el Beuzart-Plessis, Dan Ciubotaru, Eric Opdam and Jean-Loup Waldspurger for helpful correspondence. The project received funding from ERC under grant agreement No 669655; R.B. was partly supported by NSF grant DMS-1601953, the collaboration was partly supported by the US-Israel Binational Science Foundation.

Weightless functions and invariant distributions {#sec_K}
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Let ${{\mathcal H}}={{\mathcal H}}(G)$ be the Hecke algebra of compactly supported locally constant measures on $G$, the convolution product on ${{\mathcal H}}(G)$ will be denoted by $*$. For an open subsemigroup $S\subset G$ we let ${{\mathcal H}}(S)\subset {{\mathcal H}}(G)$ denote the subalgebra of measures supported on $S$. We denote by ${\mathcal D}$ the space of generalized functions on $G$ (that is the space of linear functionals on ${{\mathcal H}}(G)$) and by ${\mathcal D}^G\subset {\mathcal D}$ the subspace of invariant generalized functions. Let ${{\mathcal H}}_{cusp}$, ${\mathcal D}^G_{cusp}$ be the cuspidal part of ${{\mathcal H}}$, ${\mathcal D}^G$, i.e. ${{\mathcal H}}_{cusp}$ consists of functions acting by zero in any parabolically induced representation and ${\mathcal D}^G_{cusp}$ consists of distributions vanishing on the orthogonal complement of ${{\mathcal H}}_{cusp}$ (cf. footnote \[ftnt\] above). Until the end of the section we assume for simplicity of notation that $G$ has compact center. Then [*averaging*]{} with respect to conjugations yields 