Generalized casimir on restricted kac moody modules

Assume $A$ is symmetrizable and fix the invariant form and $\nu :\mathfrak{h}\to {\mathfrak{h}}^{\ast }$ of Invariant bilinear form for a symmetrizable kac moody algebra. A module $V$ is restricted if, for every $v\in V$, ${\mathfrak{g}}_{\alpha }v=0$ for all but finitely many positive roots $\alpha$. Fix $\rho \in {\mathfrak{h}}^{\ast }$ with $\rho (h_i)=1$. For dual Cartan bases $(u_a),(u^a)$ and opposite-root dual bases $(x_{\alpha ,s}),(y_{\alpha ,s})$, with $x$ positive and $y$ negative, define on $V$ the operator $\mathrm{\Omega }=2{\nu }^{-1}(\rho )+{\sum }_a{u}_a{u}^a+2{\sum }_{\alpha >0,s}{y}_{\alpha ,s}{x}_{\alpha ,s}$.

Products mean successive actions, as in The universal enveloping algebra as a tensor quotient. Root spaces are finite-dimensional by Kac moody root spaces are finite dimensional, and restrictedness makes the last sum finite on each vector. The tensor ${\sum }_s{y}_{\alpha ,s}\otimes x_{\alpha ,s}$ is independent of the dual bases: it corresponds to the identity map of ${\mathfrak{g}}_{\alpha }$ under its perfect pairing with ${\mathfrak{g}}_{-\alpha }$. The Cartan tensor has the same property. Thus $\mathrm{\Omega }$ is well-defined as an operator; it is not asserted to be an infinite element of $U(\mathfrak{g})$. Independence of $h_i$ permits extension of their prescribed $\rho$-values over a finite basis, and $(\rho ,{\alpha }_i)=d_i=({\alpha }_i,{\alpha }_i)\mathrm{/}2$.

Centrality is proved in Generalized kac moody casimir is central and scalar on highest weight modules.

Source comparison: Kleshchev, Definition 2.3.3 and equations (2.18)–(2.20), pp.33–34.