The Casimir comparison on a weight space

Given: The Axiom of Choice, such $\mathfrak{g}$ and $\mathfrak{h}$ with chosen positive system ${\mathrm{\Phi }}^{+}$, the Casimir element $C$, the two dual bases $h_j,h^j$ of $\mathfrak{h}$, and vectors $e_{\alpha },f_{\alpha }$ with $B(e_{\alpha },f_{\alpha })=1$ for each $\alpha \in {\mathrm{\Phi }}^{+}$; also $\lambda \in {\mathrm{\Lambda }}^{+}$ and $\mu \in {\mathfrak{h}}^{\ast }$.

The Axiom of Choice is assumed; it enters through the root-space theory supplying [F1] and [F3] and through the classification of [F6] (The Axiom of Choice).

Under Choice, the supplied Cartan subalgebra is maximal toral by Cartan subalgebras are exactly maximal toral subalgebras. Hence $\mathfrak{g}$ has the finite root-space decomposition $\mathfrak{g}=\mathfrak{h}\oplus {⨁}_{\alpha \in \mathrm{\Phi }}{\mathfrak{g}}_{\alpha }$, every root space is one-dimensional, $B$ restricts nondegenerately to $\mathfrak{h}$, pairs opposite root spaces perfectly, and pairs no other two weight spaces, and the Killing-dual vector $H_{\alpha }$ satisfies $B(H_{\alpha },h)=\alpha (h)$ for $h\in \mathfrak{h}$ (Finite semisimple Cartan, root and string structure).

For any $B$-dual bases $x_1,\dots ,x_n$ and $x^1,\dots ,x^n$ of $\mathfrak{g}$ one has $C={\sum }_i{x}_i{x}^i$, independent of the choice of dual bases (The quadratic Casimir element).

If $x\in {\mathfrak{g}}_{\alpha }$ and $y\in {\mathfrak{g}}_{-\alpha }$, then $[x,y]=B(x,y)H_{\alpha }$, and $B(x,y)=0$ whenever $x\in {\mathfrak{g}}_{\alpha }$, $y\in {\mathfrak{g}}_{\beta }$ with $\alpha +\beta \mathrm{\ne }0$ (Opposite root spaces bracket to the Killing-dual line, Under Choice, the Killing form pairs only opposite root spaces).

$V_{\mu }=\{v\in V:H\cdot v=\mu (H)v\text{ for all }H\in \mathfrak{h}\}$ for every representation $V$ of $\mathfrak{g}$ (Weight and weight space); if $x\in {\mathfrak{g}}_{\alpha }$ then $x\cdot v\in V_{\mu +\alpha }$ for $v\in V_{\mu }$ (Root vectors shift weights); and the action of $\mathfrak{g}$ extends to a unital action of $U(\mathfrak{g})$ under which a product $xy$ acts by composition of the operators (Lie representations are U(g)-modules).

The Killing form induces a pairing $(,)$ on ${\mathfrak{h}}^{\ast }$ and $\mu (H_{\alpha })=(\mu ,\alpha )$ for every $\mu \in {\mathfrak{h}}^{\ast }$ and every root $\alpha$ (The Killing form of a semisimple Lie algebra, The quadratic Casimir eigenvalue on a highest-weight module is $(\lambda ,\lambda +2\rho )$).

For $\lambda \in {\mathrm{\Lambda }}^{+}$ the module $L(\lambda )$ is a cyclic highest-weight module of highest weight $\lambda$, and $C$ acts on it by the scalar $(\lambda ,\lambda +2\rho )$; it is finite-dimensional, so each $L(\lambda {)}_{\mu }$ is finite-dimensional (Highest-weight classification, The quadratic Casimir eigenvalue on a highest-weight module is $(\lambda ,\lambda +2\rho )$, The quadratic Casimir element is central).