natl. Acad. Sci. USA 80, 1778-1782 (1983; Zbl 0512.17008)]. The authors apply their result to obtain some presentations of Chevalley groups, and conclude with an application of it that yields a decomposition of Chevalley groups over Dedekind domains that resembles the Iwasawa decomposition.
scientific article; zbMATH DE number 4122230
scientific article