\(p\)-adic norms and quadratic extensions (Q848121)

Let \(K\) denote a quadratic extension of a \(p\)-adic field \(F\). In the paper under review, the author studies the action of a subgroup \(H\) on the Bruhat-Tits building of a classical \(p\)-adic group \(G\) for the following spherical pairs \((G,H)\): {\parindent=7,5mm \begin{itemize}\item[(i)] \(G = \text{GL}(V)\), \(H=\text{GL}(W)\), where \(W\) denotes a finite dimensional \(K\)-vector space and \(V\) denotes an underlying \(F\)-vector subspace of W. \item[(ii)] \(G= \text{SO}(V, \varphi)\), \(H = U(W, \psi)\), where \((W, \psi)\) is a Hermitian \(K\)-space and \((V, \varphi)\) is a quadratic underlying \(F\)-subspace, with \(\varphi = \text{Tr}_{K / F} \psi\). \item[(iii)] \(G= \text{SO}(V, \varphi)\), \(H = U(W, \psi)\), where \((W, \psi)\) is a Hermitian \(K\)-space, while the underlying quadratic \(F\)-vector space is an \(F\)-hyperplane of a quadratic \(F\)-vector space \((V, \varphi)\). \end{itemize}} The author classifies the orbits of \(H\) and obtains some interesting explicit decompositions of the \(p\)-adic group \(G\), rather similar to a Cartan decomposition, which should have potential applications in harmonic analysis on the homogeneous spherical spaces \(G / H\).