On representations of \({\mathfrak p}\)-adic split and non-split symplectic groups and their character relations
Over a non-Archimedean local field F, the quaternionic unitary group with similitudes GUq(n) is \(\bar F\)-isomorphic to GSp(n), the symplectic group. For \(n=1\), \textit{H. Jacquet} and \textit{R. P. Langlands} [Automorphic forms on GL(2), Lect. Notes Math. 114 (1970; Zbl 0236.12010)] established a correspondence between certain of the irreducible admissible representations of GSp(1,F)\(\cong GL(2,F)\) and those of \(GUq(1,F)\cong D^{\times}\), where D is a division quaternion algebra over F. This note describes a similar correspondence, in terms of characters, for certain series of representations of GSp(2,F) and GUq(2,F), assuming the residue characteristic is odd. Both groups contain Cartan subgroups isomorphic to the multiplicative groups of quadratic extensions of F. These tori are contained in parabolic subgroups whose Levi components are essentially GL(2,F) or \(D^{\times}\), respectively. If one takes representations of these parabolic subgroups which are related by the Jacquet-Langlands correspondence and induces to GSp(2) and GUq(2) respectively, then it is reasonable to expect a relationship between the induced representations. The authors show their characters are each other's negatives, by calculating the characters explicitly.
division quaternion algebra over non-Archimedean local field
relationship between induced representations
scientific article; zbMATH DE number 3845711
scientific article