A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups
This is the detailed description of results in ``The Weyl character formula, the half-spin representations, and equal rank subgroups'' [\textit{B. Gross, B. Kostant, P. Ramond}, and \textit{S. Sternberg}, Proc. Natl. Acad. Sci. USA 95, 8441-8442 (1998; Zbl 0918.17002)]. Let \(\mathfrak{p}\) be a finite-dimensional complex vector space, \(\mathcal{B}_{\mathfrak{p}}\) be a nonsingular symmetric bilinear form on \(\mathfrak{p}\), \(\mathfrak{r}\) be a complex finite-dimensional Lie algebra, and \(\mathcal{B}_{\mathfrak{r}}\) be a nonsingular \(\text{ad }\mathfrak{r}\)-invariant symmetric bilinear form on~\(\mathfrak{r}\). Let \(\nu: \mathfrak{r}\to \text{Lie }SO(\mathfrak{p})\) be a \(\mathcal{B}_{\mathfrak{p}}\)-invariant representation of \(\mathfrak{r}\) on \(\mathfrak{p}\). Let \(\mathfrak{g} = \mathfrak{r}+\mathfrak{p}\). At first, a representation \((\nu, \mathcal{B}_{\mathfrak{p}})\) of Lie type is defined and a criterion for a Clifford algebra to be of Lie type is given. Next, a cubic Dirac operator is defined and studied for a Lie type representation \((\nu, \mathcal{B}_{\mathfrak{p}})\). Hereafter \(\text{rank }(\mathfrak{r}) = \text{rank }(\mathfrak{g})\) is assumed. \(\dim(\mathfrak{p})\) is even, so that the Clifford algebra \(C(\mathfrak{p})\) is simple and the spin module \(S\) is irreducible. For the definition of ``multiplet'' and its roles, see p. 479. Infinitesimal character values are given on p. 488. Finally, let \(G\) be a connected and simply connected compact Lie group with \(\text{Lie }(G) = \mathfrak{g}_{\mathbb{R}}\) and \(R\) be the subgroup of \(G\) with \(\text{Lie }(R) = \mathfrak{r}_{\mathbb{R}}\). Note that \(\dim X = \dim G/R\) is even. An ''equal weight assumption'' for \(\mathfrak{g}\) and \(\mathfrak{r}\) is introduced and it is shown that it is satisfied if and only if \(X = G/R\) is a spin manifold (i.e., the second Stiefel-Whitney class of X vanishes).
scientific article; zbMATH DE number 1425263
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