The Bialynicki-Birula Stratification of the Affine Grassmannian

Question

Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian Cohomology, the Principal Nilpotent and Kostant Theorem" by Ginzburg. He says that there exists a one-parameter subgroup $\nu:\mathbb{C}^*\rightarrow T$ so that the Bialynicki-Birula stratification of $\mathcal{G}r$ associated with $\mathbb{C}^*$ acting on $\mathcal{G}r$ through $\nu$ coincides with the usual stratification into Iwahori-orbits. I would appreciate a description of this one-parameter subgroup, if possible. I would also appreciate any relevant references.

Answer 1

If I understand what's written Ginzburg correctly, this claim is incorrect, but very easily fixed.

Why is this incorrect: Because there's no cocharacter into $T$ which has finite dimensional BB cells. If there were, then there would be a character whose action on tangent space at the identity coset $[e]$ had a finite dimensional space of positive weights. Since this space is $\mathfrak g((t))/\mathfrak g[[t]]$, and thus infinitely many copies of the adjoint rep of $\mathfrak{g}$, that's impossible.

How it can easily be fixed: While there's no such cocharacter into $T$, there is one into $T\times \mathbb C^*$ where $\mathbb{C}^*$ acts by loop rotation. You want to pick a cocharacter into this group where the positive weight spaces on $\mathfrak{g}((t))$ are given by the Iwahori. There are many such cocharacters (one for each way of assigning positive integers to the simple roots of the corresponding affine Lie algebra). The "most canonical" choice is to take the principal cocharacter (the one that acts with weight 1 on all simple root spaces) in $T$ and the $h+1$st power of the loop rotation (where $h$ is the Coxeter number). I might have that wrong, but anyways, you take a high enough power of loop rotation to assure you have positive weight on $t\mathfrak{g}[[t]]$.