\(R\)-groups for unitary principal series of \textit{Spin} groups
In the paper under the review, the authors study the Knapp-Stein \(R\) group for unitary principal series of split spinor groups defined over a \(p\)-adic field \(F\). Let \(T\) denote a maximal \(F\)-split torus of the spinor group \(\mathrm{Spin}_m\), and let \(\widetilde{T}\) denote a maximal \(F\)-split torus of \(\mathrm{GSpin}_m\) such that \(T = \widetilde{T} \cap \mathrm{Spin}_m\) (we note that \(\mathrm{Spin}_m\) is a derived subgroup of \(\mathrm{GSpin}_m\)). Let \(\widetilde{\chi}\) denote a unitary character of \(\widetilde{T}\), and let \(\chi\) be the restriction of \(\widetilde{\chi}\) to \(T\). Let \(R\widetilde{\chi}\) and \(R_{\chi}\) stand for the Knapp-Stein \(R\)-groups for the principal series induced from \(\widetilde{\chi}\) and \(\chi\), respectively. Author obtain a complete description of the splitting of the exact sequence \[0 \rightarrow R\widetilde{\chi} \rightarrow R_{\chi} \rightarrow R_{\chi} / R\widetilde{\chi} \rightarrow 0.\] Also, the multiplicity one theorem in the restriction of irreducible tempered non-discrete series of \(\mathrm{GSpin}_m\) to \(\mathrm{Spin}_m\) is proved. Finally, following the Keys work on \(R\)-groups, authors prove Arthur's conjecture on \(R\)-groups for the unitary principal series of quasi-split groups over \(F\).
scientific article; zbMATH DE number 7355819
scientific article