Weights for $\ell$-local compact groups

TL;DR

This paper introduces the concept of weights for $ ext{l}$-local compact groups, proposing a conjecture inspired by Alperin's Weight Conjecture and providing proofs in specific cases, along with related conjectural evidence.

Contribution

It formulates a new conjecture on weights for $ ext{l}$-local compact groups and proves it for simple cases where the index of the torus is $ ext{l}$, advancing the theory.

Findings

Conjecture relating weights to irreducible characters of the Weyl group.

Proof of the conjecture for simple $ ext{l}$-local compact groups with specific index conditions.

Evidence supporting an analogue of Robinson's Height Zero Weight Conjecture in this context.

Abstract

In this note, we initiate the study of $\mathcal{F}$-weights for an $\ell$-local compact group $\mathcal{F}$ over a discrete $\ell$-toral group $S$ with discrete torus $T$. Motivated by Alperin's Weight Conjecture for simple groups of Lie-type, we conjecture that when $T$ is the unique maximal abelian subgroup of $S$ up to $\mathcal{F}$-conjugacy and every element of $S$ is $\mathcal{F}$-fused into $T$, the number of weights of $\mathcal{F}$ is bounded above by the number of ordinary irreducible characters of its Weyl group. By combining the structure theory of $\mathcal{F}$ with the theory of blocks with cyclic defect group, we are able to give a proof of this conjecture in the case when $\mathcal{F}$ is simple and $|S:T| =\ell$. We also propose and give evidence for an analogue of the height zero case of Robinson's Ordinary Weight conjecture in this setting.