  For an arbitrary finite monoid $M$ and subgroup $K$ of the unit group of $M$,
we prove that there is a bijection between irreducible representations of $M$
with nontrivial $K$-fixed space and irreducible representations of
$\mathcal{H}_K$, the convolution algebra of $K\times K$-invariant functions
from $M$ to $F$, where $F$ is a field of characteristic not dividing $|K|$.
When $M$ is reductive and $K = B$ is a Borel subgroup of the group of units,
this indirectly provides a connection between irreducible representations of
$M$ and those of $F[R]$, where $R$ is the Renner monoid of $M$. We conclude
with a quick proof of Frobenius Reciprocity for monoids for reference in future
papers.
