## $$*$$-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 $$G=GL_n(\mathbb C)$$, $$K= U(n)$$, the authors obtain a $$U(n)$$-equivariant *-product on $$TU(n) \cong GL_n(\mathbb C)$$.
For the entire collection see [Zbl 0910.00035].

### MSC:

 2.2e+28 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)