Vector bundles on $\mathbf{P}^1$ and the Iwasawa decomposition 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?

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.
 A: 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.
