Identities of inverse Chevalley type for 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 $e^{\mu }\mathrm{gch}{V}_x^{-}(\lambda )$ of the (one-dimensional) character $e^{\mu }$, where $\mu$ is a (not necessarily dominant) minuscule weight, with the graded character $\mathrm{gch}{V}_x^{-}(\lambda )$ of the level-zero Demazure submodule $V_x^{-}(\lambda )$ over the quantum affine algebra $U_{\mathsf{q}}({\mathfrak{g}}_{\mathrm{af}})$ as an explicit finite linear combination of the graded characters of level-zero Demazure submodules. These identities immediately imply the corresponding inverse Chevalley formulas in the torus-equivariant $K$-group of the semi-infinite flag manifold ${\mathbf{Q}}_G$ 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 ${\epsilon }_k$ in the weight lattice $P$ of $G$.

29 pages