TITLE: Dyer-Lashof algebra and Steenrod algebra "duality"
QUESTION [18 upvotes]: Previously asked on math.stackexchange, but perhaps this is more appropriate for MathOverflow:
Let $X$ be a spectrum. I've heard that the action of the Steenrod algebra on $H^*(X; \mathbb{F}_p)$ and the action of the Dyer-Lashof algebra on the homology of the associated infinite loop space, $H_*(\Omega^{\infty}X; \mathbb{F}_p)$, are "dual" in some sense. 
To my knowledge, this has something to do with Koszul duality, but I've never seen this spelled out in detail. Anyone willing to enlighten me on this? Or point me toward a helpful reference?

REPLY [5 votes]: This is not the duality that was asked for in the question, but it does show a way in which the Steenrod algebra structure is related to the Dyer-Lashof algebra structure. The Dyer-Lashof algebra can act both on the homology of infinite loop spaces and $E_{\infty}$ ring spectra. If I consider the latter and look at $H_*(F(X,S^0);\mathbb{F}_p)$ then this has an action of the Dyer-Lashof algebra. It also happens to be isomorphic to the cohomology of $X$ as an algebra. In fact, the action of the Dyer-Lashof algebra "is" the action of the Steenrod algebra (there is an op that needs to be thrown in).
I learned this from chapter 3 of http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf.