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I'm reading the paper 'Bruinier, Jan Hendrik; Funke, Jens, On two geometric theta lifts. Duke Math. J. 125 (2004), no. 1, 45–90.' In p.52, they say that

We denote by $\omega=\omega_V$ the Schrödinger model of the (restriction of the) Weil representation of $G' \times O(V(\mathbb R))$ acting on $S(V(\mathbb R))$, the space of Schwartz functions on $V(\mathbb R)$.

Here I don't have the knowledge about 'Schroedinger model', 'Weil representation', and 'Schwartz function'. Could you give me a proper reference?

PinkyWay
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LWW
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  • Well.. this is hopelessly late, but I think a good reference is maybe Kudla's Notes on Local Theta Correspondence (The real numbers $\mathbb{R}$ are a local field, so you can assume in there that $F = \mathbb{R}$). Maybe you yourself can add some additional reference? – Maty Mangoo Oct 20 '23 at 16:19

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