  We study convolution semigroups of invariant/finitely satisfiable Keisler
measures in NIP groups. We show that the ideal (Ellis) subgroups are always
trivial and describe minimal left ideals in the definably amenable case,
demonstrating that they always form a Bauer simplex. Under some assumptions, we
give an explicit construction of a minimal left ideal in the semigroup of
measures from a minimal left ideal in the corresponding semigroup of types
(this includes the case of SL$_{2}(\mathbb{R})$, which is not definably
amenable). We also show that the canonical push-forward map is a homomorphism
from definable convolution on $\mathcal{G}$ to classical convolution on the
compact group $\mathcal{G}/\mathcal{G}^{00}$, and use it to classify
$\mathcal{G}^{00}$-invariant idempotent measures.
