Number Theory Seminar

A result of Bernstein from the 80s gives an equivalence between modules over the Iwahori-Hecke algebra and the category of smooth $G$-representations generated by their Iwahori-fixed vectors. Here $G$ is a $p$-adic Lie group and the coefficient field is $\Bbb{C}$. This equivalence usually fails if we take a coefficient field $k$ of characteristic $p$, as in the mod $p$ local Langlands correspondence. Schneider found a way to remedy the situation by instead relating the derived category $D(G)$ to modules over a certain differential graded variant of the (pro-$p$) Iwahori- Hecke algebra. The goal of the talk is to interpret Schneider's equivalence in the case $G=I$ as a version of Koszul duality for the Iwasawa $k$-algebra $\Omega(G)$. This makes use of Lazard's theory of $p$-valuations, which played a key role in his proof that $\Omega(G)$ is Noetherian. The rough idea is that the filtration on $\Omega(G)$ somehow corresponds to an $A_{\infty}$-structure on the Yoneda extension algebra. When $G$ is abelian the $A_{\infty}$-structure is trivial (the converse was recently shown by Carl Wang-Erickson); this generalizes a result of Schneider for $\Bbb{Z}_p$.