Spectral and Invariant Structures of the Induced Representation of SL2(R) on the Upper Half-Plane of Dual Numbers

We study representations of SL2(R) that are induced from the lower unipotent subgroup and are performed on the upper half-plane of dual numbers. We analyze the induced left action using right Lie derivatives and the Casimir operator. We find explicit joint eigenfunctions, along with spectral relations and ladder-operator structures that describe their transformation properties. We further derive explicit classes of invariant bilinear functionals and their kernels under translation, dilation, and parabolic rotation. Using these kernels and joint eigenfunctions, we build covariant transforms whose images give invariant models for the induced representation. Our results link the spectral structure of the representation with invariant distributional kernels and covariant-transform methods, offering a basis for further study of invariant subspaces and reducibility in the realization of the dual number of SL2(R).