The second part of Theorem 3.10.2 of "Introduction to representation
theory" by Etingof et al. states that
if $A$ and $B$ are $k-$algebras ($k$ an algebraically closed field) and $M$ is an irreducible finite dimensional representation of $A\otimes_k B,$ then $M\cong V\otimes_k W$ where $V$ and $W$ are finite dimensional irreducible representations of $A$ and $B$ respectively.
My question is about the first part of the remark following this theorem. This remark states that the previous proposition fails for infinite dimensional representations, "e.g. it fails when A is the Weyl algebra
in characteristic zero." I don't see how to construct an irreducible infinite dimensional representation $M$ of $A\otimes B,$
where $A$ is the Weyl algebra, such that $M\ncong V\otimes_k W$.
Best Answer
Nate's suggestion in the comments works! $M = k[x, y] e^{xy}$ is indeed both an irreducible representation of the Weyl algebra $k[x, \partial_x, y, \partial_y]$ in two variables, and not a tensor product of two irreducible representations of the Weyl algebras $k[x, \partial_x], k[y, \partial_y]$ in one variable.