Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ and positive roots, so that $B, N^\pm$ are the corresponding Borel and connected unipotent subgroups.
There is an evaluation map $\text{ev}: G(\mathbb C[t]) \to G$ mapping $\gamma \to \gamma(0)$, and we define the Iwahori subgroup to be $I=\text{ev}^{-1}(B)$. Additionally, let $J = \text{ev}^{-1}(N^+)$.
Let $X^*(T)$ be the correspondings set of coweights and $t^\lambda$ be the class of $\lambda \in X^*(T)$ in $\mathcal Gr$. The literature is abound with the fact that $\mathcal Gr$ admits a Bruhat decomposition $$ \mathcal Gr = \bigsqcup_{\lambda \in X^*(T)}e_\lambda,$$ where the $e_\lambda$ are the Schubert cells. The main proofs I have seen of this fact are usually Mitchell's Building's paper or Pressley & Segal's book Loop Groups.
Unfortunately, there seems to be some discrepencies in the statement of the Bruhat decomposition. Overwhelmingly, the literature seems to say that the Schubert cells are the orbits of the Iwahori subgroup: $e_\lambda = It^\lambda$. However, Pressley and Segal (which is rather well cited!) uses instead cells which are the orbits $e_\lambda = Jt^\lambda$. My question:
Are these decompositions topologically equivalent? Or are they instead similar notions which have been endowed with the same name. If they are the same, are there any advantages using one framework over the other. If they are different, why has the community overwhelmingly chosen the Iwahori orbit viewpoint?
