A STUDY ON JACQUET FUNCTOR AND INDUCED REPRESENTATIONS

Abstract

Let F be a non-Archimedean local field and O be its ring of integers with ϖ chosen as a fixed generator for the maximal ideal of O. Define Oℓ := O/⟨ϖℓ⟩ as the finite local ring. This thesis explores the representation theory of classical groups over F and finite local rings Oℓ. In the first part, we study the semi-simplified Jacquet module of certain representations, including generic representations of the general linear group GLn(F) with respect to a maximal Levi subgroup M := GLl(F) × GLn−l(F). Utilizing our results, we prove that the Jacquet module is multiplicity-free for a specific subcategory of representations. We extend this study to the symplectic group Sp2n(F) with maximal Levi subgroupM := GLl(F)×Sp2(n−l)(F). We conclude that for a subclass of representations of Sp2n(F), the Jacquet functor is multiplicity-free. In the second part, we construct parabolically induced representations of GL2(Oℓ) for ℓ > 1 and establish an irreducibility criterion for these representations. In cases where the representation is reducible, we determine the number of irreducible constituents. Additionally, we prove that parabolically induced representations admit a Whittaker model and explore the Kirillov model for primitive cuspidal representations of GL2(Oℓ). Our results contribute to a deeper understanding of the structure of representations of classical groups over non-Archimedean local fields and finite local rings.