Dipendra Prasad - Homological aspects of the branching problem GGP for p-adics

Abstract

Branching laws for (complex) representations of groups describe how a Representation of a group $G$ decomposes when restricted to a subgroup $H$ which in the case of compact groups is a Direct sum of irreducible representations. When dealing with infinite dimensional representations of a p-adic group $G$ restricted to a subgroup $H$, the branching laws, as they are usually studied in the subject — and make up a large body of works in the subject — describe irreducible representations of $H$ which arise as a quotient representation. It is not clear if this allows one to describe how an irreducible representation of $G$ restricted to $H$ looks like. Homological algebra methods suggest thinking not only about $HomH({\pi }_1,{\pi }_2)$, but also $ExtiH({\pi }_1,{\pi }_2)$, and $EPH({\pi }_1,{\pi }_2)$. The lecture is an attempt to discuss some of these questions.

Talk

This talk will be analogous to restriction of compact Lie group too their maximal tori.

Branching Laws: Clebsch Gordan Theorem. This has to do with tensor product of representations of $SU(2)$.

Unitary groups: Here we might be interested in the decomposition obtained from $U(n)\subseteq U(n+1)$.

Branching Laws of compact groups have some very nice properties. Those will get lost as we study non-compact groups.

Noncompact groups

Since this is a big field we focus on examples with multiplicity one (whatever that means) We discuss some stuff on parametrisations. They allow us, to give explicit descriptions. Especially interesting are the groups $G{L}_n,S{O}_n,U_n$ and their $n-1$ subgroups.

By considering many pairs simultaneously, we get additional results.

There also was some talk on Cuspidal module.

I can’t concentrate anymore so I just start writing down keywords:

Keywords

• Bessel model
• Locally finitely generated representation
• Hecke algebra
• Levi subgroup
• Bernstein decomposition
• Smooth representation
• Irreducible representation
• Bernstein block
• Noetherian ring
• Whittaker model
• Bernstein-Zelevinsky filtration
• Branching law
• Modul (Algebra)
• Projective module
• Iwahori block

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