Alladi Ramakrishnan Hall

Demazure filtrations of tensor product modules of current Lie algebra of type A1

Divya Setia

IMSc

Let g be a finite-dimensional simple Lie algebra over the complex field C and g[t] be the Lie algebra of polynomial mappings from C to g, which is its associated current Lie algebra. The notion of Weyl modules for affine Kac-Moody Lie algebras was introduced by V. Chari and A. Pressley. Subsequently it was demonstrated that for current Lie algebra of type ADE, the local Weyl modules are in fact Demazure modules of level 1 and their Demazure characters coincide with non-symmetric Macdonald polynomials, specialized at t = 0. We study the structure of the finite-dimensional representations of the current Lie algebra of type A1, sl2[t], which are obtained by taking tensor products of local Weyl modules with Demazure modules. In this talk, we shall show that these representations admit a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for sl2[t]. Using Pieri formulas, we will also express the product of two specialized Macdonald polynomials in terms of specialized Macdonald polynomials. Furthermore, we shall show that the tensor product of a local Weyl module with an irreducible sl2[t] module admits a Demazure filtration and derive graded character of such tensor product modules. This helps us express the product of a specialized Macdonald polynomial with a Schur polynomial in terms of Schur polynomial. Our findings provide evidence for the conjecture that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n.