A description of smooth vectors of the Weil representation in the geometric realization (Q883638)

The metaplectic group \(\mathrm{Mp}(2n,\mathbb{R})\) is the double covering of the symplectic group \(\mathrm{Sp}(2n,\mathbb{R})\). In this short note, the author describes the image of the Weil representation of \(\mathrm{Mp}(2n,\mathbb{R})\) in the Schwartz space \(\mathcal{S}(\mathbb{R}^n)\) of complex-valued functions and in the space \(\mathcal{S}'(\mathbb{R}^n)\) of distributions under an integral transform which maps each function on \(\mathbb{R}^n\) to a function on the Siegel upper half-plane. Notice that the paper only provides sketches of proofs of the main results.