|Ph.D Student||Maxim Gurevich|
|Subject||On Distinguished Repesentations of P-Adic General Linear|
|Department||Department of Mathematics||Supervisor||Professor Offen Omer|
|Full Thesis text|
The representation theory of reductive linear groups over non-Archimedean local fields has long been a topic of extensive research. It both complements the classical study of symmetry through Lie groups, and serves as a tool in modern formulations of number-theoretic problems.
An irreducible smooth representation of a p-adic group G is said to be H-distinguished, for a subgroup H<G, when there is a non-zero H-invariant linear form on the representation. The H-distinguished representations of G are the basic objects of harmonic analysis on the space G/H.
We treat two problems on distinguished representations in cases where G/H is a p-adic symmetric space and the group G is the general linear group.
In the first part, we let E/F to be a quadratic extension of p-adic fields. We prove that every smooth irreducible ladder representation of the group GLn(E) which is contragredient to its own Galois conjugate, possesses the expected distinction properties relative to the subgroup GLn(F). This affirms a conjecture attributed to Jacquet for a large class of representations. Along the way, we prove a reformulation of the conjecture which concerns standard modules in place of irreducible representations.
In the second part, we study the role of the mirabolic subgroup P of G = GLn(F) for smooth irreducible representations of G that are distinguished relative to a subgroup of the form Hk = GLk(F) x GLn-k(F). We show that if a non-zero H1-invariant linear form exists on a representation, then the a priori larger space of forms invariant to the intersection of P with H1 is one-dimensional. When k>1, we give a reduction of the same problem to a question about invariant distributions on the nilpotent cone tangent to the symmetric space G/Hk. Some new distributional methods for non-reductive groups are developed.