On a lifting problem of L-packets

TL;DR

This paper explores the relationship between L-packets of related reductive groups over local fields, aiming to construct L-packets of a larger group from those of a subgroup, using conjectural endoscopy theory.

Contribution

It demonstrates how conjectural properties of L-packets can be derived from endoscopy theory, providing structural insights into L-packets of larger groups based on smaller ones.

Findings

L-packets of G can be deduced from those of e G using endoscopy theory.

Structural information about L-packets of e G is obtained from that of G.

The approach applies to groups like G = Sp(2n) and e G = GSp(2n).

Abstract

Let $G \subseteq \tilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and $G_{der} = \tilde{G}_{der}$. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of that of $\tilde{G}$. Motivated by this, we hope to construct the L-packets of $\tilde{G}$ from that of $G$. The primary example in our mind is when $G = Sp(2n)$, whose L-packets have been determined by Arthur (2013), and $\tilde{G} = GSp(2n)$. As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of $\tilde{G}$ from that of $G$.