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Character and cocharacter lattices of a split torus

Statement

Let $T$ be a split torus over $k$ , so that $T ≅ D_{k} (M)$ for a free abelian group $M$ of finite rank (Groups of multiplicative type and tori, Diagonalizable groups and their character modules). Then the character group $X (T) = {Hom}_{k} (T, G_{m})$ is canonically isomorphic to $M$ , the cocharacter group $X_{*} (T) = {Hom}_{k} (G_{m}, T)$ is canonically isomorphic to the dual lattice $M^{\lor} = Hom (M, Z)$ , and the pairing $X (T) \times X_{*} (T) \to Z$ , $(χ, λ) \mapsto ⟨ χ, λ ⟩$ , defined by $χ \circ λ \in {Hom}_{k} (G_{m}, G_{m}) = Z$ , is a perfect $Z$ -bilinear pairing identifying $X_{*} (T)$ with $X (T)^{\lor}$ . Both lattices are free of finite rank equal to $\dim T$ , and the identifications and the pairing commute with extension of scalars. In particular $X (T) \otimes_{Z} Q$ and $X_{*} (T) \otimes_{Z} Q$ are $Q$ -vector spaces in perfect duality.

Facts & Assumptions

Given: A field $k$ , a free abelian group $M$ of finite rank, and a split torus $T$ with an isomorphism $T ≅ D_{k} (M)$ over $k$ .

[F1]

The diagonalizable group $D_{k} (M) = Spec k [M]$ has $D_{k} (M) (R) = Hom (M, R^{\times})$ and $G_{m} = D_{k} (Z)$ , and characters are the homomorphisms $X (G) = {Hom}_{k -groups} (G, G_{m})$ (Diagonalizable groups and their character modules).

[F2]

For abelian groups $M, N$ there are natural identifications $X (D_{k} (M)) = M$ , via $m \mapsto e_{m}$ , and ${Hom}_{k} (D_{k} (M), D_{k} (N)) = Hom (N, M)$ (Split diagonalizable groups are dual to abelian groups).

[F3]

A torus is split when it is isomorphic over $k$ to $G_{m}^{r}$ for some $r \geq 0$ (Groups of multiplicative type and tori). Its coordinate ring is $k [t_{1}, \dots, t_{r}, (t_{1} \dots t_{r})^{- 1}]$ ; it has dimension $r$ by A polynomial ring in n variables over a field has dimension n: localization cannot increase prime-chain length, and the chain $(0) ⊊ (t_{1} - 1) ⊊ \dots ⊊ (t_{1} - 1, \dots, t_{r} - 1)$ survives this localization.

Proof

1.1F1F2F3algebra

By [F3] the split torus $T$ is isomorphic over $k$ to $G_{m}^{r}$ with $r = \dim T$ , and $G_{m}^{r} = D_{k} (Z^{r})$ by [F1]; the given isomorphism $T ≅ D_{k} (M)$ and [F2] therefore identify $M$ with $Z^{r}$ . Applying [F2] to $M$ and to $Z$ gives $X (T) = {Hom}_{k} (D_{k} (M), D_{k} (Z)) = Hom (Z, M) ≅ M$ and $X_{*} (T) = {Hom}_{k} (D_{k} (Z), D_{k} (M)) = Hom (M, Z) = M^{\lor}$ . Both are free abelian of rank $r = \dim T$ ; the identifications are induced by the anti-equivalence and are compatible with any change of the splitting isomorphism, which only renames $M$ by the induced automorphism.

2.1F1F2step 1.1algebra

For $χ \in X (T)$ and $λ \in X_{*} (T)$ the composite $χ \circ λ : G_{m} \to G_{m}$ is a homomorphism, and [F1] identifies ${Hom}_{k} (G_{m}, G_{m}) = {Hom}_{k} (D_{k} (Z), D_{k} (Z)) = Hom (Z, Z) = Z$ by [F2]; so $⟨ χ, λ ⟩ : = χ \circ λ$ is an integer and composition of homomorphisms is $Z$ -bilinear. Under the identifications of step 1.1, an element of $X (T)$ is an element $m \in M$ and an element of $X_{*} (T)$ is a homomorphism $μ : M \to Z$ , and the corresponding composite is $μ (m)$ : $χ \circ λ$ is evaluation of the character at the cocharacter. Therefore the map $X_{*} (T) \to X (T)^{\lor}$ , $λ \mapsto (χ \mapsto ⟨ χ, λ ⟩)$ , is exactly the identity $Hom (M, Z) \to Hom (M, Z)$ and hence an isomorphism, so the pairing is perfect and identifies $X_{*} (T)$ with $X (T)^{\lor}$ .

3.1F2step 1.1step 2.1algebra∎

Let $k^{'} \supseteq k$ be a field extension. Base change of group algebras identifies $D_{k^{'}} (M) = D_{k} (M)_{k^{'}}$ , and [F2] applied over the field $k^{'}$ gives $X (T_{k^{'}}) = X (D_{k^{'}} (M)) = M$ and $X_{*} (T_{k^{'}}) = M^{\lor}$ with the same evaluation pairing, so the two identifications of step 1.1 commute with extension of scalars. For free lattices of finite rank the dual of a base change is the base change of the dual, so tensoring the perfect pairing of step 2.1 with $Q$ exhibits $(X (T) \otimes Q)^{\lor} = X_{*} (T) \otimes Q$ and gives a perfect $Q$ -bilinear pairing of $X (T) \otimes Q$ with $X_{*} (T) \otimes Q$ .

Depends on

Used by

Abstract root data and their Weyl groups Definition
Roots and root groups of a split reductive group Definition
Split reductive groups Definition
The root datum of a split reductive group Definition
Weights, dominant weights and the highest-weight order of a rational representation Definition
Central characters and descent along a central isogeny Lemma
Dominant characters of a torus times a split semisimple group are primitive weights Lemma
Every dominant character of a split reductive group is a highest weight Lemma
Structure of SL₂ and root coordinates Lemma
Cocharacter limit subgroups Theorem
Weight subgroups of a torus action Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Florian Herzig, Linear Algebraic Groups (University of Toronto lecture notes, 2013) (standard reference, not scraped)