Related papers: Rationality of quaternionic Eisenstein series on $…

We here revisit Fourier analysis on the Heisenberg group H^d. Whereas, according to the standard definition, the Fourier transform of an integrable function f on H^d is a one parameter family of bounded operators on L 2 (R^d), we define (by…

The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted A(E,tau), that depend on a quartic elliptic curve E in P^3 and a translation automorphism tau of E. They are graded…

We consider three families of groups: the Bianchi groups SL(2,O) where O is the ring of integers of an imaginary, quadratic field; the groups SL*(2,O) where O is a *-order of a definite, rational quaternion algebra with an orthogonal…

We identify a class of "semi-modular" forms invariant on special subgroups of $GL_2(\mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an…

Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of…

We introduce a family of algebras $\mathcal{A}_{M,N}$, $M,N\in\mathbb{Z}$, as an extension of a pair of commuting quantum toroidal $\mathfrak{gl}_1$ subalgebras $\mathcal{E}_1,\check{\mathcal{E}}_1$, wherein the parameters are tuned in a…

By using the degenerate Whittaker functions, we study the Fourier expansion of the Gan-Gurevich lifts which are Hecke eigen quaternionic cusp forms of weight $k$ ($k\geq 2$, even) on the split exceptional group $G_2$ over $\mathbb{Q}$ which…

Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $\mathcal{O}$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a…

We define Eisenstein series on rank 2 hyperbolic Kac--Moody groups over R, induced from quasi--characters. We prove convergence of the constant term and hence the almost everywhere convergence of the Eisenstein series. We define and…

We extend a result of Greenberg and Stevens on the interpolation of modular symbols in Hida families to the context of non-split rational quaternion algebras. Both the definite case and the indefinite case are considered.

A formulation of quaternionic quantum mechanics ($\mathbb{H}$QM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After…

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$.…

Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric endomorphisms on $N$, we define the extension $M$ of $N$ by $D$ to be the Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction similar to…

We prove rationality of the quotient $\mathbb{C}^n / H_n$ for the finite Heisenberg group $H_n$, any $n \ge 1$, acting on $\mathbb{C}^n$ via its irreducible representation.

Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $\Gamma$ be a non-uniform lattice in $G$. When $\Gamma$ is a…

We prove that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative…

Superspace is considered as space of parameters of the supercoherent states defining the basis for oscillator-like unitary irreducible representations of the generalized superconformal group SU(2m,2n/2N) in the field of quaternions H. The…

By explicitly calculating and then analytically continuing the Fourier expansion of the twisted double Eisenstein series $E_{s,k-s}^{*}(z,w; 1/2)$ of Diamantis and O'Sullivan, we prove a formula of the Petersson inner product of Cohen's…

The family of quaternionic quasi-unitary (or quaternionic unitary Cayley--Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_{N+1}, as well as many…

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other…