# Orthogonality of Pfaffian polynomials in $SO(2m)$

I've been struggling here to invert some integral equation involving Pfaffians, and it would be very nice if you could shed some light on the problem.

Let's go to it. Let $V=\{-1,1\}^{m}$ and $S = \left\{\oplus_{j=1}^m\begin{pmatrix} 0&v_j \\ -v_j&0\end{pmatrix} \big | v\in V\right\}$, with $v_j$ the $j$-th element of $v$, i.e., $S$ is the set of all $2^m$ skew-symmetric $2m \times 2m$ matrices with the form $\oplus_{j=1}^m\begin{pmatrix} 0&\pm 1 \\ \mp 1&0\end{pmatrix}$.

Now, let $I_m \in S$, $R\in SO(2m)$, and $g: \text{Skew}_{2m}\mapsto \mathbb{R}_+$ be a given function. I want to determine $\omega: \text{Skew}_{2m}\mapsto \mathbb{R}$, which is related to $g$ as follows: $$g(R.I_m.R^T) = \int_{R^\prime \in SO(2m)} d \mu(R^\prime)\; \sum_{S_m\in S} \; \omega( {R^\prime}^T.S_m.{R^\prime})\; \text{Pf}({R^\prime}^T.S_m.{R^\prime}+{R}^T.I_m.{R}),$$ where $\mu$ is a uniform measure in $SO(2m)$, and $\text{Pf}$ is the Pfaffian.

One approach I've been trying, up to now wihtout success, is to use the integral form of Pfaffians: $$\text{Pf}(A) = \int d\theta_1\cdots \theta_{2m} e^{-\frac{1}{2} \theta^T.A.\theta},$$ where $\theta_i$, $i\in [2m]$, are Grassmann variables, i.e., $\theta_i \theta_j + \theta_j\theta_i=0$

Would you know how to tackle that? Any ideas?

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