Introduction par Ahmed Abbes Room: Centre de conférences Marilyn et James Simons

q-Higgs Modules and the Nygaard Filtration

• Takeshi Tsuji (The University of Tokyo)

Room: Centre de conférences Marilyn et James Simons Similarly to the theory of crystalline cohomology, prismatic crystals and their cohomology over a bounded prism (R,I) are described in terms of q-Higgs modules when (R,I) is defined over the q-crystalline prism, or more generally when R/I contains a primitive pth root of unity. We see how this allows us to understand the isogeny property of Frobenius, the de Rham specialization, and the Nygaard filtration for the prismatic cohomology of a prismatic F-crystal.

Relative Poincaré Duality in Non Archimedean Geometry

• Emanuel Reinecke (IHES)

Room: Centre de conférences Marilyn et James Simons In my talk, I will explain a relative version of mod-p Poincare duality for any proper morphism of rigid-analytic varieties over a p-adic field. A key ingredient in the proof will be a novel construction of trace maps for smooth morphisms and for proper morphisms of rigid-analytic varieties. Along the way, I will sketch a new and essentially diagrammatic proof of mod-p Poincare duality for smooth and proper morphisms. Joint work with Shizhang Li and Bogdan Zavyalov.

On the Left Adjoint of mod p Smooth Parabolic Induction

• Cédric Pépin (Université Sorbonne Paris Nord)

Room: Centre de conférences Marilyn et James Simons Let $F/\mathbb{Q}_p$ be a finite field extension with ring of integers $\mathcal{O}_F$ and residue field $k_F$. Let $\mathbf{G}$ be a split connected reductive group over $\mathcal{O}_F$, and $D(G)$ be the derived category of smooth representations of $G:=\mathbf{G}(F)$ over a fixed field extension of $k_F$. For a Borel $\mathbf{B}=\mathbf{T}\mathbf{U}\subset\mathbf{G}$, the $t$-exact parabolic induction functor ${\rm Ind}_B^G:D(T)\rightarrow D(G)$ admits a left adjoint $L(U,-)$, as proved by Heyer. We study the functor $L(U,-)$ on algebraic weights $L(\lambda)$ which are $p$-small. When $F$ is unramified, we show that $L(U,ind_{G(\mathcal{O}_F)}^G(L(\lambda)))\in D(T)$ splits completely (under some mild modular assumptions). This is joint work with Karol Koziol.

Localization Theory for Harish-Chandra Bimodules in Positive Characteristic

• Simon Riche (Université Clermont Auvergne)

Room: Centre de conférences Marilyn et James Simons Given a connected reductive algebraic group G over an algebraically closed field of positive characteristic, a Harish-Chandra bimodule is a G-equivariant bimodule for the enveloping algebra of the Lie algebra of G such that the differential of the G-action coincides with the diagonal action of the Lie algebra. Such objects are interesting in particular because they realize translation functors for representations of G. We will explain how Harish-Chandra bimodules with a regular central character can be related to equivariant coherent sheaves on the Steinberg variety of G, building on the localization theory of Bezrukavnikov-Mirkovic-Rumynin. This relation is an essential step in the construction of an equivalence relating such coherent sheaves to perverse sheaves on the affine flag variety of the Langlands dual group, and opens the way to a study of certain categories of representations of the Lie algebra of G using constructible sheaves.