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For a $k \times k$ positive matrix $V=(v_{ij}), $ write $V=\Gamma'D\Gamma,$ where $D=diag(d_1,d_2,\ldots,d_k)$ with $d_1>d_2>\cdots>d_k,$ and $\Gamma$ is orthogonal matrix. From the result of Farrell (1985), $$dV=\prod_{i<j}(d_i-d_j) ~dD d\Gamma,~~\qquad(1)$$ here $dV=\prod_{i\le j} dv_{ij}$, $dD=\prod_{i=1}^k dd_{i},$ and $d\Gamma$ denotes over the space of orthonormal matrices.

Question: What is the sufficient and necessary conditions of a,b,c for the following integration is integrable? $$\int\int\frac{1}{|V|^a|W|^b|I_k+V+W|^c\prod_{i<j}(d_i-d_j)\prod_{i<j}(\lambda_i-\lambda_j)} dVdW~~\qquad(2)$$ where positive matrix $W=O'\Lambda O,$ $\Lambda=diag(\lambda_1,\lambda_2,\ldots,\lambda_k)$ with $\lambda_1>\lambda_2>\cdots>\lambda_k,$ and $O$ is orthogonal matrix.

From (1), it follows that $$\int\int\frac{1}{|D|^a|\Lambda|^b|I_k+\Gamma'D\Gamma+O'\Lambda O|^c} dDd\Lambda d\Gamma dO.$$

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