The notion of norm and the representation theory of orthogonal groups
Nondiscrete tempered spectra of classical groups over local fields are studied by using the theory of endoscopy that relates them to self-dual representations of \(GL(n)\). Let \(\tau\) and \(\tau'\) be two discrete series representations of \(SO_{2m} (F)\) and \(GL_n (F)\) over a local field \(F\), respectively, and \(\tau \otimes \tau'\) the representation of the Levi factor \(SO_{2m} (F) \times GL_n (F)\) of \(SO_{2r} (F)\) with \(r = m + n\). \(I (\tau \otimes \tau')\) is then the representation of \(SO_{2r} (F)\) unitarily induced from \(\tau \otimes \tau'\). The determination of the reducibility of \(I(\tau \otimes \tau')\) is used to study the residue of the standard intertwining operator. The latter, in turn, requires the introduction of norm correspondence from a certain subset of \(\theta\)-conjugacy classes in \(GL_n (F)\) into the set of conjugacy classes in \(SO_{2m} (F)\) where \(\theta (g) = w_n^{-1} \cdot^t g^{-1} \cdot w_n\) is the automorphism of \(GL_n(F)\) defined by means of the \(n \times n\) matrix \(w_n\) whose unique nonzero elements lie on its second diagonal.
scientific article; zbMATH DE number 763280
scientific article