Atlas: Representations of compact Lie groups and the osculating spaces of their orbits by Claudio Gorodski

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International Congress on Differential Geometry in memory of Alfred Gray (1939-1998)
September 18-23, 2000
Universidad del País Vasco
Bilbao, Spain

Organizers
M. Fernández (chairman), R. Ibáñez, M. Macho-Stadler

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Representations of compact Lie groups and the osculating spaces of their orbits
by
Claudio Gorodski
University of São Paulo, Brazil
Coauthors: Gudlaugur Thorbergsson (University of Cologne)

Oral Communication

Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating space of all of their nontrivial orbits coincide with the representation space. We say that representations with this property are of class O². Our approach is to find restrictions on the class O² and then apply them to classify variationally complete and taut representations. The known classifications of transitive actions of compact Lie groups on spheres (Borel-Montgomery-Samelson), of cohomogeneity two orthogonal representations (Hsiang-Lawson), and more generally of polar representations (Dadok) also follow easily.

Variationally complete actions were introduced by Bott. Bott and Samelson proved that isotropy representations of symmetric spaces are variationally complete. We prove the following converse of their theorem.

Theorem 1 A variationally complete representation of a compact connected Lie group is orbit equivalent to the isotropy representation of a symmetric space.

We in fact prove more than we have stated here and give a complete classification of irreducible variationally complete representations. As a corollary we get a new proof of the classification of polar representations due to Dadok since they are variationally complete (Conlon) and isotropy representations of symmetric spaces are easily seen to be polar. It follows that an orthogonal representation is variationally complete if and only if it is polar, and it is polar if and only if it is orbit equivalent to the isotropy representation of a symmetric space.

Another important result of Bott and Samelson is that the distance functions to orbits of variationally complete representations are perfect Morse functions. We say that orbits with this property are taut and call representations taut if all of their orbits are taut. We prove the following classification theorem about taut representations.

Theorem 2 A taut irreducible representation \rho of a compact connected Lie group G is either orbit equivalent to the isotropy representation of a symmetric space or it is one of the following representations:

	G		\rho
	SO(2)×Spin(9)		(standard)\otimes(spin)
	U(2)×Sp(n)		(standard)\otimes(standard)
	SU(2)×Sp(n)		(standard)³\otimes(standard)

As a consequence of this theorem, the classes of polar, variationally complete and taut irreducible representations of compact simple groups coincide. Moreover, in the case of a general compact Lie group, the taut, non-polar irreducible representations are exactly the irreducible representations of cohomogeneity three (Straume). A further related result is the classification of compact homogeneous taut submanifolds of Euclidean space with flat normal bundle due to Olmos. He shows that such a submanifold is a principal orbit of the isotropy representation of a symmetric space. It follows that the principal orbits of the three taut non-polar representations in the above theorem do not have flat normal bundle which is also easy to see directly.

Bott and Samelson proved the tautness of variationally complete representations as an application of their K-cycles which are concrete representatives of a basis for the homology of the orbits. (In their paper the group acting is denoted by K which explain the name K-cycle). We find that the three taut non-polar representations in our theorem admit generalizations of the K-cycles of Bott and Samelson; moreover we can also prove their tautness with shorter, although maybe less illuminating arguments based on a theorem of Floyd, following an idea of Duistermaat.

Kuiper observed that the second osculating space of a taut submanifold in a Euclidean space V coincides with V if the submanifold is not contained in a proper affine subspace. In fact he proved this more generally for tight submanifolds, but this is unimportant for us since an orbit is tight if and only if it is taut. Since the classes of representations we are dealing with are all taut, it follows from this observation of Kuiper that they belong to class O² if they are irreducible. The class O² is much more tractable than the other classes of representations we are dealing with since it involves an infinitesimal condition. The technique of Dadok, notably his invariant k(\lambda), turns out to be an extremely helpful tool to restrict the class O² so much that they remaining cases are accessible to more geometric methods that we develop.

We have pointed out that Dadok's classification of polar representations follows as a consequence of our first theorem. Several other classification results can also be easily proved with our methods and these include the classification of transitive actions on spheres due to Borel and Montgomery-Samelson as well as the classification of cohomogeneity two representations due to Hsiang-Lawson. The cohomogeneity two representations are polar and therefore included in Dadok's classification, but the point here is that it is very easy to see directly that they belong to class O² without referring to tautness.

One could also use our methods to classify symmetric spaces and their isotropy representations. This would not lead to a simpler proof than those existing in the literature. Insisting on the details needed to classify symmetric spaces and their isotropy representations would make the work much longer and involved than it is. We therefore use this classification in our work.

Previous to this work taut representations were studied by Galemann (who decided with some exceptions which representations of SU(n) and U(n) can be taut) and Console-Thorbergsson (who proved among other things that a compact group admitting an effective taut representation can have at most four simple factors). Eschenburg and Heintze use the theory of isoparametric submanifolds to reprove Dadok's classification of polar representations under the assumption of cohomogeneity at least three. They also use the same theory to decide which irreducible representations are orbit equivalent to isotropy representations of symmetric spaces. This also follows immediately from our work.

Date received: May 12, 2000