I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function of a **compact** Kahler variety $X$. Let $T\_0=1,T\_1, \cdots, T\_m$ be the basis of the cohomology group $H^\*(X,\mathbb{Z})$, and $T\_1, \cdots, T\_r$ be the basis of the second cohomology group $H^2(X,\mathbb{Z})$. We define the matrix $g\_{ij}=\int\_{X}T\_i\cup T\_j$, and its inverse matrix $g^{ij}=(g\_{ij})^{-1}$, which provide the dual basis 
\begin{equation}
T^a=\sum\_{b=1}^mg^{ab}T\_b~,
\end{equation}
so that $\int\_X T^i\cup T\_j=\delta^i\_j$. We denote by $\overline{\mathcal{M}}\_{g,n}(X,\beta)$ the moduli space of stable maps from connected genus
$g$ curves with $n$-marked points to $X$ representing the class $\beta\in H\_2(X,\mathbb{Z})$. Let $\mathcal{L}\_1,\cdots,\mathcal{L}\_n$ be the corresponding tautological line bundles over $\overline{\mathcal{M}}\_{g,n}(X,\beta)$. For $\gamma\_1,\cdots,\gamma\_n\in H^\*(X,\mathbb{Z})$ and non-negative integers $d\_i$, the gravitational correlation function is defined
\begin{equation}
\left\langle \tau\_{d\_1} \gamma\_1, \cdots, \tau\_{d\_n} \gamma\_n \right\rangle\_{g,\beta}=\int\_{[\overline{\mathcal{M}}\_{g,n}(X,\beta)]^{\textrm{vir}}} \prod\_{i=1}^n c\_1(\mathcal{L}\_i) ^{d\_i}\cup\textrm{ev}^\*(\gamma\_i)~.
\end{equation}
The $J$-function of $X$ is defined by using the psi class $\psi=c\_1\mathcal{L}\_1)$
\begin{equation}
J[X]=e^{\delta/\hbar}\left(1+\sum\_{\beta\in H\_2(X,\mathbb{Z})}\sum\_{a=1}^m q^\beta \left\langle\frac{T\_a}{\hbar-\psi},1 \right\rangle\_{0,\beta}~T^a\right)~,
\end{equation}
where $\delta=\sum\_{i=1}^r t\_i T\_i$ and $q^\beta=e^{\int\_\beta \delta}$.


Recently, I have conjectured in [this paper](http://arxiv.org/abs/1408.4132) the expression of hypergeometric type for $J$-function of cotangent bundle of complete flag variety $\mathrm{Fl}\_N=SL(N,\mathbb{C})/B$ by using the supersymmetric partition function on $S^2$:
\begin{eqnarray}
J[T^\*\mathrm{Fl}\_N]&=&\sum\_{\vec{k}^{(I)}} \prod\_{I=1}^{N-1}z\_I^{\vert k^{(I)}\vert } \prod\_{I=2}^{N-1}\prod\_{s\neq t}^{I} \tfrac{(1+\hbar^{-1} H\_{st}^{(I)}+\hbar^{-1}m)\_{k\_s^{(I)}-k\_t^{(I)}}}{(\hbar^{-1} H\_{st}^{(I)})\_{k\_s^{(I)}-k\_t^{(I)}}} \cr
&& \prod\_{I=1}^{N-2} \prod\_{s=1}^{I}\prod\_{t=1}^{{I+1}} \tfrac{(\hbar^{-1} H\_{s}^{(I)} -\hbar^{-1} H\_{t}^{(I+1)}-\hbar^{-1}m)\_{k^{(I)}\_s - k^{(I+1)}\_t}}{(1+\hbar^{-1} H\_{s}^{(I)} -\hbar^{-1} H\_{t}^{(I+1)})\_{k^{(I)}\_s - k^{(I+1)}\_t}}\cr
&& \prod\_{s=1}^{N-1}\prod\_{t=1}^{N} \tfrac{(\hbar^{-1} H\_{s}^{(N-1)}-\hbar^{-1} H\_{t}^{(N)}-\hbar^{-1}m)\_{k^{(N-1)}\_s}}{(1+\hbar^{-1} H\_{s}^{(N-1)}-\hbar^{-1} H\_{t}^{(N)})\_{k^{(N-1)}\_s}}~,
\end{eqnarray}
where we identify $H\_{s}^{(I)}$ $(s = 1,...,I)$ with Chern roots to the duals of the
universal bundles $\mathcal{S}\_I$:
\begin{equation}
0\subset \mathcal{S}\_1 \subset \mathcal{S}\_2 \subset \dots \subset \mathcal{S}\_{N-1} \subset \mathcal{S}\_{N} = 
{\mathbb{C}}^N \otimes {\cal O}\_{\mathrm{Fl}\_N}~.
\end{equation}
I have checked that, after multiplying an appropriate factor, it becomes an eigenfunction of trigonometric Calgero-Moser Hamiltonian
\begin{equation}
H=\hbar^2\sum\_{i}\frac{\partial^2}{\partial t\_i^2}-2m(m+\hbar)\sum\_{\alpha\in \Delta^+} \frac 1{(e^{\langle t,\alpha\rangle/2}-e^{-\langle t,\alpha\rangle/2})^2}~.
\end{equation}
This is consistent with theorem 3.2 of [Braverman-Maulik-Okounkov](http://arxiv.org/abs/1001.0056).



> 
> **Question:** The formula of $J[T^\*\mathrm{Fl}\_N]$ certainly contains the parameter $m$ that does not appear in the definition of $J$-function of a compact Kahler variety. It appears to me that one has to introduce the parameter $m$ since the cotangent bundle of the complete flag variety is non-compact. I would like to know the definition of $J[T^\*\mathrm{Fl}\_N]$ and how it incorporates the parameter $m$.
> 
> 
> 



Definition of Givental $J$-function of cotangent bundle of flag variety
The parameter $m$ appears if you study everything equivariantly with respect to the
${\mathbb C}^\*$-action which acts in the standard way on the fibers of the cotangent bundle.


