Related papers: Remarks on the theta decomposition of vector-value…
These notes contain a revisit to the matrix construction of the Weil representation by the author in the Journal of Algebra in 1972.
We show that it is possible to remove two differential operators from the standard collection of $m$ of them used to embed the space of Jacobi forms of \textit{odd} weight $k$ and index $m$ into several pieces of elliptic modular forms.…
We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules.We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors…
We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral…
We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.
We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve…
We introduce a certain differential (heat) operator on the space of Hermitian Jacobi forms of degree 1, show it's commutation with certain Hecke operators and use it to construct a lift of elliptic cusp forms to Hermitian Jacobi cusp forms.…
In this manuscript, we develope the theory of harmonic analysis on the Heisenberg group G of high dimension. We investigate the theta functions and the Weil representation related to this Heisenberg group and describe the connection among…
This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $\theta$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes).…
In the article, we study the Oberdieck derivative defined on the space of weak Jacobi forms. We prove that the Oberdieck derivative maps a Jacobi form to a Jacobi form. Moreover, we study the adjoint of Oberdieck derivative of a Jacobi cusp…
Jacobi's elliptic functions have been constructed from a deformed Lie algebra. The generators of the algebra have been obtained from a bi-orthogonal system. The deformation parameter resembles the modulus of the relevant elliptic functions.
The notion of double depth associated with quasi-Jacobi forms allows distinguishing, within the algebra of quasi-Jacobi singular forms of index zero, certain significant subalgebras (modular-type forms, elliptic-type forms, Jacobi forms).…
The aim of this paper is to derive new results about Jacobi's inversion formulas for modular forms of levels 5 and 6. For this purpose, we use Farkas and Kra's theory of theta functions with rational characteristics.
We complete the classification of the finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent elements in classical Lie algebras. This extends earlier work where this classification is…
Some explicit expressions are given for the theta series of Niemeier lattices. As an application, we present some of their congruence relations.
We continue our work on understanding Howe correspondences by using theta representations from p-adic groups to compact groups. We prove some results for unitary theta representations of compact groups with respect to the induction and…
In this paper explicit decompositions are provided of the Weyl reflections in affine Lie algebras, in terms of fundamental Weyl reflections.
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
We study flat deformations of quotients of a polynomial algebra in a class of graded commutative associative algebras. Functional equations and their solutions in terms of theta functions play important role in these studies. An analog of…
We prove the transformation laws of the four Jacobi theta functions using Gordon's proof for the transformation law of the Dedekind eta function.