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Coefficients généralisés de séries principales sphériques et distributions sphériques sur \(G_ \mathbb{C} /G_ \mathbb{R}\). (Generalized coefficients of spherical principal series and spherical distributions over \(G_ \mathbb{C} /G_ \mathbb{R}\)). (French) Zbl 0741.43010

Let \(G\) be a semi-simple complex Lie group, \({\mathfrak g}\) its Lie algebra, and \(H\subset G\) a real form of \(G\). One assumes that \(H\) admits a compact Cartan subgroup \(T\). The spherical principal series \((\pi_ \lambda,I_ \lambda)\) of \(G\) (\(\lambda\in{\mathfrak a}_ \mathbb{C}', {\mathfrak a}=i\hbox{Lie}(T)\)), admits an \(H\)-invariant distribution vector \(\xi_ \lambda\), which is meromorphic in \(\lambda\). The spherical distribution \(\Theta_ \lambda\) is defined by \(\Theta_ \lambda(\varphi)=\langle \pi_ \lambda(\varphi)\xi_ \lambda,\xi_{-\lambda}\rangle\), \(\varphi\in C^ \infty_ c(G)\). Then \(\Theta_ \lambda\) is an \(H\)-invariant distribution on \(G/H\), which is meromorphic in \(\lambda\), and eigendistribution of the \(G\)-invariant differential operators on \(G/H\). It was proved by P. Harinck that \(\Theta_ \lambda\) is a locally integrable function on \(G/H\), analytic in the open set \(HA^{\text{reg}}H\), where \(A^{\text{reg}}\) is the set of regular elements in \(A=\exp{\mathfrak a}\). The main result of the paper is an explicit formula for \(\Theta_ \lambda\).
One chooses a positive system \(\Delta^ +\) in the root system \(\Delta({\mathfrak g,a})\). Let \(W\) be the Weyl group of the pair \(({\mathfrak g,a})\), and \(W_ H\) the subgroup of the \(w\)’s in \(W\) with a representative in \(H\). Then \[ \Theta_ \lambda(\alpha H)=(C\sum_{w\in W_ H}(-1)^{\ell(w)_ a-w\lambda}) / (\prod_{\alpha\in\Delta+}\langle\lambda, \alpha\rangle\prod_{\alpha\in\Delta+} (a^{-\alpha}-a^ \alpha)) \] and, for \(w\in W\backslash W_ H\), \(\Theta_ \lambda(aw H)=0\). By a result of P. Harinck one knows that \[ \Theta_ \lambda(aH)=\sum_{s\in W}d(w,\lambda)a^{-w\lambda}/\prod_{\alpha\in\Delta+}(a^{-\alpha}- a^ \alpha). \] The problem is to compute the numbers \(d(w,\lambda)\). The idea of the proof is to study the asymptotic behaviour of \(\Theta_ \lambda\) (\(wa \exp(tX)H)\) as \(t\to\infty\), for \(X\) regular in \({\mathfrak a}\). This has to be done in the distribution sense. A crucial point is to get estimates in order to apply the Lebesgue dominated convergence theorem. The study of the asymptotic behaviour involves the action of the intertwining operators on the distribution vectors \(\xi_ \lambda\). This is done by a reduction to the rank one cases \(SL(2,\mathbb{C})/SU(2)\) and \(SL(2,\mathbb{C})/SL(2,\mathbb{R})\).
Reviewer: J.Faraut (Paris)

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E46 Semisimple Lie groups and their representations
53C35 Differential geometry of symmetric spaces
46F10 Operations with distributions and generalized functions
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References:

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