Related papers: Theta blocks related to root systems

Jacobi's theta relations among quartic products of theta functions are generalized to those of arbitrary $n$ products. Igusa's procedure of derivation is extended to prove such general theta relations, from which we obtain general addition…

Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This function has connections to Gromov-Witten theory, Hilbert schemes, symmetric…

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…

This paper explores affine Weyl groups and their associated Hecke algebras, concentrating on the Poincar\'e series with coefficients in Hecke algebra. We investigate its relationship with zeta functions on complexes and extend existing…

In this paper, we obtain some behaviours of theta sums of higher index for the Schroedinger-Weil representation of the Jacobi group associated with a positive definite symmetric real matrix of degree m.

We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$ -- biaxial double $(\beta,q)$-Grothendieck polynomials -- which specialize at $q=0$ and $v=1$ to double $\beta$-Grothendieck polynomials…

It is shown in this paper that there is a continuum set of orthogonal systems relative to the weight function $\tilde{Z}^2(t)$. The corresponding integrals cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.

We study the space of vector valued theta functions for the Weil representation of a positive definite even lattice of rank two with fundamental discriminant. We work out the relation of this space to the corresponding scalar valued theta…

We introduce `canonical' classes in the Selmer groups of certain Galois representations with a conjugate-symplectic symmetry. They are images of special cycles in unitary Shimura varieties, and defined uniquely up to a scalar. The…

Hecke expected that an explicit set of theta series obtained from maximal orders of the definite quaternion algebra over Q which is ramified at a prime N will be a basis of the space of holomorphic modular forms of weight 2 and level N.…

We find identities between theta constants with rational characteristics evaluated at period matrix of $R,$ a cyclic 3 sheeted cover of the sphere with $3k$ branch points $\lambda_1...\lambda_{3k}.$ These identities follow from Thomae…

In this paper, we compare the two definitions of Bloch group, and survey the elements in Bloch group. We confirm the Lichtenbaum conjecture on the field $Q(\zeta_p)$ under the assumption the truth of the base of the Bloch group of…

This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace…

Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight $\frac12$. Since the only modular forms of weight $\frac12$ on congruence subgroups of…

In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of `zeta element', of `Weil-\'etale cohomology complexes' and of `integral Selmer groups' for the…

We show that the blocks of category O for the Lie superalgebra q_n associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type C. This allows us to prove two…

In this paper, we examine the tower property concerning the genericity of global theta lifts between various classical groups, drawing inspiration from Rallis' tower property. By exploring the relationship between the analytic properties of…

This paper has three main objectives: (i) To establish an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) To provide an explicit formula for the Fourier coefficients of…

We introduce multivariate generalizations of the Bernstein-Szego polynomials, which are associated to the root systems of the complex simple Lie algebras. The multivariate polynomials in question generalize Macdonald's Hall-Littlewood…

We consider RoCK (or Rouquier) blocks of symmetric groups and Hecke algebras at roots of unity. We prove a conjecture of Turner asserting that a certain idempotent truncation of a RoCK block of weight $d$ of a symmetric group $\mathfrak…