Related papers: Hecke Operators on Vector-Valued Modular Forms

In this article, the authors consider the Schr\"{o}dinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to…

We compute the intertwining relation between the Hecke operators and the Siegel lowering operators on Siegel modular forms of arbitrary level $N$ and character $\chi$ by using formulas for the action of the Hecke operators on Fourier…

We derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring $\mathfrak{o}$. To this end, we develop Hecke-theoretic techniques for the enumeration, by…

We study Hecke operators on moduli spaces of ramified $G$-bundles using the combinatorial language of Hecke graphs. We introduce a general notion of $\mathcal H$-ramification in the spirit of parahoric ramification, which depends on a…

We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and…

We give a survey of recent work on the construction of differential operators on various types of modular forms (mod p). We also discuss a framework for determining the effect of such operators on the mod p Galois representations attached…

Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the…

We present a method to compute two Hecke operators acting on a space of algebraic modular forms simultaneously based on an idea of Eichler's. We show that in certain cases this method can be used to obtain the action of the full Hecke…

In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen…

We define a new category analogous to ${\bf FI}$ for the $0$-Hecke algebra $H_n(0)$ called the $0$-Hecke category, $\mathcal{H}$, indexing sequences of representations of $H_n(0)$ as $n$ varies under suitable compatibility conditions. We…

We present complete classifications of Toeplitz + Hankel operators on vector-valued Hardy spaces and classify paired operators on $L^2(\mathbb{T})$. We also study the latter class through the lens of inner functions on the disc.

Matrix representations of Hecke operators on classical holomorphical cusp forms and corresponding period polynomials are well known. In this article we define Hecke operators on period functions and show that they correspond to the Hecke…

Recently, a weak converse theorem for Borcherds' lifting operator of type $O(2,1)$ for $\G_0(N)$ is proved and the logarithmic derivative of a modular form for $\G_0(N)$ is explicitly described in terms of the values of Niebur-Poincar\'e…

This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…

In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to…

We construct a family of exact functors from the BGG category of representations of the Lie algebra sl to the category of finite-dimensional representations of the degenerate (or graded) affine Hecke algebra H of GL. These functors…

We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program. For a smooth projective curve X we introduce an algebraic stack \tilde\Bun_G of…

In this paper we extend the results in [Ra] on the representation of the Hecke algebra, determined by the matrix coefficients of a projective, unitary representation, in the discrete series of representations of the ambient group, to a more…

Let X be the group of weights of a maximal torus of a simply connected semisimple group over C and let W be the Weyl group. The semidirect product W(Q\otimes X/X) is called the extended Weyl group. There is a natural C(v)-algebra H called…

We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common ``finite approximation'' method. These spectra are simple…