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Invariant eigendistributions on a semisimple Lie algebra and homology classes on the conormal variety. I: An integral formula. (English) Zbl 0755.22004
According to a basic result of Harish-Chandra the invariant eigendistributions on a semisimple complex Lie algebra \(\mathfrak g_ 0\) are locally integrable functions, which for a regular infinitesimal character \(\lambda\) are given on a Cartan subalgebra \(\mathfrak h\) by the formula \[ \Theta(X)={1\over \pi(X)}\sum_{s\in W}m(\Theta,s)e^{\langle \lambda,s\cdot X\rangle}. \] Let \(\mathfrak g\) be the complexification of \({\mathfrak g}_ 0\), and let \(G\) be the adjoint group of \(\mathfrak g\). Let \(\mathcal B\) be the flag manifold of \(G\) and let \({\mathcal B}^*\) be the cotangent bundle realized as \({\mathcal B}^*=\{({\mathfrak b},\nu)\mid {\mathfrak b}\in {\mathcal B},\;\nu\in{\mathfrak b}^ \perp\}.\) The conormal variety \(\mathcal S\) of the \(G_ 0\)-action, where \(G_ 0\), is the adjoint group of \({\mathfrak g}_ 0\), is the subset of \({\mathcal B}^*\) with \(\nu\in i{\mathfrak g}_ 0\). For a fixed Borel subalgebra \(\mathfrak b\) in \(\mathcal B\) let \({\mathfrak h}_ 0\) be a Cartan subalgebra of \({\mathfrak g}_ 0\), such that \({\mathfrak h}\subset{\mathfrak b}\). For \(\lambda\in{\mathfrak h}^*\) regular there is a natural bijective map \(\pi_ \lambda: {\mathcal B}^*\to\Omega_ \Lambda=G\cdot\lambda\). Let \(\sigma_ \lambda\) be the canonical holomorphic 2-form on \(\Omega_ \lambda\). The author proves that each invariant eigendistribution with infinitesimal character \(\lambda\) may be expressed as a contour integral – in distribution sense – over a homology class \(\Gamma\in H_{2n}({\mathcal S})\): \[ \Theta_ \Gamma(X)={1\over (2\pi)^ nn!}\int_{\pi_ \lambda\Gamma}e^{\langle \xi,X\rangle}\sigma^ n_ \lambda(d\xi). \] For \(w \in W\) we define \({\mathfrak b}_ w = w^{-1}{\mathfrak b}\) and \(S_ w = G_ 0 \cdot {\mathfrak b}_ w\). Let \({\mathcal S}_ w\) be the conormal of \(S_ w\). Then \(\Gamma\) may be represented as a linear combination \(\Gamma=\sum_ wm_ w {\mathcal S}_ w\), so one need to know the distributions corresponding to \(\Gamma={\mathcal S}_ w\). The author shows, that the coefficient \(m(\Theta_ \Gamma,s)\) in the Harish-Chandra decomposition is up to sign the Euler number of \({\mathfrak b}_ s\) on the closure of \(\Gamma\).

22E30 Analysis on real and complex Lie groups
22E10 General properties and structure of complex Lie groups
57T10 Homology and cohomology of Lie groups
Full Text: DOI
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