Cycles of Bott-Samelson type for taut representations (Q1597184)

A representation \(\rho: G\to\text O(V)\) of a connected compact Lie group \(G\) is called taut if the Morse inequalities for the distance function to any orbit of \(G\) in \(V\) are in fact equalities. The well known examples of taut irreducible representations are the isotropy representations of irreducible symmetric spaces. It was shown by the authors (Preprint, 2000), that, besides these examples, there exist only the following ones: \(G = \text{SO}(2)\times\text{SO}(9), \rho = \rho_1\otimes_{\mathbb R}\rho_2; G = \text U(2)\times\text{Sp}(n), \rho = \rho_1\otimes_{\mathbb C}\rho_2; G = \text{SU}(2)\times\text{Sp}(n), \rho = \rho_1^3\otimes_{\mathbb H}\rho_2\), where \(\rho_1\) and \(\rho_2\) are the standard representations of the first and of the second factor, respectively. In the paper, a new proof of tautness of these exceptional representations is contained, giving explicit cycles which represent the \(\mathbb Z_2\)-homology of their orbits. It is also proved that these representations are not variationally complete.