[Paper Review] Cocenters of $p$-adic groups, I: Newton decomposition

Xuhua He|arXiv (Cornell University)|Oct 15, 2016

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper introduces the Newton decomposition for $p$-adic reductive groups, providing a canonical decomposition of the cocenter of the Hecke algebra into Newton strata. It establishes Iwahori-Matsumoto-type generators for each Newton component and proves a generalized Howe's conjecture for both ordinary and twisted invariant distributions, with explicit structure for the rigid cocenter.

ABSTRACT

In this paper, we introduce the Newton decomposition on a connected reductive $p$-adic group $G$. Based on it we give a nice decomposition of the cocenter of its Hecke algebra. Here we consider both the ordinary cocenter associated to the usual conjugation action on $G$ and the twisted cocenter arising from the theory of twisted endoscopy. We give Iwahori-Matsumoto type generators on the Newton components of the cocenter. Based on it, we prove a generalization of Howe's conjecture on the restriction of (both ordinary and twisted) invariant distributions. Finally we give an explicit description of the structure of the rigid cocenter.

Motivation & Objective

• To understand the structure of the cocenter of the Hecke algebra for $p$-adic reductive groups, particularly in the integral form over $\mathbb{Z}[p^{-1}]$.
• To extend the duality between cocenters and representations to the twisted setting via twisted endoscopy.
• To provide a decomposition of the cocenter into Newton strata that respects both the conjugation action and compact support.
• To establish a framework for studying mod-$l$ representations through the integral form of the Hecke algebra.
• To give an explicit description of the rigid cocenter and its decomposition via Newton components.

Proposed method

• Introduce the Newton decomposition of a $p$-adic group $G$ as a disjoint union $G = \coprod_{\nu} G(\nu)$, where each $G(\nu)$ is a $G$-orbit of an open compact subset $X_\nu$.
• Define the Hecke algebra $H$ and its cocenter $\bar{H}$, and show that $H = \bigoplus_\nu H(\nu)$ and $\bar{H} = \bigoplus_\nu \bar{H}(\nu)$ via the Newton decomposition.
• Use Iwahori-Matsumoto type generators to describe elements in each Newton component $\bar{H}(\nu)$, particularly for congruence subgroups $\mathcal{I}_n$.
• Analyze the twisted cocenter using $\theta$-twisted conjugacy classes and $\theta$-invariant distributions, focusing on the rigid part of the group.
• Construct a finite-dimensional model for the space of $\theta$-invariant distributions via standard pairs $(\mathcal{P}, \tau)$ and define relations via equations in the parameter set $A$.
• Prove that the kernel of the duality map is spanned by commutators $[H_R(\mathcal{P}), H_R(\mathcal{P}\dot{\tau})]_{\theta,\omega}$ and relations from the set $A$.

Experimental results

Research questions

• RQ1 How can the cocenter of the Hecke algebra of a $p$-adic group be decomposed in a way that respects both conjugation and compact support?
• RQ2 What is the structure of the twisted cocenter arising from endoscopic transfer, and how does it relate to invariant distributions?
• RQ3 Can Iwahori-Matsumoto type generators be used to describe the Newton components of the cocenter?
• RQ4 How does the rigid cocenter decompose under the Newton stratification, and what is its dual space of invariant distributions?
• RQ5 What are the defining relations for the space of $\theta$-invariant distributions supported on a $\theta$-stable subset of the rigid part of $G$?

Key findings

• The cocenter $\bar{H}$ of the Hecke algebra admits a canonical decomposition $\bar{H} = \bigoplus_\nu \bar{H}(\nu)$ indexed by Newton points $\nu$, with each $\bar{H}(\nu)$ corresponding to a Newton stratum $G(\nu)$.
• For congruence subgroups $\mathcal{I}_n$, the cocenter $\bar{H}(G, \mathcal{I}_n)$ also admits a Newton decomposition $\bar{H}(G, \mathcal{I}_n) = \bigoplus_\nu \bar{H}(G, \mathcal{I}_n; \nu)$, even though the strata are not stable under $\mathcal{I}_n$-action.
• The space of $\theta$-invariant distributions on the rigid part of $G$ decomposes as $J(G)_{\text{rig}} = J(G)_{\text{rig}}^0 \oplus J(G)_{\text{rig}}^1$, where $J(G)_{\text{rig}}^0$ is dual to the $\theta$-invariant part of the rigid cocenter.
• The kernel of the duality map from $\bar{H}_R^{\text{rig}}$ to $J(G)_{\text{rig}}$ is spanned by commutators $[H_R(\mathcal{P}), H_R(\mathcal{P}\dot{\tau})]_{\theta,\omega} \cap H_R(\mathcal{P}\dot{\tau}, \mathcal{I}_n)$ and relations from a finite set $A$ of 5-tuples.
• The image of the restriction map of $J(G)_{\text{rig}}^0$ to $\bigoplus_{(\mathcal{P},\tau)\in\text{StP}_0} H_R(\mathcal{P}\dot{\tau}, \mathcal{I}_n)$ equals the dual of the image of $\bigoplus_{(\mathcal{P},\tau)\in\text{StP}_0} H_R(\mathcal{P}\dot{\tau}, \mathcal{I}_n)$ in $\bar{H}_R^{\text{rig}}$.
• The generalized Howe's conjecture holds: the space of $\theta$-invariant distributions on $G$ is isomorphic to the dual of the $\theta$-invariant part of the rigid cocenter, with explicit relations defined by the finite set $A$.