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In a lot of fields in algebraic geometry (e.g. GIT or topics on étale cohomology) which make use of group scheme concepts (or in more tame way of algebraic groups), the class of reductive algebraic groups/group schemes seem to play a prominent role.

Presumably this is a quite broad question, but up to now I haven't found a discussion treating the following question:

What is the deeper meaning of reductive algebraic groups making them so interesting for the fields mentioned above? What are their striking features in light of applications in algebraic geometry? Is there any philosophy behind this?

Sean Lawton
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user267839
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    If a reductive group $G$ acts on a finitely generated (commutative) algebra/${\mathbb C}$ by algebra automorphisms, then the algebra of invariants is finitely generated (not for other groups). This means that the space of $G$ orbits is an affine variety; I guess this is at the heart of applications in geometry. – Venkataramana Jul 12 '19 at 03:46
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    @Venkataramana The space of $G$-orbits is more complicated than the spectrum of the algebra of invariants. For instance for $SL_{n\ge 2}$ acting linearly on the $n$-dimensional affine space, there are two orbits, the space of orbits has two points with a non-$T_1$ topology, but the GIT quotient is a point. – YCor Jul 12 '19 at 05:36
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    One should think of the "space of G orbits" as a first approximation of "taking the quotient by the G action". Then it is really helpful to have a finitely generated ring of invariants. – Wilberd van der Kallen Jul 12 '19 at 08:32
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    You might want to look up Haboush's theorem. Another interesting feature is the connection between (split) reductive groups and root data – Louis Jaburi Jul 12 '19 at 10:55
  • You can look at the answer: https://mathoverflow.net/questions/321096/what-is-the-defining-property-of-reductive-groups-and-why-are-they-important/321106#321106 – Ben McKay Jul 12 '19 at 15:04

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Hilbert's 14th question roughly asks when the ring of invariants (under the rational action of an algebraic group $G$) of a finitely generated $k$-algebra $A$ ($k$ a field) is finitely generated. Nagata answered this affirmatively if $G$ is geometrically reductive and gave a counter-example if $G$ is not reductive (Popov showed that if $A^G$ is finitely generated for all $A$ then $G$ had to be reductive). Haboush resolved Mumford's conjecture and showed reductive groups (unipotent radical is trivial) and geometrically reductive groups are equivalent. This allowed for GIT to be be extended from characteristic 0 to arbitrary characteristic. Here is the basic idea behind GIT. First, use the ring of invariants in the affine setting to construct an affine algebraic quotient (good categorical quotient). Then, in the quasi-projective case one chooses an ample $G$-linearized line bundle $L$ (essentially a choice of coordinate system) and with respect to $L$ define semistable points (essentially points that can be distinguished by invariants in terms of the coordinates). Then one covers the semistable locus with affine opens and uses the affine quotient construction to obtain a local quotients. Finally one glues together the local affine pieces to obtain a quasi-projective GIT quotient. The construction all comes down to Hilbert's 14th being true for reductive groups and potentially false otherwise.

References:

  1. To learn about GIT, where this summary mostly comes from, I recommend Lectures on Invariant Theory by Dolgachev.

  2. There is non-reductive GIT too, but it is much more complicated. See Towards non-reductive geometric invariant theory by Brent Doran, Frances Kirwan.

Now, that speaks to the practical reality of why reductive groups are important, but does not address the "philosophy" that you ask for.

Here opinions might vary. I find the following philosophy comforting: "reductive affine algebraic groups are the algebro-geometric analogue of compact Lie groups."

There are a number of reasons for this:

  1. Instead of integrating over compact groups, one has the Reynolds operator for reductive groups.
  2. A complex affine algebraic group $G$ is reductive if and only if it is complexification of a compact Lie group.
  3. Compact quotients have Mostow Slices and Complex Reductive quotients have Luna Slices.
  4. Through Kempf-Ness Theory one can actually replace (up to homeomorphism) affine GIT quotients $X/G$ with the analytic topology with quotients $Y/K$ where $Y\subset X$ and $K$ is a maximal compact subgroup of $G$.
  5. Via 4. Oliver's proof of the Conner Conjecture extends to affine GIT quotients with the analytic topology.

Reference: Schwarz, Gerald W., The topology of algebraic quotients. Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), 135–151, Progr. Math., 80, Birkhäuser Boston, Boston, MA, 1989.

Venkataramana
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Sean Lawton
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