Affine automorphism group and wavelet transform (Q2725228)

Let \(M_{n,m}\) be the space of all \(n\times m\) complex matrices and let \(M_{m,m}= M_m\). The classical domain of type I is defined by NEWLINE\[NEWLINE{\mathbf D}_I =\{Z\in M_{n+m,m}: I_m-ZZ^*\in \Omega\},NEWLINE\]NEWLINE where \(I_m\) is the unit matrix of order \(m\) and \(\Omega\) is the space of all positive defined Hermite matrices of order \(m\). Define a mapping \(\Phi:M_{n,m} \times M_{n,m}\to M_m\) by \(\Phi (u,v) =v^*u\). Then \({\mathbf D}_I\) is analytically equivalent to the Siegel domain NEWLINE\[NEWLINE{\mathbf D}(\Phi, \Omega)=\bigl\{(w,z)\in M_{n,m}\times M_m: \text{Im} (z)-\Phi (w,w) \in\Omega\bigr\}.NEWLINE\]NEWLINE The Shilov boundary of \({\mathbf D}(\Phi, \Omega)\) is NEWLINE\[NEWLINES=\bigl\{ (w,z)\in M_{n,m} \times M_m:\text{Im}(z)-\Phi (w,w)=0 \bigr\}.NEWLINE\]NEWLINE \(S\) becomes a nilpotent Lie group \({\mathcal R}\) if one defines an appropriate multiplication on \(S\). Let \({\mathcal P}\) be the affine automorphism group on \({\mathbf D}(\Phi, \Omega)\). The restriction of \({\mathcal P}\) on \({\mathcal R}\) is also an affine group on \({\mathcal R}\). Furthermore, \({\mathcal P}\) has a unitary representation \(U\) on \(L^2({\mathcal R})\). Decompose the space \(L^2({\mathcal R})\) as the direct sum of the irreducible subspaces of \(U\). Then the authors obtain an admissible wavelet characterization of \(L^2({\mathcal R})\).