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)^*.


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).


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.

  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: