GIT lecture 15-16

These lecture (GIT15-16) were given on 9/30/09 and 10/2/09.

Excerpts

First this result was proved (it was only stated last lecture)

Prop. T a torus of dim n acts on V of dim N by t.(x_1, ..., x_N) = (\chi_1(t)x_1, ..., \chi_N(t)x_N) and stabilizer of generic point is finite.  Let x \in V with supp(x) = \{\chi_1, ..., \chi_k\} then x is semi-stable iff 0 \in C(\chi_1, ..., \chi_k) and properly stable iff

  1. \dim span\{\chi_1, ..., \chi_k\} = n
  2. 0 \in interior of C(\chi_1, ..., \chi_k)

Cor. If G is a torus, then a point is properly stable iff its properly stable with respect to all 1 – parameter subgroups.

Thm (Hilbert-Mumford numerical criterion) If G is reductive then (and X quasi projetive) then x \in X is

  1. semi-stable iff is is semi-stable with repspect to any 1-parameter subroups.
  2. properly stable iff it is so for any 1-parameter subgroup.

The idea of the proof is simple.  If a points fails to be semi-stable (or properly stable) then there is a 1-parameter subgroup that realizes this failure.

Some technical results are used to prove this.  The notation is O = k[[t]], K = k((t)).

Lemma: if y \in \overline{G \cdot x} - G \cdot x then there exists \lambda \in G(K) (recall G \subset GL(V)) so this is effectively a matrix with coeff in K, hence a matrix which has entries which are Laurent series in t) such that \lim_{t \to 0} \lambda(t)\cdot x = y.

Thm(Iwahori). Let G be a reductive algebraic group.  Then each double coset G(O) \backslash G(K) /G(O) contains a unique 1-parameter subgroup. In other words, given a K point g \in G(K), then can write g(t) = A(t) \lambda(t) B(t) for A,B \in G(O) and \lambda(t) a 1-param. sub group.

This lemma is also useful.

Lemma: with the notation above, if \lim_{t \to 0} g(t) = \lim_{t\to 0} A(t) \lambda(t) B(t)x = y exists (so \lim_{t \to 0} \lambda(t) B(t)x = y':= A^{-1}(t)y then \lim_{t \to 0} \lambda(t)x = z exists and \lim_{t\to 0} \lambda^{-1}(t)y' = z.

This is proved checking everything component wise and using that \lambda(t) = diag(t^{a_1},..., t^{a_n}) for integers a_i.

Here is part of the argument for the thm above.  Suppose x is not properly stable.  This means either G\cdot x is not closed, or G\cdot x is closed by stab(x) is infinite.  Say stab(x) is infinite. I forget at the moment, but for some reason G \to G\cdot x is affine.  It it was proper, then it would be finite, but the fibers are infinite, so its not proper.  Now by the valuative criterion for properness there exist g \in G(K) - G(O) such that g(t) \in stab(x).

now use Iwahori to write \lim_{t \to 0} g(t)\cdot x = \lim_{t \to 0} A(t) \lambda(t) B(t)\cdot x = x. So by the lemma above

\lim_{t \to 0} \lambda(t)x = z

exists and \lambda(t')\cdot z = \lim_{t \to 0}\lambda(tt')x = z, so \lambda(t) \in stab(z). But z is in the orbit of x, because G\cdot x is closed, and all elements in the orbit have conjugate stabilizers, so stab(x) contains a 1 parameter subgroup as well.

In the rest of the lecture there were some applications given, but I haven’t gone through it carefully.

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