The space $\Omega SU(n)$ is homotopy-equivalent to $SL\_n(\mathbb{C}[z,z^{-1}])/SL\_n(\mathbb{C}[z])$. Using this, Steve Mitchell introduced a filtration of $\Omega SU(n)$ by subspaces $F\_k$ which can be viewed as Schubert varieties ([MR0862881](https://mathscinet.ams.org/mathscinet-getitem?mr=862881)). The homology ring is $H\_\*(\Omega SU(n))=\mathbb{Z}[b\_1,\dotsc,b\_{n-1}]$, with $|b\_i|=2i$, and $H\_\*(F\_k)$ is just the subspace spanned by monomials $b^\alpha=\prod\_{i=1}^{n-1}b\_i^{\alpha\_i}$ with $\sum\_i\alpha\_i\leq k$. It was apparently conjectured by Hopkins and Mahowald that this filtration is stably split, so $\Sigma^\infty \Omega SU(n)\_+\simeq\bigvee\_k\Sigma^\infty F\_k/F\_{k-1}$. A later paper of Mark Mahowald and Bill Richter ([MR0862881](https://mathscinet.ams.org/mathscinet-getitem?mr=1136478)) stated that there was an article in press written by Richter, titled *A partial Cartan formula for the stable splitting of $\Omega SU(n)$, and a
stable splitting of the loops on a infinite complex Stiefel manifold*, containing a proof of the conjecture. However, no such article ever appeared. I vaguely remember that there was some discussion of this situation in the late 1990s, but I do not remember the contents of any such discussions. Has there been more recent work on this question?



Stable splitting of $\Omega SU(n)$
A paper I wrote with Allen Yuan [Multiplicative structure in the stable splitting of $\Omega SL\_n(\mathbb{C})$](https://www.sciencedirect.com/science/article/abs/pii/S0001870819301525) proves a highly structured version of this splitting.


