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## The Weil representation

So I’m going to change pace for a while here and start talking about the Weil representation. This post will be an explication of the basic problem, and the next several posts will outline the answer, with as many proofs as possible.

Fix a non-Archimedean local field $F$ of characteristic not equal to $2$. Let $V$ be a finite-dimensional vector space over $F$ equipped with a symplectic form $\langle \ , \ \rangle$. The Heisenberg group $H$ associated to $V$ is $V \times F$ as a space, with the group operation

$(v,s) \cdot (w,t) = (v + w, s + t + \frac12 \langle v,w \rangle)$.

One can classify the smooth irreducible representations of $H$ as follows: the ones trivial on the center of $H$ (which is just a copy of the additive group of $F$) are all one-dimensional and may be identified with the Pontryagin dual $\widehat{V}$ of $V$. The others are all infinite-dimensional and are classified by the following well-known theorem.

Theorem (Stone-von Neumann). Fix a nontrivial continuous character $\psi : F \to \mathbb{T}$.  Then there is a smooth irreducible representation $\rho_{\psi}$ of $H$, unique up to isomorphism, with central character $\psi$.

Now consider the symplectic group $\text{Sp}(V)$ acting on $V$, so $\text{Sp}(V)$ acts on $H$ also, fixing the center pointwise. Then if $g \in \text{Sp}(V)$, we see that $\rho_{\psi} \circ g$ is still a smooth irreducible representation of $H$ with central character $\psi$, and the Stone-von Neumann theorem tells us that there is a operator $M[g]$ which intertwines $\rho_{\psi}$ and $\rho_{\psi} \circ g$. By Schur’s lemma, this $M[g]$ is well-defined up to a scalar in $\mathbb{C}^{\times}$, and it is easy to see that this yields a projective representation of $\text{Sp}(V)$.

The first question is whether $M$ lifts to a true linear representation of $\text{Sp}(V)$, and the short answer is, in our situation, no. But in some sense the next best thing is true: by abstract nonsense $M$ lifts to a linear representation of some central extension of $\text{Sp}(V)$, and in fact there is such an extension

$1 \to \mathbb{Z}/2\mathbb{Z} \to \text{Mp}(V) \to \text{Sp}(V) \to 1$

by $\mathbb{Z}/2\mathbb{Z}$, called the metaplectic group, and a true representation of $\text{Mp}(V)$which lifts $M$, called the Weil representation. This is what I will try to explain, and to do so we will have to go deep into Witt’s theory of quadratic spaces and some related topics, such as the Maslov index of a family of lagrangian subspaces of $V$ and the Weil character of the Witt group.