We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. We show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Furthermore, we find the supports of the unitary representations.

Let ${{F}_{2n,2}}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf{P}$ denote the affine automorphism group of ${{F}_{2n,2}}$. In this article the theory of continuous wavelet transform on ${{F}_{2n,2}}$ associated with $\mathbf{P}$ is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on ${{F}_{2n,2}}$ is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if $n\,=\,1$, ${{F}_{2,2}}$ is the 3-dimensional Heisenberg group ${{H}^{1}}$, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on ${{F}_{2,2}}$. This result cannot be extended to the case $n\,\ge \,2$.

Let $G$ be the abelian Lie group $\mathbb{R}^n\times\mathbb{R}^k/\mathbb{Z}^k$, acting on the complex space $X=\mathbb{R}^{n+k}\times\ri G$. Let $F$ be a strictly convex function on $\mathbb{R}^{n+k}$. Let $H$ be the Bergman space of holomorphic functions on $X$ which are square-integrable with respect to the weight $e^{-F}$. The $G$-action on $X$ leads to a unitary $G$-representation on the Hilbert space $H$. We study the irreducible representations which occur in $H$ by means of their direct integral. This problem is motivated by geometric quantization, which associates unitary representations with invariant Kähler forms. As an application, we construct a model in the sense that every irreducible $G$-representation occurs exactly once in $H$.

For the groups $G={\mathrm{Sp}}(p,q),\ \mathrm{SO}^\ast(2n)$, and $\mathrm{U}(m,n)$, we consider degenerate principal series whose infinitesimal character coincides with a finite-dimensional representation of $G$. We prove that each irreducible constituent of maximal Gelfand–Kirillov dimension is a derived functor module. We also show that at an appropriate ‘most singular’ parameter, each irreducible constituent is weakly unipotent and unitarizable. Conversely we show that any weakly unipotent representation associated to a real form of the corresponding Richardson orbit is unique up to isomorphism and can be embedded into a degenerate principal series at the most singular integral parameter (apart from a handful of very even cases in type D). We also discuss edge-of-wedge-type embeddings of derived functor modules into degenerate principal series.

Differential operators ${{D}_{x,}}\,{{D}_{y}}$, and ${{D}_{z}}$ are formed using the action of the 3-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L\,=\,D_{x}^{2}+D_{y}^{2}+D_{z}^{2}$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space ${{L}^{2}}\left( S \right)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

We obtain a decomposition formula of a representation of Sp(p, q) or SO*(2n) unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules. In particular, we show that such an induced representation is decomposed into a direct sum of irreducible unitarily induced modules from derived functor modules under some regularity condition on the parameters. In particular, representations of SO*(2n) and Sp(p, q) induced from one-dimensional unitary representations of their parabolic subgroups are irreducible.