Demazure Filtrations of Tensor Product Modules of Current Lie Algebra of type A

Abstract

Let g be a finite-dimensional simple Lie algebra over C and g[t] be its associated cur rent algebra. Due to its connection with combinatorics, number theory and mathematical physics, the theory of of finite-dimensional representations of current algebra has garnered significant attention in the past two decades. In this thesis, we investigate the structure of f inite-dimensional representations of the current Lie algebra of type A. This thesis has been divided into 2 parts. In the first part of the thesis we have considered the finite-dimensional representations of sl2[t] current Lie algebra of type A1. Motivated by the conjecture in [3] that suggest tensor products of Demazure modules of level m and n respectively has a filtration by Demazure module of level m+n in the case of sl2[t]. While it is known that the tensor products of two Demazure module does not in general have a Demazure flag. Our study here extends this question to level 1 Demazure modules with Demazure modules of arbitrary levels for current algebras of type A1. In [15, 12, 26], it was proved that for current algebra of type ADE, the local Weyl module is in fact a Demazure module of level 1 and their Demazure character coincide with the special ized Non-symmetric Macdonald polynomial at t = 0. In [10], a family of finite-dimensional quotients of the local Weyl modules often referred to as CV-modules was introduced. In [8], a necessary and sufficient condition was obtained for the existence of level l Demazure flag in an arbitrary CV-module for sl2[t] and combinatorial description of the graded multiplicities of Demazure modules in CV-modules was given. Motivated by the study initiated in [3] on the structure of the tensor product of two local x Weyl modules for sl2[t], we begin by obtaining a presentation of the representations of sl2[t] that arises from the tensor product of a local Weyl module Wloc(λ) with a CV-module V(ξ). Specializing the partition ξ, we then explore the tensor product of local Weyl modules with irreducible sl2[t]-modules and local Weyl modules for sl2[t]. In each of these cases, we ob serve that the corresponding tensor product possesses a filtration by CV-modules. Using [8], we thus conclude that each such module has a filtration by Demazure module of appropriate levels. Using this filtration, we derive an expressions for the outer multiplicities in the tensor product of Wloc(mω)⊗ev0V(nω). As a consequence we are able to express the product of a Schur polynomial with a specialized Macdonald polynomial in terms of Schur polynomials. Finally we consider the tensor product of two local Weyl modules and prove that it has a f iltration by truncated local Weyl modules of suitable weights. In [18], it had been shown that truncated local Weyl modules are isomorphic to CV-modules. We also determine the graded character of such truncated local Weyl modules in terms of level 2 Demazure module and give the graded multiplicities of level 2 Demazure modules in Wloc(n)⊗Wloc(m). In the second part of the thesis, we will generalize the results proved in the case of sl2[t] to the case of sln+1[t]. We have considered the finite-dimensional representations of sln+1[t], current Lie algebra of type An and proved that the tensor product of local Weyl modules Wloc(mωi) ⊗Wloc(kωi) has a filtration by M(ν,λ)-modules for i = 1,n. The notion of M(ν,λ)-modules was introduced in [1] and authors have proved that M(ν,λ) is isomorphic to fusion product module D(2,λ)∗Wloc(µ). In [27], it was also proved that M(ν,λ) has a filtration by level 2 Demazure modules as an sln+1[t]-module for n ≤ 3. Therefore, we have that Wloc(mωi)⊗Wloc(kωi) has a filtration by level 2 Demazure modules as an sln+1[t] module for i = 1,n and n ≤3. xi To complete the results in the case of sl3[t], we have also considered Wloc(mω1)⊗Wloc(kω2) and proved that it has a filtration by truncated local Weyl modules. To prove this result we need to construct a short exact sequence of such truncated local Weyl modules and prove that these are isomorphic to fusion product modules. In [11], Feigin and Loktev introduced the notion of fusion product module and by definition of fusion product it is dependent on the choice of parameters but it was conjectured that it is independent of the choice of parameters. This conjecture has been proved in many papers [24, 6, 10, 12, 11, 22, 25] for various particular cases. At last, we have proved that if λ3 = 0 or |λ1|≤ 1 then the fusion product module D(3,3λ3)∗D(2,2λ2)∗D(1,λ1)∗V(θ)∗j where λ1,λ2,λ3 ∈ P+ and θ is the highest root of sl3, is independent of the parameters as an sl3[t]-module. To obtain this result, we have considered a special class of CV-modules in the case of sl3[t] and denoted it by Mj(3λ1,2λ2,λ3) where j ∈ Z+ and (λ1,λ2,λ3) ∈ (P+)3. We have also constructed a short exact sequence of Mj(3λ1,2λ2,λ3) and proved that it is isomorphic to D(3,3λ3)∗D(2,2λ2)∗D(1,λ1)∗V(θ)∗j as an sl3[t]-module when λ3 = 0 or |λ1|≤ 1. This result has an important application as it helps us to compute the graded character formula for the tensor product of two local Weyl modules Wloc(mω1)⊗Wloc(nω2) in terms of truncated local Weyl modules