Home > representation theory > The Witt group and its Weil character

## The Witt group and its Weil character

For the next couple of posts I’ll be proving the existence and some properties of the Weil index, which is a numerical invariant of a quadratic space over a local field, and is an important ingredient in the construction of the metaplectic group and its (linear) Weil representation. I will treat only the case of a non-Archimedean local field $F$ with characteristic not equal to $2$. This post will probably consist mostly of setup to put us in a natural context for the proofs.

For now let $k$ be any field of characteristic not equal to $2$.

Definition By a quadratic space over $k$ we mean a finite-dimensional vector space $W$ over $k$ equipped with a nondegenerate quadratic form $q : W \to k$.

Since the characteristic of $k$ is not $2$, quadratic forms correspond bijectively with symmetric bilinear forms. An isometry between quadratic spaces $(W_1,q_1)$ and $(W_2,q_2)$ is a linear isomorphism $T : W_1 \to W_2$ satifying $q_2(T(w)) = q_1(w)$ for all $w \in W_1$. Clearly isometry is an equivalence relation on quadratic spaces.

Example For any $a \in k^{\times}$, the map $x \mapsto ax^2$ is a nondegenerate quadratic form on $k$ which we denote by $\langle a \rangle$. It is not hard to see that any one-dimensional quadratic space has this form, and $\langle a \rangle$ is isometric to $\langle b \rangle$ if and only if $a \equiv b \ (\text{mod } k^{\times 2})$.

Given two quadratic spaces $(W_1,q_1)$ and $(W_2,q_2)$, we can construct their sum in the obvious way: its underlying space is $W_1 \oplus W_2$, and the quadratic form is defined by the formula $(q_1 \oplus q_2)(w_1,w_2) = q_1(w_1) + q_2(w_2)$.

Example The quadratic space $\mathfrak{H} = \langle 1 \rangle \oplus \langle -1 \rangle$ is called the hyperbolic plane. It is isometric to the form on $k^2$ which sends $(x,y) \mapsto xy$.

Thankfully, all quadratic spaces decompose (noncanonically) into a direct sum of spaces of the form $\langle a \rangle$, so understanding arbitrary quadratic forms reduces to understanding $\langle a \rangle$.

Proposition Any quadratic space $(V,q)$ is isometric to one of the form $\langle a_1 \rangle \oplus \cdots \oplus \langle a_n \rangle$ for some $a_1,\cdots,a_n \in k^{\times}$.

Proof. We induct on the dimension $n$ of $V$: the case $n = 0$ is trivial. For the inductive step, certainly there exists $v \in V$ with $q(v) \neq 0$, and if we write $W$ for the subspace of $V$ spanned by $v$ then we have $W \cong \langle q(v) \rangle$. Denote by $W^{\perp}$ the subspace of $V$ orthogonal to $W$ with respect to $q$, so we need only show $V = W \oplus W^{\perp}$ and we will be done by the inductive hypothesis.

Clearly $W \cap W^{\perp} = 0$ since $q$ is nondegerate when restricted to $W$, so it remains to prove $V = W + W^{\perp}$. For this, one observes that the isomorphism $V \to V^*$ induced by $q$ restricts to an isomorphism $W^{\perp} \to (V/W)^*$.

$\Box$

Finally we can introduce the group of which the Weil index is a character.

Definition-Proposition The quotient of the commutative monoid formed by isometry classes of quadratic spaces by the submonoid generated by the hyperbolic plane $\mathfrak{H}$ is a group, called the Witt group of $k$ and denoted by $W(k)$.

Proof. The content of the statement is that for any quadratic space $(V_1,q_1)$, there exists another $(V_2,q_2)$ such that $(V_1 \oplus V_2,q_1 \oplus q_2)$ is isometric to a direct sum of copies of $\mathfrak{H}$. So choose a diagonalization $\langle a_1 \rangle \oplus \cdots \oplus \langle a_n \rangle$ of $(V_1,q_1)$ and take $(V_2,q_2)$ to be $\langle -a_1 \rangle \oplus \cdots \oplus \langle -a_n \rangle$. Then the resulting direct sum $(V_1 \oplus V_2,q_1 \oplus q_2)$ is isometric to $\langle a_1 \rangle \oplus \langle -a_1 \rangle \oplus \cdots \oplus \langle a_n \rangle \oplus \langle -a_n \rangle$, so it suffices to show that for any $a \in k^{\times}$ we have $\langle a \rangle \oplus \langle -a \rangle \cong \mathfrak{H} = \langle 1 \rangle \oplus \langle -1 \rangle$. This isometry is given by $(x,y) \mapsto \tfrac{a+1}{2}(x,y)$.

$\Box$

In fact, the Witt group can be given a ring structure using the tensor product of quadratic spaces, but we will not need this.

Next time we will prove the existence of the Weil character of $W(F)$ for $F$ a non-Archimedean local field of characteristic not equal to $2$ using the theory of distributions and their Fourier transforms.