6
$\begingroup$

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by $\lambda\in X_{*}(T)^{+}$. Let $Gr_{\lambda}$ such a strata and $\overline{Gr}_{\lambda}$ the closure in $Gr$.

Then, $\overline{Gr}_{\lambda}=\coprod\limits_{\mu\leq\lambda}Gr_{\mu}$, is it a Whithney stratification?

$\endgroup$
4
$\begingroup$

The point is following: $\overline{Gr^\lambda}$ is a finite dimensional variety acted upon by the pro-algebraic group $G(\mathbb{C}[[t]])$.

This action factors through the action of some finite dimensional algebraic group called $G(\mathcal{J}^l)$. The strata you describe are just the orbits of this group action and there are finitely many of them. (See explanation below)

It is well known, that if a complex algebraic group $G$ acts on a complex variety $X$ algebraically with only finitely many orbits, then the $G$-orbits form a Whitney stratification of $X$, see Algebraic Stratifications of $G$-varieties for details.

Remark: Let $\mathcal{J}^l:= \mathbb{C}[[t]]/t^l \mathbb{C}[[t]]$ for $l\in \mathbb{N}$. Then $G(\mathcal{J}^l)$ is a finite dimensional algebraic group. Let $G(\mathcal{O}^l)$ be the kernel of the canonical homomorphism $G(\mathbb{C}[[t]])\to G(\mathcal{J}^l)$. If you choose an embedding $G\hookrightarrow GL_n(\mathbb{C})$ you can think of $G(\mathcal{O}^l)$ as the subgroup of $G(\mathbb{C}[[t]])$ where all entries on the diagonal are in the form of $1+t^l\cdot f$ with $f\in \mathbb{C}[[t]]$ and all other entries are in the form of $t^l f$. Anyway it is straight forward to show that for any $\lambda$ there exists an $l\gg 0$ such that $G(\mathcal{O}^l)$ operates trivially on $\overline{Gr^\lambda}$, in other words the $G(\mathbb{C}[[t]])$ action factors through $G(\mathcal{J}^l)$

$\endgroup$
  • $\begingroup$ in your argument in "Algebraic Stratifications of G-varieties", you only talk about Whitney's B condition, but what about the A condition? $\endgroup$ – prochet Jan 15 '14 at 15:51
  • $\begingroup$ Well, Whitney's condition A is implied by condition B. A good reference is the book of Banagl, Topology of stratified spaces, Chapter 6 if I recall it correctly. $\endgroup$ – Oliver Straser Jan 15 '14 at 19:43
3
$\begingroup$

I believe the answer is yes. There are perhaps several ways to see this, but a decent reference is the book "Affine Flag Manifolds and Principal Bundles" by Schmitt. It contains an article called "Quantization of Hitchin’s Integrable System and the Geometric Langlands Conjecture" by T.L. Gomez. Your answer can be found near the beginning of Section 4.1.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.