Recipes to compute the algebraic K-theory of Hecke algebras of reductive p-adic groups

TL;DR

This work computes the algebraic $K$-theory of Hecke algebras $\\mathcal{H}(G;R)$ for td-groups $G$ linked to reductive $p$-adic groups by exploiting the Farrell–Jones conjecture. It builds a smooth, equivariant Atiyah–Hirzebruch spectral sequence with $E^2_{p,q}=S H^{G,\\mathcal{F}}_p(X;\\overline{K_q(\\mathcal{H}(?;R))})$ that converges to $K_{p+q}(\\mathcal{H}(G;R))$, and reduces computations to Bredon-type homology with coefficients coming from the Hecke algebras of open subgroups. The paper gives explicit recipes to compute $K_0$ by analyzing a $G$-CW model for $E_{\\mathcal{C}\mathrm{op}}(G)$, specializes to concrete groups such as $\\mathrm{SL}_n(F)$, $\\mathrm{PGL}_n(F)$, and $\\mathrm{GL}_n(F)$ via Bruhat–Tits buildings, and extends the framework to central extensions and central characters. The combination of assembly, spectral sequence, and the homotopy-colimit perspective yields practical, finite-datum computations of the projective class group and higher $K$-theory for these Hecke algebras, with potential impact on smooth representations and related arithmetic questions.

Abstract

We compute the algebraic K-theory of the Hecke algebra of a reductive p-adic group G using the fact that the Farrell-Jones Conjecture is known in this context. The main tool will be the properties of the associated Bruhat-Tits building and an equivariant Atiyah-Hirzebruch spectral sequence. In particular the projective class group can be written as the colimit of the projective class groups of the compact open subgroups of G.

Theorems & Definitions (20)

• Theorem 1.1: Main Theorem
• Theorem 2.1
• Definition 2.2: Bredon homology
• proof : Proof of Theorem \ref{['the:equivariant_Atyiah-Hirzebruch_spectral_sequence_smooth']}
• Remark 2.4
• Theorem 2.5
• proof
• Lemma 2.6
• Remark 2.7
• Theorem 3.1