On disjointness of linear models and degenerate Whittaker models

In this article we reprove and generalize a result of N. Matringe [12] concerning generic representations of GL_n(F) that admits a linear model with respect to a maximal Levi subgroup GL_p(F)×GL_q(F), where F is a non-archimedean field. He showed that in this case |p−q|≤1. We extend this result to the case where F is a finite field or a local field of characteristic different from 2. Further, we study non-generic representations of GL_n(F) and describe their possible linear models in terms of their rank r(π) (see Section 6). Our arguments are based on the method of Gelfand-Kazhdan and the theory of distribution and provide uniform proofs for finite, p-adic and archimedean fields.