Department of Mathematics

Indian Institute of Science

Bangalore 560 012

SEMINAR

 Speaker : Sandeep Varma Affiliation : TIFR, Mumbai Subject Area : Mathematics Venue : Department of Mathematics, Lecture Hall I Time : 04.00 p.m. Date : August 16, 2016 (Tuesday) Title : "On residues of certain intertwining operators" Abstract : Let $\G$ be a connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. Let $\P = \M \N$ be a Levi decomposition of a maximal parabolic subgroup of $\G$, and $\sigma$ an irreducible unitary supercuspidal representation of $\M(F)$. One can then consider the representation $\text{Ind}_{\P(F)}^{\G(F)} \sigma$ (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. This calls for: (a) computing residues of certain families of intertwining operators; and (b) interpreting these residues suitably. 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 $\G$ and $\P$). We will discuss (b) above for some of the cases where only (a) was known previously, and also (a) for some new cases.