Let $\mathfrak{g}\,=\,{{\oplus }_{i\in \mathbb{Z}}}{{\mathfrak{g}}_{i}}$ be an infinite-dimensional graded Lie algebra, with dim ${{\mathfrak{g}}_{i}}\,<\,\infty $, equipped with a non-degenerate symmetric bilinear form $B$ of degree 0. The quantum Weil algebra ${{\hat{\mathcal{W}}}_{\mathfrak{g}}}$ is a completion of the tensor product of the enveloping and Clifford algebras of $\mathfrak{g}$. Provided that the Kac–Peterson class of $\mathfrak{g}$ vanishes, one can construct a cubic Dirac operator $\mathcal{D}\,\in \,\hat{\mathcal{W}}(\mathfrak{g})$, whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac– Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl–Kac character formulas for suitable “equal rank” Lie subalgebras of Kac–Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras.

Nous montrons que l’invariant de Hasse-Witt de la forme de Killing d’une algèbre de Lie semi-simple $L$ s’exprime à l’aide de l’invariant de Tits de la représentation irréductible de $L$ de poids dominant $\rho \,=\,\frac{1}{2}$ (somme des racines positives), et des invariants associés au groupe des symétries du diagramme de Dynkin de $L$.