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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:
 2.2e+47 Semisimple Lie groups and their representations
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##### References:
 [1] D. BARBASCH , D.A. VOGAN : Sketch of proof of Rossmann’s character formula (Non publié) . [2] N. BERLINE et M. VERGNE : Fourier transform of orbits of the coadjoint representation. Representation theory of reductive groups , Birkhauser, Boston, 1983 . Zbl 0527.22010 · Zbl 0527.22010 [3] N. BERLINE et M. VERGNE : The equivariant index and Kirillov’s character formula , à paraître dans American Journal of Mathematics. Zbl 0604.58046 · Zbl 0604.58046 [4] N. BERLINE et M. VERGNE : Classes caractéristiques équivariantes, Formule de localisation en cohomologie équivariante , Comptes Rendus Acad. Sci. Paris, 295 ( 1982 ), pp. 539-541. MR 83m:58002 | Zbl 0521.57020 · Zbl 0521.57020 [5] R. BOTT : Vector fields and characteristic numbers , Mich. Math. Journal, 14 ( 1967 ), pp. 231-244. Article | MR 35 #2297 | Zbl 0145.43801 · Zbl 0145.43801 [6] A. CEREZO et F. ROUVIERE : Solution élémentaire d’un opérateur invariant à gauche sur un groupe de Lie compact . Ann. Sci. Ec. Norm. Sup. 2 ( 1969 ), pp. 561-581. Numdam | MR 42 #6869 | Zbl 0191.43801 · Zbl 0191.43801 [7] M. DUFLO : Construction de représentations unitaires d’un groupe de Lie . In ”Harmonic analysis and group representations.” C.I.M.E. 1980 , ed. Liguori, Naples 1982 . · Zbl 0522.22011 [8] J.J. DUISTERMAAT : Fourier integral operators . Courant Institute, New-York ( 1973 ). MR 56 #9600 | Zbl 0272.47028 · Zbl 0272.47028 [9] J.J. DUISTERMAAT et G.J. HECKMAN : On the variation in the cohomology of the symplectic form of the reduced phase space . Inventiones Math. 69 ( 1982 ), pp. 259-268. MR 84h:58051a | Zbl 0503.58015 · Zbl 0503.58015 [10] T.J. ENRIGHT : On the fundamental series of a real semi-simple Lie algebra : their irreducibility, resolutions and multiplicity formulae . Annals of Math. 110 ( 1979 ), pp. 1-82. MR 81a:17003 | Zbl 0417.17005 · Zbl 0417.17005 [11] V. GUILLEMIN et S. STERNBERG : Geometric asymptotics . Maths Surveys no 14, A.M.S. Rhode Island ( 1977 ). MR 58 #24404 | Zbl 0364.53011 · Zbl 0364.53011 [12] V. GUILLEMIN et S. STERNBERG : Convexity properties of the moment mapping . Inventiones Math. 67 ( 1982 ), pp. 491-513. MR 83m:58037 | Zbl 0503.58017 · Zbl 0503.58017 [13] HARISH-CHANDRA : The characters of semi-simple Lie groups . Trans. Amer. Math. Soc. 83 ( 1956 ), pp. 98-163. MR 18,318c | Zbl 0072.01801 · Zbl 0072.01801 [14] HARISH-CHANDRA : Invariant eigen-distributions on a semi-simple Lie group . Trans. Amer. Math. Soc. 119 ( 1965 ), pp. 457-508. MR 31 #4862d | Zbl 0199.46402 · Zbl 0199.46402 [15] HARISH-CHANDRA : Discrete series for semi-simple Lie groups I . Construction of invariant eigendistributions. Acta Mathematica, 113 ( 1965 ), pp. 242-318. MR 36 #2744 | Zbl 0152.13402 · Zbl 0152.13402 [16] HARISH-CHANDRA : Discrete series for semi-simple Lie groups II Acta Mathematica , 116 ( 1966 ), pp. 1-111. MR 36 #2745 | Zbl 0199.20102 · Zbl 0199.20102 [17] H. HECHT et W. SCHMID : A proof of Blattner’s conjecture . Inventiones math., 31 ( 1975 ), pp. 129-154. MR 53 #715 | Zbl 0319.22012 · Zbl 0319.22012 [18] H. HECHT et W. SCHMID : Characters, asymptotic and n-homology of Harish-Chandra modules . Acta Mathematica, 151 ( 1983 ), pp. 49-151. MR 84k:22026 | Zbl 0523.22013 · Zbl 0523.22013 [19] G.J. HECKMAN : Projections of orbits and asymptotic behaviour of multiplicities for compact connected Lie groups . Inventiones math. 67 ( 1982 ), pp. 333-356. MR 84d:22019 | Zbl 0497.22006 · Zbl 0497.22006 [20] M.S. KHALGUI : Caractères des groupes de Lie . Journal of functional analysis, 47 ( 1982 ), pp. 64-77. MR 84f:22020 | Zbl 0507.22009 · Zbl 0507.22009 [21] A.A. KIRILLOV : Eléments de la théorie des représentations . Ed. M.I.R Moscou, ( 1974 ). MR 52 #14134 [22] G. LION et M. VERGNE : The Weil representation , Maslov index and theta series. Birkhäuser, Boston ( 1980 ). Zbl 0444.22005 · Zbl 0444.22005 [23] S. PANEITZ : Communication personnelle ( 1983 ). [24] W. ROSSMANN : Kirillov’s character formula for reductive Lie groups . Inventiones math., 48 ( 1978 ), pp. 207-220. MR 81g:22012 | Zbl 0372.22011 · Zbl 0372.22011 [25] W. SCHMID : On the characters of discrete series (the Hermitian symmetric case) . Inventiones math., 30 ( 1975 ), pp. 47-144. MR 53 #714 | Zbl 0324.22007 · Zbl 0324.22007 [26] P. TORASSO : Sur le caractère de la représentation de Shale-Weil de Mp(n, \Bbb R) et Sp (n, \Bbb C) . Math. Ann., 252 ( 1980 ), pp. 53-86. MR 82a:22019 | Zbl 0452.22015 · Zbl 0452.22015 [27] M. VERGNE : On Rossmann’s character formula for discrete series . Inventiones Math., 54 ( 1979 ), pp. 11-14. MR 81d:22017 | Zbl 0428.22010 · Zbl 0428.22010
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