Symplectic supercuspidal representations and related problems
In this work, the authors summarize results (originally obtained by them, and many other people) related to symplectic supercuspidal representations of \(\text{GL}(2n)\) over a non-archimedean field of characteristic zero. In more detail, they discuss the existence of certain models (Shalika models, linear models, Gelfand-Graev models) on the representations of the general linear groups and related classical and metaplectic groups (related to general linear groups via parabolic induction or arising as a stabilizers of certain functionals on unipotent radicals). The existence of these models is closely related with, in part proved, in part conjectural Langlands functoriality transfer between general linear and classical (or metaplectic) groups and theta correspondences. The symplecticity condition (given initially in terms of the existence of the pole of the exterior square \(L\)-function for \(s=0\)) ensures that these cuspidal representations of general linear groups are in the image of the appropriate functorialities. We now briefly describe the content of the paper. In the first section some historical and introductory material is presented. In the second section the Shalika model (for an irreducible admissible representation of \(\text{GL}(2n)\)) is introduced (as a possible embedding in a representation induced from the character of the Shalika subgroup), and the local uniqueness of this model is stated. Then the generalized Shalika model for \(\text{SO}(4n)\) is introduced and these two are related by the parabolic induction, at least for supercuspidal representations and induction in the Siegel case, where the Levi subgroup is isomorphic to \(\text{GL}(2n)\) (as explained in Theorem 2.3; reducibility outside of zero in this induction is required). Then, another model for \(\text{GL}(2n)\) is introduced-namely the linear model, where an irreducible admissible representation of \(\text{GL}(2n)\) has a linear model if it can be embedded in the representation induced from the trivial representation of the Levi subgroup \(\text{GL}(n)\times \text{GL}(n).\) The uniqueness statement also holds. With the opposite direction proved by Jacquet and Rallis, the authors have proved that an irreducible admissible supercuspidal representation of \(\text{GL}(2n)\) has a nonzero Shalika model if and only if it has a non-zero linear model. On the other hand, linear models for \(\text{GL}(2n)\) are related with so-called linear symplectic models for \(\text{Sp}(4n)\), and both are related similarly (via parabolic induction) as in the Shalika models-case. Now the authors state Theorem 2.8 where all the different characterizations (through \(L\)-functions, existence of certain models and functoriality) of symplecticity are stated. In the third section the authors describe local descents from \(\text{GL}(2n)\) to \(\text{SO}(2n+1)\) and to \(\widetilde{\text{Sp}}(2n)\). In the first case the local descent is given in terms of Gelfand-Graev model on \(\text{SO}(4n)\) (a twisted Jacquet functor there; in this case an odd-orthogonal group \(\text{SO}(2n+1)\) appears as the stabilizer of certain character of the unipotent radical-so the twisted Jacquet functor (as a representation of \(\text{SO}(2n+1)\)) is well-defined). So, starting from a supercuspidal representation of \(\text{GL}(2n)\) with a non-zero Shalika model, we obtain, as in the previous section, by a parabolic induction, a representation of \(\text{SO}(4n)\) with a non-zero generalized Shalika model; and then applying twisted Jacquet module on (the Langlands quotient of ) that representation, we get an irreducible generic supercuspidal representation of \(\text{SO}(2n+1)\); for this representation it is known that its Langlands functorial transfer is exactly the representation of \(\text{GL}(2n)\) we started with. Analogous construction for the local descent from \(\text{GL}(2n)\) to \(\widetilde{\text{Sp}}(2n)\) is constructed via the Fourier-Jacobi model (parabolic induction from \(\text{GL}(2n)\) to \(\text{Sp}(4n)\) and then taking certain twisted Jacquet module of that representation and where a representation of \(\widetilde{\text{Sp}}(2n)\) arises). In the fourth section the authors present a diagram giving a conceptual understanding of symplecticity (unifying all the results from the previous section-all the descents from \(\text{GL}(2n)\) to \(\widetilde{\text{Sp}}(2n)\) and \(\text{SO}(2n+1)\), and relating the representations thus obtained via theta correspondence). In the fifth section the authors give some further applications of generalized Shalika models including the theory of endoscopy.
symplectic supercuspidal representations
scientific article; zbMATH DE number 5712117
scientific article