,\H_\muhat(-z,-w)=\H_\muhat(z,w). \label{HHduality}$$

We may recover $\Omega(z,w)$ from the $\H_\muhat(z,w)$’s by the formula:

(z,w)=(\_[\^k]{}m\_). \[exp\]

From Formula (\[eulermac\]) and Formula (\[H-specializ\]) we have:

With the specialization $y_i=q^{i-1}$,

$$\Omega\left(\sqrt{q},\frac{1}{\sqrt{q}}\right)=\sum_{\lambda\in\calP}q^{(1-q)|\lambda|}\left(q^{-n(\lambda)}H_\lambda(q)\right)^{2g+k-2}\prod_{i=1}^ks_\lambda(\x_i\y).$$

\[specializ\]

The rational function $\H_\muhat(z,w)$ is a polynomial with integer coefficients. It has degree

$$d_\muhat:=n^2(2g-2+k)-\sum_{i,j}(\mu^i_j)^2+2$$ in each variable and the coefficients of $\H_\muhat(-z,w)$ are non-negative. \[conjH\]

The function $\H_\muhat(z,w)$ is computed in many cases in [@hausel-letellier-villegas §1.5].

Characters and Fourier transforms {#finite-groups}
---------------------------------

### Characters of finite general linear groups {#charGL}

For a finite group $H$ let us denote by ${\rm Mod}_H$ the category of finite dimensional $\C[H]$ left modules. Let $K$ be an other finite group. By an *$H$-module-$K$* we mean a finite dimensional $\C$-vector space $M$ endowed with a left action of $H$ and with a right action of $K$ which commute together. Such a module $M$ defines a functor $R_K^H:{\rm Mod}_K\rightarrow{\rm Mod}_H$ by $V\mapsto M\otimes_{\C[K]} V$. Let $\C(H)$ denotes the $\C$-vector space of all functions $H\rightarrow\C$ which are constant on conjugacy classes. We continue to denote by $R_K^H$ the $\C$-linear map $\C(K)\rightarrow\C(H)$ induced by the functor $R_K^H$ (we first define it on irreducible characters and then extend it by linearity to the whole $\C(K)$). Then for any $f\in\C(K)$, we have

$$R_K^H(f)(g)=|K|^{-1}\sum_{k\in K}{\rm Trace}\,\left((g,k^{-1})\,|\, M\right)f(k).\label{bimod}$$

Let $G=\GL_n(\F_q)$ with $\F_q$ a finite field. Fix a partition $\lambda=(\lambda_1,\dots,\lambda_r)$ of $n$ and let $\calF_\lambda=\calF_\lambda(\F_q)$ be the variety of partial flags of $\F_q$-vector spaces $$\{0\}=E^r\subset E^{r-1