  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.
