A remark on universal connections (Q1160454)

We determine the algebra of \(U(k+N)\)-invariant forms on the Stiefel manifold \(U(k+N)/U(N)\) as \(N\) tends to \(\infty\) (\(k\) is fixed). Actually, we show that the `universal' \(U(k)\)-connection in the Stiefel bundle, given by Narasimhan-Ramanan, induces an isomorphism from the Weil algebra of the Lie algebra of \(U(k)\) onto the algebra of \(U(k+N)\)-invariant forms on \(U(k+N)/U(N)\) as \(N\to\infty\). Exactly the same results are proved in the case of the other classical groups \(\mathrm{SO}(k)\) and \(\mathrm{Sp}(k)\). In the process, we determine the algebra of \(G_F(N)\)-invariants in the exterior algebra \(\vee_{\mathbb R}\left(F^N\oplus\ldots\oplus F^N\right)\) as \(N\to\infty\), where \(F\) is the field \(\mathbb R\), \(\mathbb C\) and \(Q\) (the skew field of quaternions) and \(G_F(N)\) is the corresponding rotation group, i.e. \(\mathrm{SO}(N)\), \(\mathrm{U}(N)\) or \(\mathrm{Sp}(N)\), respectively.