[Paper Review] Cocenter of $p$-adic groups, II: induction map

Xuhua He|arXiv (Cornell University)|Nov 21, 2016

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper constructs an induction map between Newton components of the cocenter of $p$-adic groups' Hecke algebras, proving it is surjective and adjoint to the Jacquet functor. In characteristic zero, the map is shown to be an isomorphism, leading to a Bernstein-Lusztig-type presentation of the cocenter and establishing compatibility with the Newton decomposition.

ABSTRACT

In this paper, we study some relation between the cocenter $\bar H(G)$ of the Hecke algebra $H(G)$ of a connected reductive group $G$ over an nonarchimedean local field and the cocenter $\bar H(M)$ of its Levi subgroups $M$. Given any Newton component of $\bar H(G)$, we construct the induction map $\bar i$ from the corresponding Newton component of $\bar H(M)$ to it. We show that this map is surjective. This leads to the Bernstein-Lusztig type presentation of the cocenter $\bar H(G)$, which generalizes the work \cite{HN2} on the affine Hecke algebras. We also show that the map $\bar i$ we constructed is adjoint to the Jacquet functor and in characteristic $0$, the map $\bar i$ is an isomorphism.

Motivation & Objective

• To define and study the induction map $\bar{i}_{M,R}$ from the cocenter of a Levi subgroup $M$ to the cocenter of a reductive $p$-adic group $G$, compatible with the Newton decomposition.
• To establish that this map is adjoint to the Jacquet functor on the level of representations, extending known results from characteristic zero.
• To show that the map preserves integral structures over $\mathbb{Z}[1/p]$, enabling applications to mod-$l$ representations.
• To prove that the map is an isomorphism in characteristic zero, generalizing the Bernstein-Lusztig presentation for affine Hecke algebras.
• To demonstrate compatibility of the Newton decomposition with the induction map, ensuring the structure is preserved across Levi subgroups.

Proposed method

• Construct the induction map $\bar{i}_{v}: \bar{H}(M;v) \to \bar{H}(\bar{v})$ on Newton components using the Newton decomposition $\bar{H}(G) = \oplus \bar{H}(v)$, where $v$ is a dominant rational coweight.
• Use Bushnell's construction of $P_v$-positive elements and the injective algebra homomorphism $j_{v,\mathcal{K}}: H^v(M,\mathcal{K}_M) \to H(G,\mathcal{K})$ to define the map on the level of Hecke algebras.
• Lift the map to the cocenter via the quotient $\bar{H}(G) = H(G)/[H(G), H(G)]$, ensuring compatibility with the trace map and the Jacquet functor.
• Apply the Mackey formula and analyze the positive part of algebraic functions on the space of characters $\Psi(M)_R$ to relate traces on $G$ and $M$.
• Use the spectral density theorem over $\mathbb{C}$ to show that if the trace of $\bar{i}_v(f)$ vanishes on all induced representations, then $f$ lies in the kernel of the trace on $M$, proving surjectivity.
• Prove that in characteristic zero, the map $\bar{i}_\nu$ is injective by using the spectral density theorem over $\mathbb{C}$ and freeness of the cocenter over $\mathbb{Z}[1/p]$.

Experimental results

Research questions

• RQ1 Is there a well-defined induction map $\bar{i}_{M,R}$ from the cocenter of a Levi subgroup $M$ to the cocenter of $G$ that is compatible with the Newton decomposition?
• RQ2 Can the induction map be constructed over the integral form $\bar{H}(G)$ with coefficients in $\mathbb{Z}[1/p]$, preserving structure for mod-$l$ representation theory?
• RQ3 Is the induction map adjoint to the Jacquet functor on the level of cocenters?
• RQ4 Does the map $\bar{i}_{M,R}$ become an isomorphism in characteristic zero?
• RQ5 Is the Newton decomposition compatible with the induction map, i.e., does $\bar{i}_v$ map $\bar{H}(M;v)$ into $\bar{H}(\bar{v})$?

Key findings

• The induction map $\bar{i}_v: \bar{H}(M;v) \to \bar{H}(\bar{v})$ is surjective for any Levi subgroup $M$ and Newton component $v$.
• The map $\bar{i}_v$ is adjoint to the Jacquet functor $r_{M,R}$, meaning $\mathrm{Tr}^G(\bar{i}_v(f), \pi) = \mathrm{Tr}^M(f, r_{M,R}(\pi))$ for all $\pi \in \mathfrak{R}(G)_R$.
• In characteristic zero, the map $\bar{i}_\nu: \bar{H}(M;\nu) \xrightarrow{\cong} \bar{H}(\bar{\nu})$ is an isomorphism, establishing a strong structural result.
• The kernel of the trace map on $\bar{H}(G)_R$ decomposes as $\bigoplus_{v \in V_+} (\ker \mathrm{Tr}^G_R \cap \bar{H}_R(v))$, showing compatibility with Newton components.
• The map $\bar{i}_v$ satisfies $\bar{i}_v^{-1}(\ker \mathrm{Tr}^G_R \mid_{\bar{H}_R(\bar{v})}) = \ker \mathrm{Tr}^M_R \mid_{\bar{H}_R(M;v)}$, confirming compatibility with kernels of trace maps.
• The construction preserves integral structures over $\mathbb{Z}[1/p]$, enabling applications to mod-$l$ representations and extending results beyond characteristic zero.