Identities of Inverse Chevalley Type for the Graded Characters of Level-Zero Demazure Submodules over Quantum Affine Algebras of Type C

Abstract

We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type C. These identities express the product \(e^{\mu } \text {gch} ~V_{x}^{-}(\lambda )\) of the (one-dimensional) character \(e^{\mu }\), where \(\mu \) is a (not necessarily dominant) minuscule weight, with the graded character gch\(V_{x}^{-}(\lambda )\) of the level-zero Demazure submodule \(V_{x}^{-}(\lambda )\) over the quantum affine algebra \(U_{\textsf{q}}(\mathfrak {g}_{\textrm{af}})\) as an explicit finite linear combination of the graded characters of level-zero Demazure submodules. These identities immediately imply the corresponding inverse Chevalley formulas for the torus-equivariant K-group of the semi-infinite flag manifold \(\textbf{Q}_{G}\) associated to a connected, simply-connected and simple algebraic group G of type C. Also, we derive cancellation-free identities from the identities above of inverse Chevalley type in the case that \(\mu \) is a standard basis element \({\varepsilon }_{k}\) in the weight lattice P of G.