Twisted Demazure modules, fusion product decomposition and twisted 𝑄-systems

Abstract

In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the twisted current algebras. These modules are indexed by an | R + | |R^+| -tuple of partitions ξ = ( ξ α ) α ∈ R + \xi =(\xi ^{\alpha })_{\alpha \in R^+} satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of ξ \xi these modules become isomorphic to Demazure modules in various levels for the twisted affine algebras. As a consequence we see that the defining relations of twisted Demazure modules can be greatly simplified. Furthermore, we investigate the notion of fusion products for twisted modules, first defined by Feigin and Loktev in 1999 for untwisted modules, and use the simplified presentation to prove a fusion product decomposition of twisted Demazure modules. As a consequence we prove that twisted Demazure modules can be obtained by taking the associated graded modules of (untwisted) Demazure modules for simply-laced affine algebras. Furthermore we give a semi-infinite fusion product construction for the irreducible representations of twisted affine algebras. Finally, we prove that the twisted Q Q -sytem defined by Hatayama et al. in 2001 extends to a non-canonical short exact sequence of fusion products of twisted Demazure modules.