Given a semidirect product $G = N times H$ where $N$ is%%nilpotent, connected, simply connected and normal in $G$ and where$H$ is a vector group for which $ad(h)$ is completely reducible and$mathbf R$-split, let $ au$ denote the quasiregular representation of$G$ in $L^2(N)$. An element $psi in L^2(N)$ is said to be admissibleif the wavelet transform $f mapsto langle f, au(cdot)psi angle$defines an isometry from $L^2(N)$ into $L^2(G)$. In this paper we givean explicit construction of admissible vectors in the case where $G$is not unimodular and the stabilizers in $H$ of its action on $hat N$are almost everywhere trivial. In this situation we proveorthogonality relations and we construct an explicit decomposition of$L^2(G)$ into $G$-invariant, multiplicity-free subspaces each of whichis the image of a wavelet transform . We also show that, with theassumption of (almost-everywhere) trivial stabilizers,non-unimodularity is necessary for the existence of admissiblevectors.
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机译：给定一个半直接乘积$ G = N乘以H $，其中$ N $是％G，在$ G $中是幂零，连通，简单连通和正态的，其中$ H $是向量组，其中$ ad（h）$是完全可约的和$ mathbf R $分割，让$ au $表示$ G $在$ L ^ 2（N）$中的准规则表示。如果小波变换$ f mapto langle f，au（cdot）psi angle $将等轴测图从$ L ^ 2（N）$定义为$ L ^ 2，则L ^ 2（N）$中的元素$ psi被认为是可接受的（G）$。在本文中，我们给出了在$ G $不是单模且在$ H $作用于$ hat N $的$ H $中的稳定器几乎无处不在的情况下可允许向量的显式构造。在这种情况下，我们证明了正交关系，并构造了将$ L ^ 2（G）$显式分解为$ G $不变的无多重子空间，每个子空间都是小波变换的图像。我们还表明，假设（几乎到处都是）琐碎的稳定器，对于允许向量的存在，非单模性是必要的。
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