Related papers: Log $p$-divisible groups and semi-stable represent…

We show that the category of logarithmic prismatic F-crystals on $(\mathcal{O}_K, \varpi^{\mathbb{N}})$ is equivalent to the category of $\mathbb{Z}_p$-lattices in semistable $\text{Gal}_K$-representations. We then apply our method to…

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic $p$ in terms of…

The regular representation of an essentially finite 2-group $\mathbb{G}$ in the 2-category $\mathbf{2Vect}_k$ of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that…

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad…

We prove an analog of Deligne's theorem for finite symmetric tensor categories $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $\mathcal{C}$…

Let $G$ be a reductive group scheme over the $p$-adic integers, and let $\mu$ be a minuscule cocharacter for $G$. In the Hodge-type case, we construct a functor from nilpotent $(G,\mu)$-displays over $p$-nilpotent rings $R$ to formal…

Let $F$ be a local or global field and let $G$ be a linear algebraic group over $F$. We study Tannakian categories of representations of the Kottwitz gerbes $\text{Rep}(\text{Kt}_{F})$ and the functor $G\mapsto B(F, G)$ defined by Kottwitz…

We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial…

We consider mod $p$ Hilbert modular forms for a totally real field $F$, viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic $p$. For a Hecke eigenform of arbitrary weight, we prove the…

Under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. The Tannakian reconstruction theorem provides another example where a geometric object can be reconstructed from an associated category, in…

We undertake the study of complex rank analogues of parabolic category O defined using Deligne categories. We regard these categories as a family over an affine space, introduce a stratification on this parameter space, and formulate…

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p>0$. Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$ and suppose that the induced extension of residue…

We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka-Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is…

Let $k$ be an algebraically closed field of characteristic $0$. For a log curve $X/k^{\times}$ over the standard log point, we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part…

We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the…

We prove an analogue of the Tate isogeny conjecture and the semi-simplicity conjecture for overconvergent crystalline Dieudonn\'e modules of abelian varieties defined over global function fields of characteristic $p$. As a corollary we…

The principle of tannakian duality states that any neutral tannakian category is tensorially equivalent to the category Rep_k G of finite dimensional representations of some affine group scheme G and field k, and conversely. Originally…

Let G be the group of rational points of a connected reductive group over a finite field. Based on work of Lusztig and Yun, we make the Jordan decomposition for irreducible G-representations canonical. It comes in the form of an equivalence…

Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of…

Let $K$ be an unramified extension of $\mathbb{Q}_p$, and $E$ a finite extension of $K$ with ring of integers $\mathcal{O}_E$. We associate to every finite type continuous $\mathcal{O}_E$-representation $\rho$ of…