Picard-Lefschetz theory and characters of a semisimple Lie group
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 \textit{D. Barbasch} and \textit{D. Vogan} [J. Funct. Anal. 37, 27-55 (1980; Zbl 0436.22011)].
scientific article; zbMATH DE number 871785