On Arthur packets containing a fixed tempered representation

TL;DR

This work studies the local Arthur packets for split classical $p$-adic groups and counts how many packets contain a fixed irreducible tempered representation, using the extended multi-segment parameterization and intersection theory. The authors develop a block-decomposition framework for integral extended multi-segments, and prove a recursive counting formula by analyzing interactions within and between blocks via row-exchange, union-intersection, and dual operations. They establish precise counts for blocks starting at zero and blocks not starting at zero, demonstrate the independence of blocks, and introduce a suite of raising operations (S, M, U, D) that control all equivalences inside type $Y_{\mathcal M}$ blocks, with a culminating product formula for the total count. The results provide new structural insight into Arthur packets in the tempered setting and have potential applications to theta lifts and broader Langlands-type correspondences.

Abstract

We determine the number of local Arthur packets containing a certain fixed tempered representation for classical $p$-adic groups. More specifically, given a tempered extended multi-segment supported in the integers, we determine a count for all extended multi-segments which arise from it through applications of the operators arising from the theory of intersections of local Arthur packets.

Figures (5)

• Figure 1: The $\mathcal{S}$ data for the above multi-segment is $(\{0, 1\}, \{1, 2, 3\}).$ Since $\mathcal{S}_1 \cap \mathcal{S}_2 = \{1\},$ the chain with support $[3, 1]$ is a $z$-chain.
• Figure : Type 1
• Figure : Type 1
• Figure : Type 2
• Figure : Type 3

Theorems & Definitions (178)

• Theorem 1.2: Theorem \ref{['thm-count-temp']}
• Remark 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Moe11b
• Definition 2.4
• Example 2.5
• Remark 2.6
• Theorem 2.7
• Corollary 2.8