Sufficient Condition for Adjoint Representation Kernel to be Lie Group Center

[kakarukeys]

what is the sufficient condition for the kernel of an adjoint representation to be the center of the Lie group?

Does the Lie group have to be non-compact and connected, etc?

 

[LorenzoMath]

If the Lie group is connected, the claim holds. So I heard. Is there any chance that you are studying automorphic representations and their L-functions? If so, please let me know good references to start with. Also, references for representation theory. Thanks.