Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle $T^*Fl_n$ of the complete flag variety? Negut mentioned in the first page of the paper that the partition function $Z(m)$ in the paper is closely related to the $J$-function of $T^*Fl_n$. Does it mean that the $J$-function of $T^*Fl_n$ is an eigenfunction of the Calogero-Sutherland Hamiltonian $L(m)$ written in p.5 of the paper? Or does it satisfy some integrable differential equations which is closely related to Calogero-Sutherland?

I have done simple calculation for the case of $n=2$. For $Fl_2=\mathbb{P}^1$, the $J$-function is written as
\begin{equation}
J(\mathbb{P}^1;\hbar)=e^{\frac{tx}{\hbar}}\sum_{d\ge0} \frac{e^{td}}{\prod_{k=1}^d(x+k\hbar)^2}~.
\end{equation}
It is easy to check that
\begin{equation}
\left[\hbar^2\frac{\partial^2}{\partial t^2}-e^t\right]J(\mathbb{P}^1;\hbar)=0~.
\end{equation}
On the other hand, the $J$-function of $T^*\mathbb{P}^1$ takes the form
\begin{equation}
J(T^*\mathbb{P}^1;\hbar,m)\propto e^{\frac{tx}{\hbar}}\sum_{d\ge0} \frac{e^{td}\prod_{k=0}^{d-1}(x+m+k\hbar)^2}{m^{2d}\prod_{k=1}^d(x+k\hbar)^2}~,
\end{equation}
where we introduce $m$ in such a way that $J(T^*\mathbb{P}^1;\hbar,m) \to J(\mathbb{P}^1;\hbar)$ as $m\to\infty$. Essentially, $J(T^*\mathbb{P}^1;\hbar,m)$ satisfy the Gauss hypergeometric differential equation since it is of ${}_2F_1$ form. However, I cannot see that $J(T^*\mathbb{P}^1;\hbar,m)$ (up to a certain factor) is an eigenfunction of the Calogero-Sutherland Hamiltonian $L(m)$. Is there any relation between the Calogero-Sutherland Hamiltonian of $A_1$-type and the Gauss hypergeometric differential equation? Or is $J(T^*\mathbb{P}^1;\hbar,m)$ NOT an eigenfunction of the Calogero-Sutherland Hamiltonian?

This post imported from StackExchange MathOverflow at 2014-10-13 09:17 (UTC), posted by SE-user Satoshi Nawata