ON REPRESENTATIONS OF GL(n,D) WITH A SYMPLECTIC MODEL

Abstract

Let F be a non-Archimedean local field of characteristic zero, and let D be the unique quaternion division algebra over F. For n ∈ N, let Gn = GL(n,D). The subgroup Hn = Sp(n,D) of Gn denotes the unique non-split inner form of the symplectic group Sp(2n,F). A smooth admissible complex representation (π,V) of Gn is said to have a symplectic model (or to be Hn-distinguished) if there exists a linear functional φ on V such that φ(π(h)v) = φ(v) for all h ∈ Hn and v ∈ V. This thesis provides a complete list of irreducible admissible representations of G3 and G4 having a symplectic model. We demonstrate that induced representations from finite-length representations preserve the symplectic model. We also show that the Steinberg representations of Gn do not admit a symplectic model. Furthermore, we classify those ladder representations of Gn that admit a symplectic model. In addition, we prove a part of Prasad’s conjecture which provides a family of irreducible unitary representations with a symplectic model.