more precise information.

\[distributions\]

-   We denote by ${\mathcal G}_{G ,e}$ the space of germs of $Ad$-invariant distributions near $0$ on ${\mathfrak g}$ which are restrictions of linear combinations of the Fourier transforms of invariant measures on a nilpotent orbits. Using the bijection $\phi$ we consider ${\mathcal G}_{G,e}$ as a space of germs of $Ad$-invariant distributions on $G$ near the identity.

-   For a semisimple element $s\in G$ we denote by $ {\mathcal G}_{Z_G(s),s}$ the space of germs of distributions on $Z_G(s)$ at $s$ obtained from the space ${\mathcal G}_{Z_G(s),e}$ by the shift by $s$.

-   Let $s\in G$ be a semisimple element, $ X_s=G/Z_G(s), r:G\to  X_s$ be the natural projection and ${\gamma}:X_s\to G$ be a continuous section. We denote by $dz$ a $G$-invariant measure on $X_s$.

-   We denote by ${\tilde}{\kappa}:Z_G(s)\times X_s \to G$ the map given by $(z,x)\to {\gamma}(x)sz({\gamma}(x))^{-1}$.

Let ${\mathfrak z}\subset {\mathfrak g}$ be the Lie algebra of $Z_G(s)$. By assumption there exists a complementary ${\mathfrak z}$-invariant subspace $W\subset {\mathfrak g}$ and the map ${\kappa}_0:{\mathfrak z}\oplus W\to {\mathfrak g},(z,w)\to z+Ad(s)w$ is a bijection. Therefore there exists an open neighborhood $R\subset Z_G(s)$ of $e$ such that the restriction ${\kappa}$ of ${\tilde}{\kappa}$ on $R\times X_s$ is an open embedding.

