An irreducible representation of the extended affine Lie algebra of type $A_{n-1}$ coordinatized by a quantum torus of ν variables is constructed by using the Fock space for the principal vertex operator realization of the affine Lie algebra ${\widetilde{gl}_{n}$.

Let V be a finite-dimensional real vector space on which a root system Σ is given. Consider a meromorphic function φ on V$\Bbb C$=V+iV, the singular locus of which is a locally finite union of hyperplanes of the form {λ ∈ V$\Bbb C$[mid ]〈 λ, α 〉 = s}, α ∈ Σ, s ∈ $\Bbb R$. Assume φ is of suitable decay in the imaginary directions, so that integrals of the form ∫η +iV φ λ, dλ make sense for generic η ∈ V. A residue calculus is developed that allows shifting η. This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.

Let X be a proper, smooth, connected curve, defined over an algebraically closed field of characteristic p>0 and of genus g ≥ 2. We show that there exists a finite solvable group G, of order prime to p, and a Galois étale cover Y → X, with Galois group G, which is not ordinary.

Let F a locally compact non-Archimedean field, of residue characteristic p, and ψ a nontrivial additive character of F. Let σ, σ′ be irreducible representations of the absolute Weil group of F, each of degree a power of p and not induced from a nontrivial unramified extension of F. We give a formula for the value at $s=½ of the ϵ-factor ϵ (σ ⊗ σ ',ψ,s)$, modulo roots of unity in ${\Bbb C}$ of order a power of p. Via the Langlands correspondence, we get an analogous formula for supercuspidal representations of ${\rm GL}_n(F)$.

Using multiple point spaces some new examples of perverse sheaves on images of maps are described. Furthermore, suppose $f:X → Y$ is a finite and proper map of complex analytic manifolds of dimension n and $n+1$ such that every multiple point space is nonsingular and has the dimension expected of a generic map. Then we can describe the composition series for the constant sheaf on the image in the category of perverse sheaves.