An inductive Ext non-vanishing theorem for the $p$-adic general linear group

TL;DR

This work analyzes the interplay between Ext-groups and parabolic induction for GL_n(F) within a non-semi-simple, smooth representation category. It extends a full-faithfulness framework to completed categories and proves an embedding theorem: for suitable $\tau_1,\tau_2$, there is an injection $\mathrm{Ext}^i_{\widehat{\mathcal{C}}(\mathcal{J})}(\tau_1,\tau_2) \hookrightarrow \mathrm{Ext}^i_{\widehat{\mathcal{C}}(\mathcal{J}')} (\omega\times\tau_1,\omega\times\tau_2)$, enabling inductive control of Ext under parabolic induction. Building on this, the paper defines and utilizes strong Ext relevance for Arthur-type representations, and proves an Ext-vanishing/non-vanishing dichotomy in non-tempered GGP-type branching: if $\pi_1,\pi_2$ are strong Ext relevant then $\mathrm{Ext}^i_{\mathrm{GL}_{n-1}(F)}(\pi_1,\pi_2)\neq 0$ for some $i\ge0$. The approach combines Bernstein–Zelevinsky theory, affine Hecke algebras, dualities, and a sequence of reduction lemmas (Künneth, transfer, reduction) to reduce the Ext-branching problem to manageable constituents. The results provide a structural framework for Ext-branching among Arthur-type representations and advance the understanding of homological branching laws in the p-adic GL setting, with potential implications for broader Langlands-type questions.

Abstract

We study some homological properties of the parabolic induction functor for the $p$-adic general linear group. We obtain an embedding theorem of Ext-groups in the context of parabolic induction. As an application, we establish and prove a variation of the non-tempered Gan-Gross-Prasad conjecture in homological branching laws for $p$-adic general linear groups.

Theorems & Definitions (47)

• Example 1.1
• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.3
• Remark 1.6