What is the Bruhat decomposition of the affine Grassmannian?

Question

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

$\left(\begin{array}{cc} t^{-1} & 0 \\ 0 & t^{-1} \\ \end{array}\right)$

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

$\left(\begin{array}{cc} t & 0 \\ 0 & t \\ \end{array}\right).$

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

Answer 1

Your identification of the Affine Weyl Group with elements of $ G(\mathbb{C}((t)) $ is wrong. Any co-character $\lambda:\mathbb{C}^*\to T$ where $T$ is a chosen maximal torus can be regarded as an element of $ G(\mathbb{C}((t))) $. If you choose $P$ to be the abelian group generated by the dominant coroots (which is acted upon by W) the semi-direct product with $W$ is isomorphic to the Affine Weyl Group. Now these elements, for example $t\mapsto (t^n,t^{-n})$, give you the desired Bruhat decomposition.

This is explained in the paper of Iwahori and Matsumoto: on some bruhat decomposition... from page 24 on.