Non-weight modules over the mirror Heisenberg-Virasoro algebra

In this paper, we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra $\mathcal{D}$, including Whittaker modules, $\mathcal{U}(\mathbb{C} d_0)$-free modules, and their tensor products. More precisely, we give the necessary and sufficient conditions for the Whittaker modules to be irreducible. We determine all $\mathcal{D}$-module structures on $\mathcal{U}(\mathbb{C} d_0)$, and find the necessary and sufficient conditions for these modules to be irreducible. At last we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and $\mathcal{U}(\mathbb{C} d_0)$-free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and $\mathcal{U}(\mathbb{C} d_0)$-free modules are isomorphic. These lead to many new irreducible non-weight modules over $\mathcal{D}$.