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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 $$\Gamma$$.

##### MSC:
 22E30 Analysis on real and complex Lie groups 22E10 General properties and structure of complex Lie groups 57T10 Homology and cohomology of Lie groups
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