\(*\)-product on the tangent bundle of the group \(U(n)\). (Produit \(*\) sur le fibré tangent du groupe \(U(n)\).) (French) Zbl 0930.22010

Séminaire de mathématique de Luxembourg. Luxembourg: Centre Universitaire de Luxembourg. Trav. Math. 10, 1-14 (1998).
For the complex connected reductive Lie group \(G\) the diffeomorphism \(K \times A^* \times N^+ \to G; (k,a,n) \mapsto kan,\) associated with the Iwasawa decomposition is well-known. Because of the existence of a nondegenerate bilinear Killing form on reductive groups one can identify the tangent space \(T_kK\) to \(K\) at a point \(k\in K\) with the sum \(i{\mathfrak a} + \mathfrak n^+\) and therefore identify \(G\) with \(TK\) (§3.1).
Equipping \(TK\) with the group structure of \(G\), the authors associate the *-product to \(TK\) from \(G\). In particular, for the case \(G=\mathrm{GL}_n(\mathbb C)\), \(K= U(n)\), the authors obtain a \(U(n)\)-equivariant *-product on \(TU(n) \cong \mathrm{GL}_n(\mathbb C)\).
For the entire collection see [Zbl 0910.00035].


22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)