In the setting of algebraic/reductive group, there is a notion of dual group (defined from the dual root system). Is there an explicit way to see it, or to describe it?
For instance if $T \simeq k^\times \times k^\times \subset G = GL(2)$, as algebraic groups over a field $k$, what does $\hat{T}$ look like? Is it a group of characters on $T$ or so?
