Martin's Blog : Representations of algebraic tori

Representations of algebraic tori

Posted by martin on Wednesday, 03 March 2010 at 10:08

I am finally ready to finish my series on algebraic tori, by talking about their representations. I shall show that these representations can be classified by a grading on the vector space of the representation, after extending scalars to the separable closure. I will describe this classification explicitly in a simple case.

Representations of split tori

A representation of an algebraic group $G$ over $k$ is a finite-dimensional $k$ -vector space $V$ together with a homomorphism of algebraic groups $G \to \mathop{\mathrm{GL}}(V)$ .

The representations of a split torus are easy to understand: every representation of a split torus is a direct sum of characters. This is analogous to the fact that every representation of a finite abelian group is a direct sum of characters. It can be proved from the fact that the characters form a basis for the Hopf algebra.

Recall that the character group of a split torus of rank $r$ is $\mathbb{Z}^r$ . So a representation $(V, \rho)$ of a split torus is just a $\mathbb{Z}^r$ -graded $k$ -vector space:

$V = \bigoplus_{(n_1, \ldots, n_r) \in \mathbb{Z}^r} V^{n_1, \ldots, n_r}$

where $(z_1, \ldots, z_r) \in \mathbb{G}_m^r$ acts on $V^{n_1, \ldots, n_r}$ as multiplication by $z_1^{-n_1} \ldots z_r^{-n_r}$ . (The minus signs here are just a convention.)

Conversely, any $\mathbb{Z}^r$ -graded $k$ -vector space gives rise to a representation of $\mathbb{G}_m^r$ , by the above formula.

We call the graded pieces $V^\chi$ (for $\chi \in X^*(T)$ ) weight spaces and the characters $\chi$ for which $V^\chi$ is non-zero weights of $(V, \rho)$ .

Representations of non-split tori

Now let $G$ be an arbitrary torus over any field $k$ , and let $(V, \rho)$ be a representation of $G$ .

Just as with representations of finite groups, to properly understand representations over non-algebraically closed fields, you need to analyse them over an algebraic (or at least separable) closure to ensure that all the eigenvalues exist.

So we will extend scalars and look at $\rho_{k^s} : G_{k^s} \to \mathop{\mathrm{GL}}(V_{k^s})$ .

As we saw in the previous section, such a $\rho_{k^s}$ corresponds to a $\mathbb{Z}^r$ -grading on $V_{k^s}$ .

Of course not all $\mathbb{Z}^r$ -gradings on $V_{k^s}$ give rise to representations defined over $k$ . $\rho_{k^s}$ is defined over $k$ iff it commutes with the semilinear $\mathop{\mathrm{Gal}}(k^s/k)$ -actions on $G_{k^s}$ and $\mathop{\mathrm{GL}}(V_{k^s})$ .

The Galois action on $\mathop{\mathrm{GL}}(V_{k^s})$ is induced by the semilinear Galois action on $V_{k^s}$ .

Chasing the actions around, we find that if $\rho_{k^s}$ is defined over $k$ and $\rho_{k^s}(g)$ acts on $v \in V$ as $\chi(g)^{-1}$ , then $\rho_{k^s}(g)$ must act on $\sigma(v)$ as $\sigma(\chi)(g)^{-1}$ . Hence the action of $\mathop{\mathrm{Gal}}(k^s/k)$ on $V_{k^s}$ must permute the weight spaces according to the action of $\mathop{\mathrm{Gal}}(k^s/k)$ on $X_{k^s}^*(T)$ :

$\sigma(V_{k^s}^\chi) = V_{k^s}^{\sigma(\chi)}$

Conversely, if the above relation holds for all $\chi$ , then because the weight spaces span $V_{k^s}$ it follows that the representation is defined over $k$ .

So we get an equivalence of categories between { representations of $T$ (defined over $k$ ) } and
{ finite-dimensional $k$ -vector spaces $V$ with a $X^*_{k^s}(T)$ -grading on $V_{k^s}$ such that $\sigma(V_{k^s}^\chi) = V_{k^s}^{\sigma(\chi)}$ for all $\chi \in X^*_{k^s}(T), \sigma \in \mathop{\mathrm{Gal}}(k^s/k)$ }.

An example: the circle group

The category I have just described of vector spaces with a grading sounds rather complicated so I had better give an example. Really I should have given some examples of character groups of non-split tori already.

For these examples I shall take $k = \mathbb{R}, k^s = \mathbb{C}$ . The non-trivial element of $\mathop{\mathrm{Gal}}(\mathbb{C}/\mathbb{R})$ will be denoted $\sigma$ .

There are two rank 1 tori over $\mathbb{R}$ , corresponding to the two actions of $\mathop{\mathrm{Gal}}(\mathbb{C}/\mathbb{R})$ on $\mathbb{Z}$ . Either $\sigma$ acts as the identity, and we get the split torus $\mathbb{G}_{m,\mathbb{R}}$ ; or $\sigma$ acts as -1, and we get a torus whose $A$ -points are $\{ z \in (A \otimes_\mathbb{R} \mathbb{C})^\times : \sigma(z) = z^{-1} \}$ . The latter torus is sometimes called the circle group, because its real points form a circle, and is denoted $\mathbb{U}_1$ .

Let $(V, \rho)$ be a representation of $\mathbb{U}_1$ . Then $V_\mathbb{C} = \bigoplus_{n \in \mathbb{Z}} V^{(n)}$ with $\sigma(V^{(n)}) = V^{(-n)}$ .

For $n$ a positive integer, let $V[n]$ be the subspace of $V^{(n)} \oplus V^{(-n)}$ fixed by $\sigma$ , and let $V[0]$ be the the subspace of $V^{(0)}$ fixed by $\sigma$ .

Then $V[n]$ are real vector spaces, with $V = \bigoplus_{n \geq 0} V[n]$ and $V^{(n)} \oplus V^{(-n)} = V[n] \otimes_\mathbb{R} \mathbb{C}$ .

Fix a basis $\{ v_j \}$ of $V^{(n)}$ and let $e_j = v_j + \sigma(v_j)$ and $e_j' = i (v_j - \sigma(v_j))$ . Then the $\{ e_j, e_j' \}$ form a real basis of $V[n]$ , and $\rho(e^{i\theta})$ acts on each 2-dimensional subspace $\mathbb{R}e_j \oplus \mathbb{R}e_j'$ as rotation by $n\theta$ .