measures on $X$. Also, we will work with spaces $Y_P=(G/U_P\times G/U_P)/L$ for a parabolic subgroup $P=LU_P\subset G$. In this case ${\mathcal S}(Y_P)$ will denote the space of integral kernels of operators ${\mathcal S}(G/U_P)\to {\mathcal S}(G/U_P)$, i.e. locally constant compactly supported sections of the $G$-equivariant locally constant sheaf $pr_1^*(\mu)$, where $pr_1:Y_P\to G/P$ is the first projection.

In particular, we fix a Haar measure $dg$ on $G$ and identify the space ${\mathcal S}(G)={{\mathcal H}}(G)$ with the space of compactly supported locally constant functions on $G$. Then ${\Delta}$ is identified with the space of distributions, we also get the $L^2$ pairing $\langle\ ,\ \rangle$ on ${{\mathcal H}}(G)$.

Let ${\mathfrak g}$ be the Lie algebra of $G$. We will assume that ${\mathfrak g}$ has a finite number of nilpotent conjugacy classes and that there exists a $G$-equivariant $F$-analytic bijection $\phi$ between a neighbourhood of $0$ in ${\mathfrak g}$ and a neighbourhood of $e$ in $G$. We also assume that for a semisimple element $s\in G$ the Lie algebra ${\mathfrak z}$ of its centralizer $Z_G(s)$ admits a $Z_G(s)$-invariant complement. These assumptions are well known to hold if $char(F)=0$ or if $char(F)> N$ for some $N$ depending on the rank of $G$, see [@KV §1.8] for 