Simple 𝔰𝔩 d+1-modules from Witt algebra modules

Abstract

Abstract Let d ≥ 1 {d\geq 1} be an integer and let 𝒲 d {\mathcal{W}_{d}} be the Witt algebra. For any admissible 𝒲 d {\mathcal{W}_{d}} -module P and any 𝔤 ⁢ 𝔩 d {\mathfrak{gl}_{d}} -module V, one can form a 𝒲 d {\mathcal{W}_{d}} -module ℱ ⁢ ( P , V ) {\mathcal{F}(P,V)} , which as a vector space is P ⊗ V {P\otimes V} . Since 𝒲 d {\mathcal{W}_{d}} has a natural subalgebra isomorphic to 𝔰 ⁢ 𝔩 d + 1 {\mathfrak{sl}_{d+1}} , we can view ℱ ⁢ ( P , V ) {\mathcal{F}(P,V)} as an 𝔰 ⁢ 𝔩 d + 1 {\mathfrak{sl}_{d+1}} -module. Taking P = Ω ⁢ ( 𝝀 ) {P=\Omega(\boldsymbol{\lambda})} , the rank-1 U ⁢ ( 𝔥 ) {U(\mathfrak{h})} -free 𝒲 d {\mathcal{W}_{d}} -module, and V = V ⁢ ( 𝐚 , b ) {V=V({\mathbf{a}},b)} , the simple cuspidal module over 𝔤 ⁢ 𝔩 d {\mathfrak{gl}_{d}} , we get the special 𝔰 ⁢ 𝔩 d + 1 {\mathfrak{sl}_{d+1}} -modules ℱ ⁢ ( 𝝀 ; 𝐚 , b ) = ℱ ⁢ ( Ω ⁢ ( 𝝀 ) , V ⁢ ( 𝐚 , b ) ) \mathcal{F}(\boldsymbol{\lambda};{\mathbf{a}},b)=\mathcal{F}(\Omega(% \boldsymbol{\lambda}),V({\mathbf{a}},b)) which are U ⁢ ( 𝔥 ) {U(\mathfrak{h})} -free modules of infinite rank. We determine the necessary and sufficient condition for the 𝔰 ⁢ 𝔩 d + 1 {\mathfrak{sl}_{d+1}} -module ℱ ⁢ ( 𝝀 ; 𝐚 , b ) {\mathcal{F}(\boldsymbol{\lambda};{\mathbf{a}},b)} to be simple, and for the non-simple case we construct their proper submodules explicitly. At last, using the above results, we deduce an explicit simplicity criterion for the generalized Verma modules induced from V ⁢ ( 𝐚 , b ) {V({\mathbf{a}},b)} and obtain a family of simple affine modules from ℱ ⁢ ( 𝝀 ; 𝐚 , b ) {\mathcal{F}(\boldsymbol{\lambda};{\mathbf{a}},b)} , which can be viewed as the non-weight version of loop modules.