Models of representations of general linear groups over p-adic fields

Abstract

Let F denote a p-adic field and D a quaternion division algebra over F. Ginzburg-Rallis models (abbreviated as G-R models), were discovered in "The exterior cube L-function for GL(6)" by D. Ginzburg and S. Rallis when they computed exterior cube L-functions for GL6. They made a conjecture about the relation between nonvanishing of the central value of exterior cube L-function for GL6 and the realization of G-R models on quaternion algebras. This conjecture motivates the investigation of G-R models on GL6(.') and GL3(D). In the first part of this paper, the proofs of uniqueness of G-R models on GL6(.') and GL3(D) are given. Klyachko models, also known as Whittaker-symplectic models, were first established by A.A. Klyachko in `Models for the complex representations of the groups GL(n, q)' over finite fields, and then realized by M. J. Heumos and S. Rallis in "Symplectic-Whittaker models for GLn" over p-adic fields. They showed that the existence of a unique Klyachko model for each irreducible unitary representation of GL4(F). The second part of this paper extends the result to GL5 (.T') and explicitly classified a unique Klyachko model for each irreducible unitary representation of GL5(F).