Decomposition Rules for the Ring of Representations of Non-Archimedean GLn

Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the sequence of p-adic groups $\{GL_n(F)\}_{n=0}^\infty $, with multiplication defined through parabolic induction. We study the problem of the decomposition of products of irreducible representations in $\mathcal{R}$. We obtain a necessary condition on irreducible factors of a given product by introducing a width invariant. Width $1$ representations form the previously studied class of ladder representations. We later focus on the case of a product of two ladder representations, for which we establish that all irreducible factors appear with multiplicity one. Finally, we propose a general rule for the composition series of a product of two ladder representations and prove its validity for cases in which the irreducible factors correspond to smooth Schubert varieties.

Models of Representations and Langlands Functoriality

In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.

On two questions concerning representations distinguished by the Galois involution

Let {E/F} be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the irreducible smooth representations of {\mathrm{GL}_{n}(E)} that are distinguished by its subgroup {\mathrm{GL}_{n}(F)} . One relates this class to representations which come as base change lifts from a quasi-split unitary group over F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps, we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose {\mathrm{GL}_{n}(F)} -distinction, and thus exhibit the limitations of these two approaches.