Dual Lattice

A lattice is a free abelian group $L\cong \mathbb{Z}^r$.

Its dual lattice is $L^\vee := \mathrm{Hom}_\mathbb{Z}(L,\mathbb{Z}).$

A pairing $\langle\cdot,\cdot\rangle: L\times L^\vee\to \mathbb{Z}$ is given by evaluation: $\langle \lambda,\mu\rangle=\mu(\lambda)$.

In the letter: “conjugate lattice” $cL$ is a dual lattice used to build the dual root datum and the Satake parameter space.