Multiplicative Structure in the Stable Splitting of $\Omega SL_n(\mathbb{C})$

The space of based loops in $SL_n(\mathbb{C})$, also known as the affine Grassmannian of $SL_n(\mathbb{C})$, admits an $\mathbb{E}_2$ or fusion product. Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum. We prove that the Mitchell--Richter splitting is coherently multiplicative, but not $\mathbb{E}_2$. Nonetheless, we show that the splitting becomes $\mathbb{E}_2$ after base-change to complex cobordism. Our proof of the $\mathbb{A}_\infty$ splitting involves on the one hand an analysis of the multiplicative properties of Weiss calculus, and on the other a use of Beilinson--Drinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstruction-theoretic computations.