Reducibility points for representations induced from cuspidal ones
Let \(G(n)\) be a symplectic or orthogonal group of rank \(n\) defined over a \(p\)-adic field \(F\). Let \(\pi\) be an irreducible cuspidal representation of \(G(n,F)\). The subject of the paper is the reducibility of the representations \(\rho\times\pi\), where \(\rho\) is an irreducible cuspidal representation of some \(\text{GL}(d,F)\) and \(\rho\times\pi\) denotes the representation of \(G(n+d,F)\) parabolically induced from the representation \(\rho\otimes\pi\) of \(\text{GL}(d,F)\times G(n,F)\) (\(G(n+d)\) is obtained from \(G(n)\) by adding \(d\) hyperbolic planes to the vector space on which \(G(n)\) acts). There are results concerning the reducibility points of \(\rho||^x \times\pi\) (\(\rho\) unitary, \(x\in {\mathbb R}\)). The connection with Jordan blocks, defined in \textit{C. Moeglin} [J. Eur. Math. Soc. 4, 143-200 (2002; Zbl 1002.22009)], is discussed. A particular case is treated, in which reducibility for some \(x > 0, x \neq {1\over 2}\), occurs only if \(\rho\) is a quadratic character. Then \(\pi\) is a ``quadratic unipotent representation''. An important tool in the proofs is the wave front of representations.
scientific article; zbMATH DE number 1995423
scientific article