The principal block of a $\mathbb{Z}_\ell$-spets and Yokonuma type algebras

TL;DR

This paper formulates conjectures about the dimension of the principal block of a $ ext{Z}_ ext{ell}$-spets, supported by partial proofs, and introduces Yokonuma type algebras for torus normalisers in $ ext{ell}$-compact groups.

Contribution

It proposes new conjectures on $ ext{Z}_ ext{ell}$-spets principal block dimensions and introduces Yokonuma type algebras for $ ext{ell}$-compact groups.

Findings

Conjectures are verified in specific cases.

Introduction of Yokonuma type algebras for torus normalisers.

Potential relevance for representation theory of $ ext{Z}_ ext{ell}$-spets.

Abstract

We formulate conjectures concerning the dimension of the principal block of a ${\mathbb Z}_\ell$-spets (as defined in our earlier paper), motivated by analogous statements for finite groups. We show that these conjectures hold in certain situations. For this we introduce and study a Yokonuma type algebra for torus normalisers in $\ell$-compact groups which may be of independent interest.