Stable splitting of $\Omega SU(n)$

Question

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). 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) 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?

Answer 1

A paper I wrote with Allen Yuan Multiplicative structure in the stable splitting of $\Omega SL_n(\mathbb{C})$ proves a highly structured version of this splitting.