Unitary Correlations and the Fejér Kernel

Let M be a unitary matrix with eigenvalues t _j , and let f be a function on the unit circle. Define X _f (M)=f(t _j ). We derive exact and asymptotic formulae for the covariance of X _f and X _g with respect to the measures |(M)|²dM where dM is Haar measure and an irreducible character. The asymptotic results include an analysis of the Fejér kernel which may be of independent interest.     

Nonvanishing theorems for L-functions of modular forms and their derivatives

Inventiones mathematicae -     

On the cubic Shimura lift for PGL3

Two types of evidence are presented for the conjecture that the image of the cubic Shimura correspondence for PGL₃ is characterized by the nonvanishing of a certain period. Specifically, if a cusp from occurs in the pullback of the automorphic theta function on SO₈ under the adjoint homomorphism PGL₃ → SO₈, then the cusp form is conjectured to be a Shimura lift.     

Schur Polynomials and The Yang-Baxter Equation

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map \({g \mapsto R (g)}\) from \({ \rm{GL}(2, \mathbb{C}) \times \rm{GL}(1, \mathbb{C})}\) to End \({(V \otimes V)}\) , where V is a two-dimensional vector space such that if \({g, h \in G}\) then R ₁₂(g)R ₁₃(gh) R ₂₃(h) = R ₂₃(h) R ₁₃(gh)R ₁₂(g). Here R _{i j} denotes R applied to the i, j components of \({V \otimes V \otimes V}\) . The image of this map consists of matrices whose nonzero coefficients a ₁, a ₂, b ₁, b ₂, c ₁, c ₂ are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a ₁ a ₂ + b ₁ b ₂ ? c ₁ c ₂ = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial s _λ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.     

Explicit formulas for the waldspurger and bessel models

This paper studies certain models of irreducible admissible representations of the split special orthogonal group SO(2n+1) over a nonarchimedean local field. Ifn=1, these models were considered by Waldspurger. Ifn=2, they were considered by Novodvorsky and Piatetski-Shapiro, who called them Bessel models. In the works of these authors, uniqueness of the models is established; in this paper functional equations and explicit formulas for them are obtained. As a global application, the Bessel period of the Eisenstein series on SO(2n+1) formed with a cuspidal automorphic representation π on GL(n) is computed—it is shown to be a product of L-series. This generalizes work of Böcherer and Mizumoto forn=2 and base field ?, and puts it in a representation-theoretic context. In an appendix by M. Furusawa, a new Rankin-Selberg integral is given for the standardL-function on SO(2n+1)×GL(n). The local analysis of the integral is carried out using the formulas of the paper.