As everyone knows, every vector bundle on $\mathbf{P}^1$ splits as a direct sum of line bundles $\mathcal{O}(a\_1)\oplus\cdots\oplus\mathcal{O}(a\_n)$. This means that in the Weil-uniformisation description of vector bundles
$$\text{Bun}\_{\text{GL}\_n}(\mathbf{P}^1)\ =\ \text{GL}\_n(k(\mathbf{P}^1))\backslash \text{GL}\_n(\mathbf{A}\_{\mathbf{P}^1})/\text{GL}\_n(\mathbf{O}\_{\mathbf{P}^1})$$
the diagonal matrices $\text{diag}(t^{a\_1},...,t^{a\_n})$ form a complete set of double coset representatives, where $t$ is a uniformiser of some point in $\mathbf{P}^1$.


**Question:** Pretending that we knew nothing about vector bundles on $\mathbf{P}^1$, is it possible to get this result on double quotient representatives directly using the Iwasawa decomposition of $\text{GL}\_n(\mathbf{A}\_{\mathbf{P}^1})$, or something similar?




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**Clarification:** The question is “the right side is confusing, how can we massage it algebraically to make it obvious that it's just $\mathbf{Z}$"? I really love David Speyer's answer, a (different, non-adelic) way to compute $\text{Bun}\_{\text{GL}\_n}(\mathbf{P}^1)$, but it doesn't address that confusion. In my mind the **question is still open**.



Vector bundles on $\mathbf{P}^1$ and the Iwasawa decomposition
**Disclaimer:** I am not fully confident in my understanding of the terminology here. Corrections are welcome.


$\def\GL{\mathrm{GL}}\def\PP{\mathbb{P}}\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\Spec{\mathrm{Spec}\ }$My understanding is that this is Bruhat decomposition for the loop group of $\GL\_n$.


Rather than your adelic formalism, I will cover $\PP^1$ with two open sets $\Spec \CC[t]$ and $\Spec \CC[t^{-1}]$, overlapping in $\Spec \CC[t, t^{-1}]$. Since every projective module over $\CC[t]$ is free, any vector bundle trivializes one each of our open sets (and hence on their overlap), and so vector bundles are classified by double cosets
$$\GL\_n(\CC[t]) \backslash \! \GL\_n(\CC[t, t^{-1}]) / \GL\_n(\CC[t^{-1}]).$$


Now (I might be using terminology wrongly here) the ind-group $\GL\_n(\CC[t, t^{-1}]$ is a Kac-Moody group, called the loop group of $\GL\_n$. A pair of opposite Borels can be taken to be
$$B\_- := \{ g \in \GL\_n(\CC[t]) : g \bmod t \ \mbox{is upper triangular} \}$$
$$B\_+ := \{ g \in \GL\_n(\CC[t^{-1}]) : g \bmod t^{-1} \ \mbox{is lower triangular} \}.$$
The Weyl group is $W:=S\_n \ltimes \mathbb{Z}^n$, and can be represented by permutation matrices whose nonzero entries are monomials in $t$. We have Bruhat decomposition, $\GL\_n(\CC[t, t^{-1}] = \bigsqcup\_{w \in W} B\_- w B\_+$. So the elements of $W$ are double coset representatives for $B\_- \backslash \GL\_n(\CC[t,t^{-1}])/B\_+$.


We want to quotient not by the Borels but by the parabolics, $P\_- := \GL\_n(\CC[t])$ and $P\_+ = \GL\_n(\CC[t^{-1}])$. So the elements of $W$ still represent all cosets, but redundantly. More specifically, the parabolic subgroup $W\_P$ of $W$ corresponding to $P\_{\pm}$ is $S\_n$. Standard theory tells us that we can get a family of representatives for $B\_- \backslash \GL\_n(\CC[t,t^{-1}])/B\_+$ by taking representatives for $W\_P \backslash W / W\_P$.


We compute $$W\_P \backslash W / W\_P = S\_n \backslash (S\_n \ltimes \ZZ^n) / S\_n = S\_n \backslash \ZZ^n$$
$$ = \{ (a\_1, a\_2, \ldots, a\_n) \in \ZZ^n : a\_1 \geq a\_2 \geq \cdots \geq a\_n \}.$$
Thus, isomorphism classes of vector bundles are indexed by $n$-tuples of decreasing integers, which is correct.


