On residues of certain intertwinig operators

Abstract: Let $\rm G$ be a connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. Let $\rm P = M N$ be a Levi decomposition of a maximal parabolic subgroup of $\rm G$, and $\sigma$ an irreducible unitary supercuspidal representation of ${\rm M}(F)$. One can then consider the representation $\text{Ind}_{{\rm P}(F)}^{{\rm G}(F)} \sigma$ (normalized parabolic induction). This induced representation is known to be either irreducible or of length two. The question of when it is irreducible turns out to be (conjecturally) related to local $L$-functions, and also to poles of a family of so called intertwining operators.

Because of this, one would like to:

(a) get expressions describing residues of certain families of intertwining operators; and

(b) interpret these residues suitably, using the theory of endoscopy when applicable.

There is an approach pioneered by Freydoon Shahidi to implement such a programme, which was developed further by him as well as by David Goldberg, Steven Spallone, Wen-Wei Li and Xiaoxiang Yu, in several cases (i.e., for various choices of $\rm G$ and $\rm P$). We will discuss (a) in the cases where $\rm G$ is an almost simple group whose absolute root system is of exceptional type or of types $B_n$ or $D_n$ with $n > 3$, and where $\rm P$ is a `Heisenberg parabolic subgroup'. If time permits, partial results towards and speculations concerning (b) will be discussed.

number theoryrepresentation theory

Audience: researchers in the topic

Number Theory and Representations in Valparaiso