Lifting of characters on orthogonal and metaplectic groups
This is a fundamental paper on the lifting of characters between special orthogonal groups \(\text{SO}(2n+1)\) over \(\mathbb{R}\) and the nonlinear metaplectic group \(\widetilde{\text{SP}} (2n,\mathbb{R})\) It is a continuation of the work of the author with D. Barbasch, D. Vogan and J. Johnson. This work is closely interconnected to the examples of endoscopic transfer and theta lifting and is an aspect of the duality between root systems of type \(B_n\) and \(C_n\). The main result of the paper is the following theorem, proved as theorem 4.6. 1) The map \(\Gamma: \theta\to \theta'= \Gamma(\theta)\) is a bijection between stably invariant eigendistributions on \(\text{SO}(2n+1)\) and genuine stably invariant eigendistributions on \(\widetilde{\text{SP}} (2n,\mathbb{R})\). 2) \(\theta'\) is a virtual character if and only if \(\theta\) is a virtual character, where virtual character means a linear combination of irreducible characters with integral coefficients. 3) \(\theta'\) is tempered if and only if \(\theta\) is tempered. 4) \(\Gamma\) takes the (stable) discrete series of \(\text{SO}(2n+1)\) to the (stable) genuine discrete series of \(\widetilde{\text{SP}} (2n,\mathbb{R})\). 5) \(\Gamma\) commutes with parabolic induction. 6) If \(\theta\) is a tempered virtual character, then \(\theta'\) is the (normalized) theta lift of \(\theta\). The map \(\Gamma\) is such that the numerators in the character formulas for \(\theta\) and \(\theta'\) are the same on all Cartan subgroups, and this result is proved as proposition 9.3. The proof of the theorem is an application of T. Hirai's work on matching conditions and the induced character formula. The theorem also uses the fundamental work of Harish-Chandra for the construction of characters of representations. As an example the case of \((\text{SO}(3), \widetilde{\text{SL}} (2,\mathbb{R}))\) is discussed in detail in the final section of the paper, and this is closely related to the celebrated Shimura correspondence.
construction of characters of representations
scientific article; zbMATH DE number 1425089
scientific article