Heisenberg double of the generalized quantum euclidean group and its representations

The generalized quantum Euclidean group $\text{oq}({\mathfrak{b}}_{m,n})$ is a natural generalization of the quantum Euclidean group $\text{oq}({\mathfrak{b}}_{1,1})$. The Heisenberg double $\text{od}({\mathfrak{b}}_{m,n})$ of $\text{oq}({\mathfrak{b}}_{m,n})$ is the smash product of $\text{oq}({\mathfrak{b}}_{m,n})$ with its Hopf dual $\text{ou}({\mathfrak{b}}_{m,n})$. In this paper, we study the weight modules, the prime spectrum and the automorphism group of the Heisenberg double $\text{od}({\mathfrak{b}}_{m,n})$.