Haar measure

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1: 35.1 Special Notation
 $a,b$ complex variables. … normalized Haar measure on $\mathbf{O}(m)$. …
2: 35.4 Partitions and Zonal Polynomials
35.4.3 $Z_{\kappa}\left(\mathbf{T}\right)=Z_{\kappa}\left(\mathbf{I}\right)\,\left|% \mathbf{T}\right|^{k_{m}}\*\int\limits_{\mathbf{O}(m)}\prod_{j=1}^{m-1}|(% \mathbf{H}\mathbf{T}\mathbf{H}^{-1})_{j}|^{k_{j}-k_{j+1}}\mathrm{d}{\mathbf{H}},$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure $\mathrm{d}{\mathbf{H}}$. …
35.4.7 $\int_{\mathbf{O}(m)}Z_{\kappa}\left(\mathbf{S}\mathbf{H}\mathbf{T}\mathbf{H}^{% -1}\right)\mathrm{d}{\mathbf{H}}=\frac{Z_{\kappa}\left(\mathbf{S}\right)Z_{% \kappa}\left(\mathbf{T}\right)}{Z_{\kappa}\left(\mathbf{I}\right)}.$
3: 35.5 Bessel Functions of Matrix Argument
35.5.8 $\int_{\mathbf{O}(m)}\mathrm{etr}\left(\mathbf{S}\mathbf{H}\right)\mathrm{d}{% \mathbf{H}}=\frac{A_{-1/2}\left(-\frac{1}{4}\mathbf{S}\mathbf{S}^{\mathrm{T}}% \right)}{A_{-1/2}\left(\boldsymbol{{0}}\right)},$ $\mathbf{S}$ arbitrary.