The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Groupes et algébres de lie

Harmonic analysis on semi‐simple lie groups—i

Lie Groups and Lie Algebras

Théorèmes de finitude pour la cohomologie des espaces complexes

Dopo aver definito il complesso di Gauchy-Riemann tangenziale su sottovarieta reali generiche, di codimensione qualsiasi, di una varieta eomplessa, per mezzo di una opportuna successione di Mayer-Vietoris se ne studiano i gruppi di coomologia locali. Si ottengono inoltre risultati relativi al problema dell' estensione di «distribuzioni di Cauchy-Riemann» e alla coomologia globale di sottovarieta compatte e di domini compatti con frontiera regolare a tratti.

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Tangential Cauchy-Riemann complexes on distributions

1 Preliminary notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 2 Essential pseudoconcavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3 Examples of pseudoconcave and essentially pseudoconcave CR manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 4 A regularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 5 A weak unique continuation theorem . . . . . . . . . . . . . . . . . . . . . . . . 273 6 The maximum modulus principle . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7 Complex CR vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8 Applications to the compact case . . . . . . . . . . . . . . . . . . . . . . . . . . 279

A weak pseudoconcavity condition for abstract almost CR manifolds

Poincaré lemma for tangential Cauchy Riemann complexes

Let $G$ be a real form of a complex semisimple Lie group $\widehat G$Gˆ, $P\subset\widehat G$P⊂Gˆ a parabolic subgroup and $V=\widehat G/P$V=Gˆ/P the corresponding flag manifold of $\widehat G$Gˆ. As was shown by J. A. Wolf in [Bull. Amer. Math. Soc. 75 (1969), 1121--1237; MR0251246 (40 #4477)], there exists exactly one $G$G-orbit $M$M on $V$V that is closed (and hence compact) in $V$V. This orbit is connected, of minimal dimension amongst the $G$G-orbits and it is naturally endowed with a $G$G-invariant CR structure, namely the one induced by the $\widehat G$Gˆ-invariant complex structure of $V$V. Let us call a homogeneous CR manifold which occurs as a closed orbit of this kind a minimal parabolic orbit. Observe that any minimal parabolic orbit $M$M is (up to equivalences) uniquely determined just by a pair $(G,P)$(G,P), where $G$G is a real form of a complex semisimple Lie group $\widehat G$Gˆ and $P$P is a parabolic subgroup of $\widehat G$Gˆ.
 The authors single out a set of properties satisfied by the pairs of Lie algebras $(\germ g={\rm Lie}(G),\germ p={\rm Lie}(P))$(g=Lie(G),p=Lie(P)) associated with minimal parabolic orbits and prove that, conversely, any pair satisfying such properties (called an effective minimal parabolic CR algebra) is associated with a minimal parabolic orbit in some flag manifold.
 After this, they show that for a fixed real semisimple Lie algebra $\germ g$g, the effective minimal parabolic CR algebras of the form $(\germ g,\germ p)$(g,p) are in one-to-one correspondence with a special class of cross-markings of the Satake diagram of $\germ g$g. Then, they study some topological properties of the homogeneous CR manifolds associated with effective minimal parabolic CR algebras and give complete characterizations of those, whose associated minimal parabolic orbits satisfy some special conditions on their CR structure. Such characterizations determine immediately new families of examples of CR manifolds that are either CR separable or of finite type or that satisfy nondegeneracy conditions of various kinds.

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The cr structure of minimal orbits in complex flag manifolds

After showing that the partial complex structure is defined by an inner derivation, we give a complete classification of semisimple Levi-Tanaka algebra.

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Mauro Nacinovich

Papers

Tangential Cauchy-Riemann complexes on distributions

A weak pseudoconcavity condition for abstract almost CR manifolds

Poincaré lemma for tangential Cauchy Riemann complexes

The cr structure of minimal orbits in complex flag manifolds

Classification of Semisimple Levi-Tanaka Algebras (*).