Forgive me for asking what is undoubtedly an elementary question.


The Weil representation (defined below) of the metaplectic group $\operatorname{Mp}\_2(\mathbb{Z})$ can be defined in terms of the generators traditionally denoted $(T,1)$ and $(S,\sqrt{\tau})$. On a superficial level the images of these generators seems to depend *only* on the discriminant form $(L'/L,Q)$ of a lattice $(L,Q)$ (defined below), however, much of the theory seems to depend on the lattice itself (in particular, its signature). For example, in her [thesis](https://tuprints.ulb.tu-darmstadt.de/4458/) Alfes-Neumann works with theta functions defined with respect to a lattice of signature $(1,2)$, and in [later work](https://arxiv.org/abs/1607.02701) Bruinier and his students [Klein and Kupka](https://pubmed.ncbi.nlm.nih.gov/34840518/) use lattices of signature $(1,0)$ while citing Alfes-Neumann's formulas for the lattice of signature $(1,2)$.


While this isn't stated in the papers, it seems that the reason this can be justified is because there is an isomorphism of discriminant groups which preserves the quadratic form. I should also mention the lattices both satisfy $b\_--b\_+=\pm1$, where the signature of the lattice is $(b\_+,b\_-)$ (there is a duality between a lattice of signature $(b\_-,b\_+)$ and a lattice of signature $(b\_+,b\_-)$ that we can use).


**The question is then: Does the Weil representation $ \rho\_L $ depend only on the discriminant form $ (L'/L,Q) $?** That is, if $ M $ is another lattice with quadratic form $ R $ and there exists a group isomorphism $ \phi:L'/L\to M'/M $ with $ R(\phi(x)) = Q(x) $ for all $ x $ in $ L'/L $, is it the case that the Weil representations $ \rho\_L $ and $ \rho\_M $ are equivalent?


It seems trivial that this should be the case, but then why use the more complicated lattice of Alfes' thesis instead of a simple one? Moreover, there are other results in the literature which work for lattices of certain signature. Do these results extend to any lattice with an appropriate isomorphism?




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Recall a lattice is a finitely generated $\mathbb{Z}$-module $L$ together with a quadratic form $Q$. Specifying a quadratic form on $L$ is equivalent to specifying a symmetric bilinear form on $L$. An even lattice is one for which $Q(\lambda)\in \mathbb{Z}$ for all $\lambda$. The dual lattice of $L$ is the set $\\{y\in L\otimes\_\mathbb{Z}\mathbb{Q}:(x,y)\in\mathbb{Z}\text{ for all }x\in L\\}$. Here $Q$ is extended in the natural way.


Let $ (L,Q) $ be an even nondegenerate lattice of signature $ (b\_+,b\_-). $ The finite dimensional group algebra $ \mathbb{C}[L'/L] $ has its standard basis vectors denoted by $ \mathfrak{e}\_h, $ $ h\in L'/L. $
There is a unitary representation $ \rho\_L $ of $ \operatorname{Mp}\_2(\mathbb{Z})$ in $\operatorname{GL}(\mathbb{C}[L'/L])$ defined on the generators by


$$\rho\_L(T,1)\mathfrak{e}\_h = e(Q(h))\mathfrak{e}\_h$$


$$\rho\_L(S,\sqrt{z})=\frac{\sqrt{i}^{b\_--b\_+}}{\sqrt{|L'/L|}}\sum\_{h'\in L'/L}e((-h,h'))\mathfrak{e}\_{h'}.$$


This is the Weil representation associated to $ L. $



Does the Weil representation depend only on the discriminant group?
The Weil representation depends only on the discriminant form, as you already observed.


The thesis of Alfes-Neumann and the paper that you cite use various theta lifts, which do *not* depend only on the discriminant form. The lattice $L$ needs to be given, not because the Weil representation depends specifically on $L$ rather than $L'/L$ (it does not), but because theta lifts involve integrating against kernels (essentially, a theta function) that depends crucially on $L$ itself.


