Generalized symmetric pairs, Lie superalgebras, Dirac operators and Kostant constants

In this paper, we present a geometric construction of contragredient Lie superalgebras, with the exceptions of $A(m,n)$ $(m,n>0)$ and $G(3)$. Let $\mathfrak{g}$ be a complex simple Lie algebra with an order 4 automorphism $\sigma$, the eigenspace decomposition being $\mathfrak{g}=\mathfrak{g}(1)+\mathfrak{g}(i)+\mathfrak{g}(-1)+\mathfrak{g}(-1)$. When $\mathfrak{g}(i)$ admits a $\mathfrak{g}(1)$-invariant symplectic structure, $(\mathfrak{g},\sigma )$ is called a \emph{generalized symmetric pair with symplectic subspace}. In this paper, we classify all such pairs and construct a natural Lie superalgebra structure on $L=L_{\bar{0}}+L_{\bar{1}}$ with $L_{\bar{0}}=\mathfrak{g}(1)$ and $L_{\bar{1}}=\mathfrak{g}(i)$. Furthermore, we investigate the constants arising from the squares of two distinct Dirac operators: $D$ for $\mathfrak{g}={\mathfrak{g}}^{{\sigma }^2}+{\mathfrak{g}}^{-{\sigma }^2}$, and $D_s$ for $L=\mathfrak{g}(1)+\mathfrak{g}(i)$. We prove that the constant $C$ in $D^2$ admits a remarkable factorization $$C=J^{\ast }({\mathrm{\Omega }}_{L_{\bar{0}}})\cdot \dim L_{\bar{1}}\cdot h_{\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}^{\vee },$$ where $h_{\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}^{\vee }$ is the signed dual Coxeter number of $L$ and $J^{\ast }({\mathrm{\Omega }}_{L_{\bar{0}}})$ is a mapped Casimir constant of its even part. This formula reveals an unexpected unification of symmetric space geometry and Lie superalgebra invariants. Finally, we prove that the Kostant constant ${\mathrm{K}}_{\mathrm{s}}={{\mathrm{\Omega }}_{L_{\bar{0}}}\mid }_{L_{\bar{1}}}=\frac{1}{2}$ is universal for all contragredient Lie superalgebras with non-degenerate Killing form, which implies that the constant $C_s$ in $D_s^2$ satisfies $C_s=\frac{1}{16}\dim L_{\bar{0}}$. We then establish a bijection between such superalgebras and solutions of the superadmissibility index equations $E_s^{I\text{\hspace{0.17em}⁣}I}=\frac{1}{2},E_s^I=\frac{1}{2}$, recovering the classification from an invariant-theoretic perspective.