On residues of certain intertwining operators

Abstract: Let $G$ be a connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. Let $P = MN$ be a Levi decomposition of a maximal parabolic subgroup of $G$, and $\pi$ an irreducible unitary supercuspidal representation of $M(F)$. One can then consider the representation $Ind_{P(F)}^{G(F)} \pi$ (normalized parabolic induction). Assume that $P$ is conjugate to an opposite by an element $w_0 \in G(F)$ that normalizes $M$, and which fixes the isomorphism class of $\pi$ (i.e., $\pi \cong \,^{w_0}\pi$). Then, by the work of Harish-Chandra, $Ind_{P(F)}^{G(F)} \pi$ is irreducible if and only if a certain family $A(s, \pi, w_0)$ of so called intertwining operators has a pole at $s = 0$. In this case, after making certain choices, the residue of $A(s, \pi, w_0)$ at $s = 0$ can be captured by a scalar $R(\tilde \pi) \in \mathbb{C}$, which has a conjectural expression in terms of some gamma factors related to Shahidi's local coefficients, as described by Arthur's local intertwining relation.

Following a program pioneered by Freydoon Shahidi, and furthered by him as well as David Goldberg, Steven Spallone, Wen-Wei Li, Li Cai, Bin Xu, Xiaoxiang Yu etc., one seeks to:

(a) get explicit expressions to describe $R(\tilde \pi)$ ; and (b) interpret the resulting expression for $R(\tilde \pi)$ suitably, using the theory of endoscopy when applicable.

So far, these questions have been studied mostly for classical (including unitary) groups, or in some simple situations. We will discuss (a) above in a non-classical and slightly "less simple" situation, in the cases where $G$ is an almost simple group whose absolute root system is of exceptional type or of type $B_n$ with $n \geq 3$ or $D_n$ with $n \geq 4$, and where $P$ is a "Heisenberg parabolic subgroup". We will then comment on what we can say of (b) above in the $G_2$, $B_3$ and $D_4$ cases. Though the reducibility results and the $R(\tilde \pi)$ values are more or less already known in these cases by the Langlands-Shahidi method and related results (e.g., the work of Henniart and Lomeli and Caihua Luo in the case of $D_4$), our investigations also suggest the existence of harmonic analytic expressions for certain gamma values, which in some cases just amount to the formal degree conjecture of Ichino, Ikeda and Hiraga, but in other cases seem slightly unwieldy and perhaps intriguing.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

( slides )

Comments: Zoom link: us02web.zoom.us/j/81501452154?pwd=OWVtRmU0bFpoMEY3OUxrVW04STFJQT09

Zoom number: 815 0145 2154

Zoom password: 363804

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POINTS - Peking Online International Number Theory Seminar

Series comments: Description: Seminar on number theory and related topics

This seminar series is sponsored by the Beijing International Center of Mathematical Research (BICMR) and the School of Mathematical Sciences of Peking University.

The conference number and password are available on the external website. See also the announcements on bicmr.pku.edu.cn

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