Dual lattice - Knowlpedia

A lattice is a finite free abelian group $L\simeq\mathbb Z^r$ . Its integral dual lattice is

L^\vee=\operatorname{Hom}_{\mathbb Z}(L,\mathbb Z).

Evaluation gives a perfect pairing

L\times L^\vee\longrightarrow\mathbb Z.

The bidual map $L\to L^{\vee\vee}$ is an isomorphism.

Torus lattices

For the character and cocharacter lattices of an algebraic torus $T$ ,

X_*(T)\simeq X^*(T)^\vee.

If $T$ is defined over a nonsplit field, both lattices carry an absolute Galois-group action and the evaluation pairing is Galois-equivariant.

The coordinate ring of a complex torus with character lattice $L$ is the group algebra $\mathbb C[L]$ . Thus passing to a dual torus exchanges its character and cocharacter lattices.

Distinctions

The integral dual $L^\vee$ is not the same object as:

the real or complex linear dual of $L\otimes\mathbb R$ ;
the Pontryagin dual of $L$ as a discrete topological group;
the discriminant dual of a lattice equipped with a bilinear form.

These constructions can be related after extra choices but should not be identified by notation alone.

Relation to the letter

The letter's “conjugate lattice” $cL$ participates in the root datum of the Langlands dual group. Modern notation records both character and cocharacter lattices explicitly, which avoids hiding the isogeny form.

References

T. A. Springer, Linear Algebraic Groups, second edition, Birkhäuser, 1998.