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This paper gives insight into intriguing connections between two apparently unrelated theories: the theory of skein modules of 3-manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. Let R be a…

Fontaine's $D_{\mathrm{cris}}$ functor allow us to associate an isocrystal to any crystalline representation. For a reductive group $G$, we study the reduction of lattices inside a germ of crystalline representations with $G$-structure $V$…

Using the theory of $(\varphi, \Gamma)$-modules we generalizes Greenberg's construction of the $\Cal L$-invariant to semistable representations

In this paper we describe a theory of (branched) crystals which is adapted to the study of representations in the BGG category $\cal O$ and which generalizes the theory of normal crystals of Kashiwara. In the case of $sl_2$ we show that one…

We show that Penkov's approach to a superanalog of Borel-Bott-Weil theorem for $G=GL(m|n)$ over a field of zero characteristic can be extended for a perfect field of arbitrary odd characteristic. We also prove some partial version of…

Representations of four dimensional superconformal groups are constructed as fields on many different superspaces, including super Minkowski space, chiral superspace, harmonic superspace and analytic superspace. Any unitary irreducible…

The purpose of this paper is to prove necessary and sufficient criteria for a $GL(m|n)$-supermodule to have a good or Weryl filtration. We also introduce the notion of a Steinberg supermodule analogous to the classical notion of Steinberg…

We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.

We consider irreducible lowest-weight representations of Cherednik algebras associated to certain classes of complex reflection groups in characteristic p. In particular, we study maximal graded submodules of Verma modules associated to…

We provide an explicit combinatorial realization of all simple and injective (hence, and projective) modules in the category of bounded $\mathfrak{sp}(2n)$-modules. This realization is defined via a natural tableaux correspondence between…

Let G be a subgroup of GL(V), where V is a finite dimensional vector space over a finite field of characteristic p >0. If det(g-1) = 0 for all g \in G then we call G a fixed-point subgroup of GL(V). Motivated in parallel by questions in…

With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre…

In this paper, we use geometric methods to study the relations between admissible representations of $\mathbf{GL}_n(\mathbb{C})$ and unramified representations of $\mathbf{GL}_m(\mathbb{Q}_p)$. We show that the geometric relationship…

Let $G$ be a finite group of Lie type and $\ell$ be a prime which is not equal to the defining characteristic of $G$. In this note we discuss some open problems concerning the $\ell$-modular irreducible representations of $G$. We also…

The main result of this article is an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie supergroups. It also includes an exposition of recent results of the second author on…

We identify a class of "semi-modular" forms invariant on special subgroups of $GL_2(\mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an…

We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural…

We determine semisimple reductions of irreducible, 2-dimensional crystalline representations of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_{p^f})$. To this end, we provide explicit representatives for the…

We study branching multiplicity spaces of complex classical groups in terms of GL(2) representations. In particular, we show how combinatorics of GL(2) representations are intertwined to make branching rules under the restriction of GL(n)…

In our previous paper math.QA/0412192 the Cayley-Hamilton identity for the GL(m|n) type quantum matrix algebra was obtained. Here we continue investigation of that identity. We derive it in three alternative forms and, most importantly, we…