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 :)



Which philosophy for reductive groups?
Harish-Chandra told the following (paraphrased) little story that he heard from Chevalley: "When God and the Devil were creating the universe, God gave the Devil a free hand in building things but told him to keep off certain objects to which He would attend. Semisimple groups were among those special items."


From the modern perspective the class of (connected) reductive groups is more natural than that of (connected) semisimple groups for the purposes of setting up a robust general theory, due to the fact that Levi factors of parabolics in reductive groups are always reductive but generally are not semisimple when the ambient group is semisimple. However, after some development of the basic theory one learns that reductive groups are just a fattening of semisimple groups via a central torus (e.g., GL$\_n$ versus SL$\_n$), so Harish-Chandra had no trouble to get by in the semisimple case by just dragging along some central tori here and there in the middle of proofs. 


So your question appears to be basically the same as asking: why are (connected) semisimple groups such a natural class on which to focus so much effort? It is the rich class of examples from representation theory and number theory that historically motivated subsequent developments and inspired the search for a *single uniform perspective* with to understand disparate phenomena in a common manner.


(i) The possibility to prove theorems about reductive groups via inductive techniques, building up from Levi factors in proper parabolic subgroups, is very powerful and characteristic of the beauty of the subject; one cannot fathom this from the mundane definition with (geometric) unipotent radicals, much as I think it is inconceivable from the raw definition of "(connected) compact Lie group" that there should be any rich structure theory at all. This makes learning how to think about the subject a tricky matter, since it either looks like a grab-bag of explicit examples (which may or may not be to one's liking) or else one has to suspend patience and get somewhat far into the theory to see the rich structure that unifies all of the examples. 


Harish-Chandra used the inductive method all the time to prove theorems of sweeping generality about all reductive groups without ever needing to compute explicitly beyond SL$\_2$, almost as if there is only "one" reductive group, $G$. One can really prove a huge amount about these groups without ever computing with matrices at all. (Not that explicit computations are bad, but one can get much insight without them.) It is only by treating them all in the same way, and maintaining a completely general outlook, that Harish-Chandra's induction could succeed. This led him to make an analogy with finance: "If you don't borrow enough, you have cashflow problems. If you borrow too much, you can't pay the interest."


(ii) There are numerous contexts in which the unified structure theory is the answer to many prayers. Consider "finite simple groups of Lie type": classically these were studied case-by-case, but the general theory of reductive groups provides a completely uniform approach to their internal subgroup structure as well as proofs of simplicity and even formulas for their size. Consider connected compact Lie groups: these are *functorially* "the same" as connected reductive $\mathbf{R}$-groups that do not contain ${\rm{GL}}\_1$ as an $\mathbf{R}$-subgroup, and share many similarities with the theory of connected complex Lie groups with a semisimple Lie algebra, so the algebraic theory over general fields unifies those similarities.


For "algebraic" questions about a finite-dimensional representation of an arbitrary group $\Gamma$ on a vector space $V$ over a field $k$ of characteristic 0, one can often replace $\Gamma$ with its Zariski closure $G$ in ${\rm{GL}}(V)$. If $\Gamma$ is acting irreducibly (or just semi-simply) on $V$ then such $G$ has *reductive* identity component! So once it is known that there is a rich structure theory of connected reductive groups then it opens the door to solving problems that don't seem to mention such groups at all (e.g., to prove that if $V$ and $V'$ are semisimple finite-dimensional representations of $\Gamma$ over a field $k$ of characteristic 0 then $V \otimes V'$ is also a semisimple representation). 


(iii) It is a general fact that a smooth connected affine group over a field of characteristic 0 is reductive if and only if all of its algebraic representations are completely reducible, but this is utterly non-obvious (in the more interesting "only if" direction) just from the definition of "reductive" via the unipotent radical. Another characterization, due to Borel and Richardson, has perhaps less apparent significance (but is actually quite important): if $G$ is a smooth affine subgroup of ${\rm{GL}}\_n$ over a field (any smooth affine group can be realized in that way) then ${\rm{GL}}\_n/G$ is affine if and only if $G^0$ is reductive!


(iv) Suppose you are handed a smooth connected affine group $G$ over a perfect field $k$ and that you know nothing else about $G$ and you'd like to prove some general theorem. The unipotent radical $U$ has a (canonical) composition series whose successive quotients are vector groups, so $U$ is often "easy" to analyze. What about $G/U$? Tautologically this is reductive, but so what? That is useful only if we know something serious about the structure of reductive groups beyond the opaque definition involving triviality of the unipotent radical. It seems scarcely believable from just the definition and knowledge of some basic examples that there should be a uniform method to analyze the internal structure and basic representation theory of all reductive groups, but it is true.


It is also an undeserved miracle that the structure theory (including the role of root systems) works in an essentially characteristic-free manner, even over arbitrary fields (no need to limit to the perfect case; see the books by Borel, Humphreys, and Springer), as well as over rings such as $\mathbf{Z}$ (see SGA3). And many ideas from the theory of quadratic forms (e.g., mass formulas, Hasse Principle) can be extended and understood better by putting them into a broader group-theoretic framework via the structure theory of reductive groups (going far beyond the case of special orthogonal and spin groups). 


(v) The early decades of the theory of automorphic forms exhibited an abundance of similar-looking features, such as for Hilbert modular forms, Siegel modular forms, theta-series in the study of quadratic forms, and so on. But how to unify these into a single framework, treat both Maass forms and holomorphic forms on equal footing, understand the significance of Eisenstein series, etc.? It is precisely the *unified* perspective on the internal structure of all reductive groups and the analogous unification for the representation theory in Harish-Chandra's work that makes them an ideal framework for subsuming all such considerations into a uniform group-theoretic framework. 


In much contemporary research there is a focus on specific classes of groups (e.g., unitary groups, symplectic similitude groups, etc.) in order to prove a result (even if the dream is to treat all reductive groups at once!), much as in contemporary algebraic geometry there is more focus on the study of special classes of structures (e.g., toric varieties, K3 surfaces, etc.) than was the case in Grothendieck's work. But the uniform inductive approach to the internal structure of all reductive groups is very powerful to keep in the back of one's mind. So if you see the theory of reductive groups as the study of some list of explicit examples then that is a big mistake (yet part of the appeal is the balance between the concreteness of the building blocks in the classification and the possibility to nonetheless study them in a manner which often avoids needing the classification).


