Related papers: Rankin-Cohen Deformations and Representation Theor…
A fundamental question, first raised by Langlands, is to know whether the Rankin-Selberg product of two (not necessarily holomorphic) cusp forms f and g is modular, i.e., if there exists an automorphic form f box g on GL(4)/Q whose standard…
This expository paper is based on the lectures given at the program `Modular Representation Theory of Finite and $p$-adic Groups' at the National University of Singapore. We are concerned with recent results on representation theory and…
Let $A = \Bbbk Q / I$ be the path algebra of any finite quiver $Q$ modulo any two-sided ideal $I$ of relations and let $R$ be any reduction system satisfying the diamond condition for $I$. We introduce an intrinsic notion of deformation of…
Based upon new global class field concepts leading to Langlands two-dimensional global correspondences,a modular representation of cusp forms is proposed in terms of global elliptic (bisemi)modules which are (truncated) Fourier series over…
In this article, we study the Zariski closure of modular points in the two-dimensional universal deformation space when the residual Galois representation is reducible. Unlike the previous approaches in the residually irreducible case from…
Whenever a given Poisson manifold is equipped with discrete symmetries the corresponding algebra of invariant functions or the algebra of functions twisted by the symmetry group can have new deformations, which are not captured by…
We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is…
Inspired by the work of Wang and Zhou [4] for Rota-Baxter algebras, we develop a cohomology theory of Rota-Baxter systems and justify it by interpreting the lower degree cohomology groups as formal deformations and as abelian extensions of…
This paper uses previous results of the authors on vector-valued modular forms to study certain non-congruence modular forms. We prove that these forms have unbounded denominators, and in certain cases we verify congruences of…
The Lie algebra su(2) of the classical group SU(2) is built from two commuting quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the…
We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy…
For any positive integer $n$, we introduce a quasi-homogeneous vector field $\textsf{D}$ of degree $2$ on a moduli space $\textsf{T}$ of enhanced Calabi-Yau $n$-folds arising from the Dwork family. By Calabi-Yau quasi-modular forms for…
An elementary introduction to Hilbert modular forms, with a particular attention to their differential properties: Rankin-Cohen brakets, structure of differential rings... This text will appear in SMF Seminaires et Congres.
In this article we study quotients of deformations of simple singularities, and attempt to characterize them in terms of subsystems of simple root systems. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous…
This paper outlines a covariant theory of operators defined on groups and homogeneous spaces. A systematic use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is…
We study the representation theory of the rook-Brauer algebra RB_k(x), also called the partial Brauer algebra. This algebra has a basis of "rook-Brauer" diagrams, which are Brauer diagrams that allow for the possibility of missing edges.…
We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In…
This paper develops the fundamentals of modular representation theory for finite monoids, introducing the decomposition matrix and exploring its connection to Brauer characters. We define modular characteristic and explain how the…
In order to obtain a classification of all possible quantum deformations of the two-photon algebra $h_6$, we introduce its corresponding general Lie bialgebra, which is a coboundary one. Two non-standard quantum deformations of $h_6$,…
In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an $\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of…