Parameterizing conjugacy classes of maximal unramified tori via Bruhat-Tits theory
Let \(k\) be a field with nontrivial discrete valuation \(\nu\). \(k\) is complete with perfect residue field \(f\) and \(\bar{k}\) is a fixed algebraic closure of \(k\). Let \(K\) be the maximal unramified extension of \(k\) in \(\bar{k}\). Let \(G\) denote the group of \(k\)-rational points of a reductive linear algebraic \(k\)-group {\textbf{G}} and \(G^o\) the group of \(k\)-rational points of the identity component {\textbf{G}}\(^o\). Define \(B(G)\) as the (enlarged) Bruhat-Tits building of \(G^o\). A subgroup of \(G\) is called unramified torus if it is the group of \(k\)-rational points of a \(k\)-torus in {\textbf{G}}\(^o\) that splits over an unramified extension of \(k\). The aim of this work is to classify \(G\)-conjugacy classes of maximal unramified tori in \(G\) in terms of equivalence classes of pairs (G\(_F\),\; T) where \(F\) is a facet in the building, G\(_F\) is the connected reductive \(f\)-group associated to \(F\), and T is an \(f\)-minisotropic maximal torus in G\(_F\). For this purpose, the author introduces the set \(I^t\) of pairs (\(F\),\; T) where \(F\) is a facet in \(B(G)\), defines an equivalence relation \(\sim\) on \(I^t\), and confines the treatment to its subset \(I^m\) of minisotropic pairs in \(I^t\). To each element (\(F\),\; T)\(\; \in I^t\), there corresponds a \(G\)-conjugacy class C(\(F\),\; T) of maximal unramified tori in \(G\), and C\(^T\) denotes the set of \(G\)-conjugacy classes of maximal unramified tori in \(G\). The main result of the present work is that there is a bijective correspondence between \(I^m/\sim\) and C\(^T\) given by the map that sends (\(F\),\; T) to C(\(F\),\; T).
scientific article; zbMATH DE number 5140515
scientific article