Extended affine Lie algebras, vertex algebras and equivariant $phi$-coordinated quasi modules

For any nullity $2$ extended affine Lie algebra $mathcal{E}$ of maximal type and $ellinmathbb{C}$, we prove that there exist a vertex algebra $V_{mathcal{E}}(ell)$ and an automorphism group $G$ of $V_{mathcal{E}}(ell)$ equipped with a linear character $chi$, such that the category of restricted $mathcal{E}$-modules of level $ell$ is canonically isomorphic to the category of $(G,chi)$-equivariant $phi$-coordinated quasi $V_{mathcal{E}}(ell)$-modules. Moreover, when $ell$ is a nonnegative integer, there is a quotient vertex algebra $L_{mathcal{E}}(ell)$ of $V_{mathcal{E}}(ell)$ modulo by a $G$-stable ideal, and we prove that the integrable restricted $mathcal{E}$-modules of level $ell$ are exactly the $(G,chi)$-equivariant $phi$-coordinated quasi $L_{mathcal{E}}(ell)$-modules.

$(G,chi_phi)$-equivariant $phi$-coordinated modules for vertex algebras

To give a unified treatment on the association of Lie algebras and vertex algebras, we study $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for vertex algebras, where $G$ is a group with $chi_phi$ a linear character of $G$ and $phi$ is an associate of the one-dimensional additive formal group. The theory of $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for nonlocal vertex algebra is established in cite{JKLT}. In this paper, we concentrate on the context of vertex algebras. We establish several conceptual results, including a generalized commutator formula and a general construction of vertex algebras and their $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules. Furthermore, for any conformal algebra $mathcal{C}$, we construct a class of Lie algebras $widehat{mathcal{C}}_phi[G]$ and prove that restricted $widehat{mathcal{C}}_phi[G]$-modules are exactly $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for the universal enveloping vertex algebra of $mathcal{C}$. As an application, we determine the $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for affine and Virasoro vertex algebras.

Toroidal extended affine Lie algebras and vertex algebras

In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $phi$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $phi$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.

Twisted vertex operators and unitary Lie algebras

A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.

Uniqueness of Rankin-Selberg periods

Let $k$ be a local field of characteristic zero. Rankin-Selbergs local zeta integrals produce linear functionals on generic irreducible admissible smooth representations of $GL_n(k)times GL_r(k)$, with certain invariance properties. We show that up to scalar multiplication, these linear functionals are determined by the invariance properties.

Twisted $Gamma$-Lie algebras and their vertex operator representations

Let $Gamma$ be a generic subgroup of the multiplicative group $mathbb{C}^*$ of nonzero complex numbers. We define a class of Lie algebras associated to $Gamma$, called twisted $Gamma$-Lie algebras, which is a natural generalization of the twisted affine Lie algebras. Starting from an arbitrary even sublattice $Q$ of $mathbb Z^N$ and an arbitrary finite order isometry of $mathbb Z^N$ preserving $Q$, we construct a family of twisted $Gamma$-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted $Gamma$-Lie algebras. As application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type $A$, trigonometric Lie algebras of series $A$ and $B$, unitary Lie algebras, and $BC$-graded Lie algebras.