Seminar of August 16th

Let N be a connected, simply connected nilpotent Lie group, which is provided with a left invariant riemannian metric.  The isometry group of N is the semi-direct product $ K\ltimes N $, where K is the group of orthogonal automorphisms of N. We consider all the  Gelfand pairs $(K \ltimes N , K)$ determined by Jorge Lauret in [«Gelfand pairs attached to representations of compact Lie groups» Transformation Groups, 5(4):307-324 (2000)]. In these cases, the real-valued functions over N that are integrable and K-invariant form a commutative algebra with convolution as a product. For each irreducible representation W of K, we will determine whether the  \(End(W)\)-valued functions over N that are integrable and bi-W-equivariant form a commutative algebra.