We define the affine Grassmannian to be the quotient $Gr = GL\_n(\mathbb{C}((t)))/GL\_n(\mathbb{C}[[t]])$ where $\mathbb{C}((t))$ is the field of formal Laurent series and $\mathbb{C}[[t]]$ is the ring of formal power series. (The affine Grassmannian can be defined more generally, but here we restrict to a special case.) If we let $B$ be the Borel subgroup of upper triangular matrices in $GL\_n(\mathbb{C})$, $T$ a maximal torus, then the Weyl group $W=N(T)/T$ is just $S\_n$. Let $\widetilde{W} = \mathbb{Z}^{n-1} \rtimes W$ denote the affine Weyl group. Then for $i = 1, 2,...,n-1$ the affine permutations in $\widetilde{W}$ correspond to the usual permutation matrices in $GL\_n(\mathbb{C})$, namely the identity matrix with columns $i$ and $i+1$ interchanged. The matrix for the affine permutation $s\_0$ has ones along the diagonal in rows $2, 3, ..., n-1$, has $t$ in the right hand corner, and $t^{-1}$ in the bottom left corner. Let $I$ denote the Iwahori subgroup, that is, the inverse image of $B$ under the reduction map $GL\_n(\mathbb{C}((t))) \rightarrow GL\_n(\mathbb{C})$. Then $I$ is the set of upper triangular matrices mod $t$. I read somewhere that $GL\_n(\mathbb{C}((t)))$ has a decomposition


$GL\_n(\mathbb{C}((t))) = \cup IwGL\_n(\mathbb{C}[[t]])$


where $w$ varies across the affine permutation matrices. This decomposition is supposed to induce the Bruhat decomposition of the affine Grassmannian into Schubert cells. Now something here is wrong. Since $I \subset GL\_n(\mathbb{C}[[t]])$, and the determinant of any affine permutation matrix is 1 or -1, we have that for any $w \in \widetilde{W}$ the determinant of any matrix in $IwGL\_n(\mathbb{C}[[t]])$ has power series determinant, but the matrix


t^{-1} 0 


0 t^{-1}


is in $GL\_n(\mathbb{C}((t)))$ with determinant $t^{-2}$ and inverse


t 0 


0 t 


So, my question is, what is wrong here? What is the correct decomposition and indexing set?



What is the Bruhat decomposition of the affine Grassmannian?
There are two discrepancies: first, the "strict" affine Bruhat decomposition applies to SL(n), not GL(n); second, the decomposition you're wanting, in the *strict* case, would be $G=\bigcup\_w IwI$, that is, with Iwahori on both sides. (This would make it a disjoint union.)


To adapt to GL(n), the affine Weyl group needs to be enlarged to include the obvious extra diagonal elements.


This doesn't depend so much on $\mathbb C((t))$ and $\mathbb C[[t]]$, as it applies to the field of fractions of a (maybe complete) discrete valuation ring.


