We establish the existence of Demazure flags for graded local Weyl modules for hyper current algebras in positive characteristic. If the underlying simple Lie algebra is simply laced, the flag has length one, i.e., the graded local Weyl modules are isomorphic to Demazure modules. This extends to the positive characteristic setting results of Chari-Loktev, Fourier-Littelmann, and Naoi for current algebras in characteristic zero. Using this result, we prove that the character of local Weyl modules for hyper loop algebras depend only on the highest weight, but not on the (algebraically closed) ground field, and deduce a tensor product factorization for them.

Given an algebraically closed field k of characteristic zero, a Lie superalgebra g over k and an associative, commutative k-algebra A with unit, a Lie superalgebra of the form g ⊗ k A is known as a map superalgebra. Map superalgebras generalize important classes of Lie superalgebras, such as, loop superalgebras (where A = k[t, t −1 ]), and current superalgebras (where A = k[t]). In this paper, we define Weyl functors, global and local Weyl modules for all map superalgebras where g is either sl(n, n) with n ≥ 2, or a finite-dimensional simple Lie superalgebra not of type q(n). Under certain conditions on the triangular decomposition of these Lie superalgebras we prove that global and local Weyl modules satisfy certain universal and tensor product decomposition properties. We also give necessary and sufficient conditions for local (resp. global) Weyl modules to be finite dimensional (resp. finitely generated).2010 Mathematics Subject Classification. 17B65, 17B10.

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form g ⊗ F F[t1, . . . , t ℓ ]/I, where g is a finite-dimensional Lie algebra over a field F of characteristic zero, and I is a finite-codimensional ideal of F[t1, . . . , t ℓ ] generated by monomials. In particular, when g is semisimple and F is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in g ⊗ F F[t1, . . . , t ℓ ]/I. As an application of our main result, we show that the center of the universal enveloping algebra of g ⊗ F F[t1, . . . , t ℓ ]/I acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two.

Let Γ be a group acting on a scheme X and on a Lie superalgebra g. The corresponding equivariant map superalgebra M (g, X) Γ is the Lie superalgebra of equivariant regular maps from X to g. In this paper we complete the classification of finite-dimensional irreducible M (g, X) Γmodules when g is a finite-dimensional simple Lie superalgebra, X is of finite type, and Γ is a finite abelian group acting freely on the rational points of X. We also describe extensions between these irreducible modules in terms of extensions between modules for certain finite-dimensional Lie superalgebras. As an application, when Γ is trivial and g is of type B(0, n), we describe the block decomposition of the category of finite-dimensional M (g, X) Γ -modules in terms of spectral characters for g.