Example of an algebra finite over a commutative subalgebra with infinite dimensional simple modules

Let $A$ be an algebra over an algebraically closed field $k.$ Recall that if $A$ is
a finitely generated module over its center, and if its center is a finitely generated
algebra over $k,$ then by the Schur's lemma all simple $A$-modules are finite dimensional
over $k.$

Motivated by the above, I would like an example of a $k$-algebra $A,$ such that:

1) $A$
has a simple module of infinitie dimension over $k,$

2) $A$ contains
a commutative finitely generated subalgebra over which $A$ is a finitely generated
left and right module.

Thanks in advance.