The following observation between the spaces of global sections of line bundles on the nilpotent cone and the Hall-Littlewood polynomials is made in a recent physics preprint [1403.0585](http://arxiv.org/abs/1403.0585). Is this a known mathematical fact?


Pick a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$. Given an element $w$ in the positive Weyl chamber of the weight lattice of $\mathfrak{g}$, there is a natural ad $\mathfrak{g}$-equivariant line bundle $L\_w$ on $\mathcal{N}$. Let $V\_w=H^0(\mathcal{N},L\_w)$. $V\_w$ has a natural action of $G\times \mathbb{C}^\times$ (where $G$ is a Lie group for $\mathfrak{g}$).


Then the character of $V\_w$ as a $G\times \mathbb{C}^\times$ representation is (essentially) given by the Hall-Littlewood polynomial $P\_w(x,q)$ of type $\mathfrak{g}$ where $(x,q)\in G\times \mathbb{C}^\times$.


I must confess that I don't know how to construct $L\_w$ for general $\mathfrak{g}$. For type $A$, it goes as follows. Let $\mathfrak{g}=\mathfrak{sl}(N)$. The nilpotent cone $\mathcal{N}$ is a holomorphic symplectic quotient of a linear space (in particular, a quiver variety in the sense of Nakajima). In particular, it has the form


$\mathcal{N}=X / GL(1)\times GL(2) \times \cdots GL(N-1)$


Then the line bundle $L\_w$ is 


$L\_w = (X\times \mathbb{C} )/GL(1)\times GL(2) \times \cdots GL(N-1)$


where $GL(1)\times GL(2) \times \cdots GL(N-1)$ acts on $\mathbb{C}$ by a one-dimensional representation $\chi\_w: GL(1)\times GL(2) \times \cdots GL(N-1)\to \mathbb{C}^\times$ determined by $w$ (a weight vector in the positive Weyl chamber of the weight lattice of $\mathfrak{sl}(N)$.


I would also appreciate if you could tell me how $L\_w$ can be constructed for other $\mathfrak{g}$, as part of the question.



Hall-Littlewood functions and functions on the nilpotent cone
I took notes on a lecture that Mark Haiman gave explaining how to do a similar thing on the cotangent bundle on the flag variety: (see 2.1 of <http://math.berkeley.edu/~monks/seminars/Notes.pdf> )


This is close to what you are asking if we consider the Springer resolution of the cotangent bundle to the nilpotent cone, but I do not know how to get your construction on the nose.


