Consider a semisimple Lie group $G$. We define the Langlands dual $\hat{G}$ of $G$ as the group which has as a root system, the root system generated by the coroots of $G$. Recall that given a root $\alpha$, its coroot is defined by $$\hat{\alpha}=2\dfrac{\alpha}{(\alpha,\alpha)}$$
Now I wonder, which groups are self-dual?
I could work the exercise for any group, just computing the coroot system. However I was wondering if there is an easier way to state which group is L-dual to which other, and if it is self-Langlands-dual. Does it exist a reference with a list of the couples of dual groups?
The Langlands dual group is defined for reductive groups, not only for semisimple Lie groups. For example, $SL(n)$ is dual to $PGL(n)$, $SO(2n+1)$ is dual to $Sp(n)$ and $SO(2n)$ is self-dual. The group $GL(n)$ is self-dual, too. Passing to the level of Lie algebras, the Langlands duality changes the types of simple factors of the Lie algebra by taking the transpose of the corresponding Cartan matrices. For a reference see J.W. Cogdell's article Dual groups and Langlands Functoriality, section $1$ and table $1$.