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.
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.
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.
