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Projection d’orbites, formule de Kirillov et formule de Blattner. (Projection of orbits, Kirillov formula, and Blattner formula). (French) Zbl 0575.22014
This paper concerns the Kirillov theory of semisimple Lie groups. If G is a semisimple connected Lie group, let \({\mathfrak g}\) be its Lie algebra and \({\mathfrak g}={\mathfrak k}\oplus {\mathfrak p}^ a \)Cartan decomposition. One hypothesis at this point is that \({\mathfrak k}\) and \({\mathfrak g}\) have the same rank, i.e. that there is a compactly embedded Cartan algebra. Let \(\Omega\) denote a regular, elliptic orbit on the dual \({\mathfrak g}^*\) with its symplectic manifold structure and its Liouville measure \(\beta =\beta_{\Omega}\). The direct decomposition \({\mathfrak g}^*={\mathfrak k}^*\oplus {\mathfrak p}^*\) gives a projection \(J: {\mathfrak g}^*\to {\mathfrak k}^*\). The coadjoint action of the analytic group K generated by \({\mathfrak k}\) on \(\Omega\) is Hamiltonian, and J is the associated momentum map.
The focus of the first part of the paper is a calculation of the direct image measure \(J_*(\beta)\). The first principal result in this work is a formula for \(\int_{\Omega}\phi (Jf) d\beta (f)\) for rapidly decreasing smooth functions \(\phi\). In fact, if \({\mathfrak t}\) is a fixed Cartan algebra of \({\mathfrak k}\) then this integral is \(\int_{{\mathfrak t}^*}A^+(\phi)(\xi) dB_{\lambda}(\xi)\), where \(A^+\) and \(B_{\lambda}\) need to be explained. Indeed \(A^+(\phi)(\xi)=\pm (W:1)^{-1}\int_{K_{\xi}}\phi (f) d\beta_{K_{\xi}}(f)\) where W is the Weyl group attached to the situation, and where the sign \(\pm\) is the sign of an element of the Weyl group suitably picked for \(\xi\). Further, \(B_{\lambda}\) is a measure on \({\mathfrak t}^*\) obtained as a sum over the Weyl group of measures concocted from certain Heaviside measures, translated by a fixed element \(\lambda\) from \(\Omega\) \(\cap {\mathfrak t}^*\) which happens to be an orbit of the Weyl group.
In the second chapter one considers the generalized function \(\theta\) on G associated to the orbit \(\Omega\) which is the character of a suitable irreducible representation. If one abbreviates \(j_{{\mathfrak g}}(X)=\det ((ad X)^{-1}\) \((e^{(ad X)/2}-e^{-(ad X)/2}))\), then one has the formula of Kirillov valid for all \(X\in {\mathfrak g}\) in a sufficiently small neighborhood of 0 in \({\mathfrak g}\), which reads \(j_{{\mathfrak g}}(X)^{1/2} j_{{\mathfrak g}}(X)={\hat \beta}(X)\). This formula is generalized in such a way, that for a fixed element \(s\in K\) one defines a generalized function \(\theta_ s\) near 0 in \({\mathfrak z}=\ker (1-ad s)\), and \(\theta_ s(X)\) is calculated for X near zero in \({\mathfrak z}\) in terms of \(\beta \Omega_ s\) in the spirit of a Fourier transform.
The restriction of \(\theta\) to K is computed in the third chapter; the authors give a new proof of a formula conjectured by Blattner and proved by Hecht and Schmid, and independently by Enright with a different proof. Even the formulation of this formula resists the compact format of a review. For 60 pages of substantial mathematics, the reader anyhow has to resort to the source. The authors aspire successfully to a clear presentation of a powerful machinery.
Reviewer: K.H.Hofmann

MSC:
22E46 Semisimple Lie groups and their representations
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