Related papers: Zak transform, Weil representation, and integral o…
We consider the operator $\mathcal R$, which sends a function on $\mathbb R^{2n}$ to its integrals over all affine Lagrangian subspaces in $\mathbb R^{2n}$. We discuss properties of the operator $\mathcal R$ and of the representation of the…
Let F be the usual real field. Let W be a symplectic vector space over F. It is known that there are two different Weil representations of a Meteplectic covering group $\widetilde{Sp}(W)$. By some twisted actions, we reorganize them into a…
A description is given of the image of the Weil representation of the symplectic group in the Schwartz space and in the space of tempered distributions under the Gaussian integral transform. We also discuss the problem of infinite…
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…
The transformation behaviour of the vector valued theta function of a positive-definite even lattice under the metaplectic group $\mathrm{Mp}_2(\mathbb{Z})$ is described by the Weil representation. We show that the invariants of this…
We construct, by contraction of a suitable complex vector bundle, the Weil representation of the finite symplectic group $Sp(A)$. We give an explicit description of the space of all lagrangian subspaces, which we use to compute the cocycle…
The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group, known as Weil's representation. We prove that there exists an appropriate choice of…
We obtain explicit formulas for the spinor representation $\rho$ of the real orthosymplectic supergroup $\mathrm{OSp}(2p|2q,\mathbb{R})$ by integral 'Gauss--Berezin' operators. Next, we extend $\rho$ to a complex domain and get a…
We suppose that $G$ is a locally compact abelian group, $Y$ is a measure space, and $H$ is a reproducing kernel Hilbert space on $G\times Y$ such that $H$ is naturally embedded into $L^2(G\times Y)$ and it is invariant under the…
In this paper we construct a new variant of the Weil representation, associated with a symplectic vector space V defined over a finite field of characteristic two. Our variant is a representation of a bigger group than that of Weil. In the…
We discuss a fine tuning of the co- and contra-variant transforms through construction of specific fiducial and reconstructing vectors. The technique is illustrated on three different forms of induced representations of the Heisenberg…
The Weil representation is used to construct a minimal type of the two-fold central extension of $\operatorname{Sp}_{2n}(\mathbb{Q}_2)$. The corresponding Hecke algebra is shown to be isomorphic to the classical affine Hecke algebra of the…
To a finite quadratic module, that is, a finite abelian group D together with a non-singular quadratic form Q:D --> Q/Z, it is possible to associate a representation of either the modular group, SL(2,Z), or its metaplectic cover, Mp(2,Z),…
We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where…
Let $R$ be a commutative $\mathbb{Z}[1/p]$-algebra, let $m \leq n$ be positive integers, and let $G_n=\text{GL}_n(F)$ and $G_m=\text{GL}_m(F)$ where $F$ is a $p$-adic field. The Weil representation is the smooth $R[G_n\times G_m]$-module…
We define Hecke operators on vector valued modular forms transforming with the Weil representation associated to a discriminant form. We describe the properties of the corresponding algebra of Hecke operators and study the action on modular…
The Jacobi group is the semi-direct product of the symplectic group and the Heisenberg group. The Jacobi group is an important object in the framework of quantum mechanics, geometric quantization and optics. In this paper, we study the Weil…
We study some explicit Siegel modular forms from Weil representations. For the classical theta group $\Gamma_m(1,2)$ with $m > 1$, there are some eighth roots of unity associated with these modular forms, as noted in the works of Andrianov,…
In a previous work the authors gave a conceptual explanation for the linearity of the Weil representation over a finite field k of odd characteristic: There exists a canonical system of intertwining operators between the Lagrangian models…
Let k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack \hat G over k, the metaplectic extension of the Greenberg realization of Sp_{2n}(R). We also…