zbMATH — the first resource for mathematics

Picard-Lefschetz theory and characters of a semisimple Lie group. (English) Zbl 0851.22013
Let $$\mathfrak g$$ be a complex semisimple Lie algebra, $$\mathfrak h$$ a Cartan subalgebra and $$W$$ the Weyl group. Let $$q : {\mathfrak g}^* \mapsto {\mathfrak h}^*/W$$ be the quotient map onto the coadjoint orbits. For a regular $$\lambda \in {\mathfrak h}^*$$ let $$p_\lambda : \Omega \to \Omega_\lambda$$ be a homeomorphism from a standard fibre of $$q$$ onto the fibre over $$\lambda$$ and $$p_0 : \Omega \mapsto {\mathcal N}$$ its limit map into the nilpotent variety $$\mathcal N$$. Given a real subalgebra $${\mathfrak g}_0$$ and its Lie group $$G_0$$ denote by $$\mathcal S$$ the inverse image of $$\mathcal N \cap {\mathfrak g}^*_0$$ under $$p_0$$. Let $$s = s_\alpha$$ in $$W$$ be a simple reflection and let $$\lambda \in {\mathfrak h}^*$$ be orthonormal to $$\alpha$$ and no other simple roots. The author proves that the monodromy representation of $$s$$ in the top homology $$H_{2n}({\mathcal S})$$ is a reflection along the subspace $$H_{2n} ({\mathcal S}_0)$$, where $${\mathcal S}_0$$ is the inverse image under $$p_0$$ of the orbit $$G_0 \cdot \lambda$$.
The author also gives a graded filtration of $$H_{2n} ({\mathcal S})$$ in terms of the homologies of inverse images under $$p_0$$ of nilpotent orbits of $$G_0$$.
In the second part the author studies the character of representations of the Lie group $$G_0$$ with semisimple Lie algebra $${\mathfrak g}_0$$. Let $$\pi$$ be an admissible representation of $$G_0$$. It has been proved by the author [In: A. Connes et al. (eds), Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honor of J. Dixmier, Paris 1989, Prog. Math. 92, 263-287 (1990; Zbl 0744.22012)] that the character of $$\pi$$ on $${\mathfrak g}_0$$ is an integration $$\Theta(\Gamma, \lambda)$$ of the exponential function over $$p_\lambda \Gamma$$ for some $$\lambda \in {\mathfrak h}^*$$ and a cycle $$\Gamma$$ in $$H_{2n}({\mathcal S})$$. The author proves that the wave front set of the representation $$\pi$$ is equal to the support of $$\Gamma$$ under the Springer resolution $$p_0 : {\mathcal S} \mapsto {\mathcal N} \cap {\mathfrak g}_0$$, to the generic wave front set at 0 of $$\Theta$$ on $${\mathfrak g}_0$$, and to the asymptotic support at $$\infty$$ of the Fourier transform on $$\Theta$$. This also proves a conjecture of D. Barbasch and D. Vogan [J. Funct. Anal. 37, 27-55 (1980; Zbl 0436.22011)].