TITLE: Which philosophy for reductive groups?
QUESTION [46 upvotes]: I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be reductive.
Hence my question : why is reductive interesting?
Of course I have already seen some formal definitions, like to have a trivial unipotent radical or some discussions about root systems. But it does not seem very satisfactory to me: why is this assumption the one that seems to be the Grail of all generalizations done in this field?  why are those reductive groups so deeply interesting? what are the implied or equivalent properties making them so desirable? (I have heard of the complete reducibility of representations -- is it true? without other assumption on the representations? is it an equivalent or a weaker statement?)
Any clue helping me to cloth off reductive groups will be of great use :)

REPLY [15 votes]: Here is a natural interpretation of complex reductive groups:
A complex algebraic group is reductive if and only if it is the complexification of a compact Lie group.
More precisely, every compact Lie group $G $ admits a (unique?) real algebraic structure and thus we can speak of its complexification $ G \otimes_\mathbb R \mathbb C $.  The groups obtained this way are precisely the complex reductive groups.  In fact, there is an equivalence of categories between compact Lie groups and complex reductive groups.