If we just consider central representations, i.e., those for which $z$ acts by a nonzero scalar, then up to isomorphism there is a unique irreducible representation of the algebra. It is infinite dimensional, given by polynomials in one variable. If the action of $z$ is multiplication by $\lambda \in K$, then we can write the representation as $K[x]$, where the action of $x$ is multiplication by $x$, and the action of $y$ is $\lambda \frac{\partial}{\partial x}$.

If you want the algebra specified more rigidly (i.e., you want to remember the specific actions of the elements $x,y,z$ instead of just the algebra structure), then there is a parameter space of irreducible representations, with one coordinate describing the action of $z$ (i.e., taking values in nonzero elements of $K$), and the rest of the coordinates describing a line in the span of $x,y$. This generalizes to the $2n+1$ dimensional setting, where the representation is given by polynomial functions on a Lagrangian subspace of a $2n$ dimensional symplectic vector space. The parameter space of representations involves a torus and the Lagrangian Grassmannian, but I don't remember if it has the geometric structure of a product or some kind of fibration.

The finite length modules are in bijection with finitely generated holonomic D-modules on the affine line (resp. affine $n$-space), since the central condition endows the modules with an action of the Weyl algebra, which is a quotient of the universal enveloping algebra.

For non-central representations, things are considerably messier. When $z$ acts trivially, you're asking for pairs (or $2n$-tuples) of commuting matrices, and when $z$ is arbitrary, you can have finite dimensional representations look a lot like representations of arbitrary nilpotent lie algebras.

Regarding references, I guess anything about D-modules should work, but depending on your background, they may be hard to read. Howe has a paper called "A century of Lie theory", and Rosenberg has a paper called "A selective history of the Stone-von-Neumann theorem", both of which could be useful.

If we just consider central representations, i.e., those for which $z$ acts by a nonzero scalar, then up to a certain kind of equivalence (given by conjugation with algebra isomorphisms) there is a unique irreducible representation of the algebra. It is infinite dimensional, given by polynomials in one variable. If the action of $z$ is multiplication by $\lambda \in K^\times$, then we can write the representation as $K[x]$, where the action of $x$ is multiplication by $x$, and the action of $y$ is $\lambda \frac{\partial}{\partial x}$.

If you want to rigidify the equivalence to isomorphism, (i.e., you want to remember the specific actions of the elements $x,y,z$ instead of just the algebra structure), then there is a parameter space of irreducible representations, with one coordinate describing the action of $z$ (i.e., taking values in nonzero elements of $K$), and the rest of the coordinates describing a line in the span of $x,y$. The construction generalizes to the $2n+1$ dimensional setting, where the representation is given by polynomial functions on a Lagrangian subspace of a $2n$ dimensional symplectic vector space, but you lose uniqueness. The parameter space of these representations involves a torus and the Lagrangian Grassmannian, but I don't remember if it has the geometric structure of a product or some kind of fibration.

The finite length modules are in bijection with finitely generated holonomic D-modules on the affine line, since the central condition endows the modules with an action of the Weyl algebra, which is a quotient of the universal enveloping algebra. As zamanjan notes in a comment here, when $n > 1$, this fails rather spectacularly. Stafford (1983) showed that there are irreducible modules with characteristic cycle of dimension $2n-1 > n$, and Bernstein-Lunts showed that in the $n=2$ case, this is in some sense a property of the generic irreducible module.

For non-central representations, things are considerably messier. When $z$ acts trivially, you're asking for pairs (or $2n$-tuples) of commuting matrices, and when $z$ is arbitrary, you can have finite dimensional representations look a lot like representations of arbitrary nilpotent lie algebras.

Regarding references, I guess anything about D-modules should work, but depending on your background, they may be hard to read. Howe has a paper called "A century of Lie theory", and Rosenberg has a paper called "A selective history of the Stone-von-Neumann theorem", both of which could be useful.

If you want something more general than central representations, you might consider (the elder) Kirillov's work on the orbit method. I don't know much about it.

If we just consider central representations, i.e., those for which $z$ acts by a nonzero scalar, then up to isomorphism there is a unique irreducible representation of the algebra. It is infinite dimensional, given by polynomials in one variable. If the action of $z$ is multiplication by $\lambda \in K$, then we can write the representation as $K[x]$, where the action of $x$ is multiplication by $x$, and the action of $y$ is $\lambda \frac{\partial}{\partial x}$.

If you want the algebra specified more rigidly (i.e., you want to remember the specific actions of the elements $x,y,z$ instead of just the algebra structure), then there is a parameter space of irreducible representations, with one coordinate describing the action of $z$ (i.e., taking values in nonzero elements of $K$), and the rest of the coordinates describing a line in the span of $x,y$. This generalizes to the $2n+1$ dimensional setting, where the representation is given by polynomial functions on a Lagrangian subspace of a $2n$ dimensional symplectic vector space. The parameter space of representations involves a torus and the Lagrangian Grassmannian, but I don't remember if it has the geometric structure of a product or some kind of fibration.

The finite length modules are in bijection with finitely generated holonomic D-modules on the affine line (resp. affine $n$-space), since the central condition endows the modules with an action of the Weyl algebra, which is a quotient of the universal enveloping algebra.

Regarding references, I guess anything about D-modules should work, but depending on your background, they may be hard to read. Howe has a paper called "A century of Lie theory", and Rosenberg has a paper called "A selective history of the Stone-von-Neumann theorem", both of which could be useful.

If we just consider central representations, i.e., those for which $z$ acts by a nonzero scalar, then up to isomorphism there is a unique irreducible representation of the algebra. It is infinite dimensional, given by polynomials in one variable. If the action of $z$ is multiplication by $\lambda \in K$, then we can write the representation as $K[x]$, where the action of $x$ is multiplication by $x$, and the action of $y$ is $\lambda \frac{\partial}{\partial x}$.

If you want the algebra specified more rigidly (i.e., you want to remember the specific actions of the elements $x,y,z$ instead of just the algebra structure), then there is a parameter space of irreducible representations, with one coordinate describing the action of $z$ (i.e., taking values in nonzero elements of $K$), and the rest of the coordinates describing a line in the span of $x,y$. This generalizes to the $2n+1$ dimensional setting, where the representation is given by polynomial functions on a Lagrangian subspace of a $2n$ dimensional symplectic vector space. The parameter space of representations involves a torus and the Lagrangian Grassmannian, but I don't remember if it has the geometric structure of a product or some kind of fibration.

The finite length modules are in bijection with finitely generated holonomic D-modules on the affine line (resp. affine $n$-space), since the central condition endows the modules with an action of the Weyl algebra, which is a quotient of the universal enveloping algebra.

For non-central representations, things are considerably messier. When $z$ acts trivially, you're asking for pairs (or $2n$-tuples) of commuting matrices, and when $z$ is arbitrary, you can have finite dimensional representations look a lot like representations of arbitrary nilpotent lie algebras.

Regarding references, I guess anything about D-modules should work, but depending on your background, they may be hard to read. Howe has a paper called "A century of Lie theory", and Rosenberg has a paper called "A selective history of the Stone-von-Neumann theorem", both of which could be useful.