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We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the…

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular…

Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for…

In the context of Covariant Quantum Mechanics for a spin particle, we classify the ``quantum vector fields'', i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the…

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra.…

We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a…

While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group $\Gamma$ and an arbitrary representation $R$ of $\Gamma$ in GL$(n,{\mathbb C})$, their existence has been established in the literature only when…

We introduce a generalization of degenerate affine Hecke algebra, called wreath Hecke algebra, associated to an arbitrary finite group G. The simple modules of the wreath Hecke algebra and of its associated cyclotomic algebras are…

This paper describes the vector bundle on the elliptic modular curve that is associated to a vertex operator algebra $V$ (VOA) or more generally a quasi-vertex operator algebra (QVOA), with a view towards future applications aimed at…

Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of…

In this paper, we construct a quantization functor, associating a complex vector space H(V) to a finite dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil…

This paper shows that the cyclotomic quiver Hecke algebras of type $A$, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying all seminormal bases and then give an explicit…

In this paper, we study mathematical functions of relevance to pure gravity in AdS3. Modular covariance places stringent constraints on the space of such functions; modular invariance places even stronger constraints on how they may be…

In this work we construct an eigencurve for p-adic modular forms attached to an indefinite quaternion algebra over Q. Our theory includes the definition, both as rules on test objects and sections of line bundle, of p-adic modular forms,…

In this note we are interested in labelling the irreducible representations of non-semisimple specialisations of Hecke algebras of complex reflection groups. We will use category O for the rational Cherednik algebra and the KZ functor…

We study various relations governing quasi-automorphic forms associated to discrete subgroups of ${\rm SL}(2,\mathbb{R}) $ called Hecke groups. We show that the Eisenstein series associated to a Hecke group ${\rm H}(m)$ satisfy a set of $m$…

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in…

We present the general framework of \'Ecalle's moulds in the case of linearization of a formal vector field without and within resonances. We enlighten the power of moulds by their universality, and calculability. We modify then \'Ecalle's…

Heisenberg groups over algebras with central involution and their automorphism groups are constructed. The complex quaternion group algebra over a prime field is used as an example. Its subspaces provide finite models for each of the real…

We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be…