# A fractional Quantum Hall state is translational invariant on a Riemann sphere

In this note (I guess presentation) which is about the relation between certain Jack polynomials and FQH states, Haldane states the following (paraphrasing slightly)

Generic incompressible FQHE droplets covering an area $2\pi N_\Phi$ are described by polynomial wavefunctions $f(z_1, \cdots, z_N)$ ($N$ number of particles) obeying both \begin{aligned} &L^+f:=\left[\sum_{i=1}^N \frac{\partial}{\partial z_i}\right]f(z_1, \cdots, z_N)=0 && \text{highest weight condition}\\ &L^-f:=\left[N_\Phi(z_1+\cdots, z_N)-\sum_{i=1}^N z_i^2\frac{\partial}{\partial z_i}\right]f(z_1, \cdots, z_N)=0 && \text{lowest weight condition} \end{aligned} This is the condition that the state is translationally-invariant when put on the Riemannn sphere.

In this paper section II, Haldane & Bernevig furthermore say that

Uniform states on the sphere satisfy the conditions $L^\pm \psi = 0$.

In the same paper the origin of why these condition are called highest/lowest becomes clear.

I'm looking for a reference for/proof of all of these facts:

• Generic incompressible FQHE droplets covering an area $2\pi N_\Phi$ are described by polynomial wavefunctions $f(z_1, \cdots, z_N)$ ($N$ number of particles) obeying $L^\pm f = 0$.
• The fact that the above condition is equivalent to saying the state is translationally-invariant when put on the Riemannn sphere.

• And finally an explanation of what is meant by "Uniform states on the sphere satisfy the conditions $L^\pm \psi = 0$." (I'm guessing the single particle density obtained from the respective wavefunction is uniform on the sphere?) and furthermore why does $L^\pm \psi=0$ imply this assertion?

One thing that I know is a polynomial $f$ satisfies the highest weight condition if and only if for all $c\in \mathbb{C}$ we have $$f(z_1+c, \cdots, z_N+c) = f(z_1, \cdots, z_N)$$ This is not hard to prove with a simple Taylor explansion and this implies translational invariance over $\mathbb{C}$. It is not exactly clear to me what's the lowest weight condition is about however. Neither I know how the rest of the conclusions are made.