# Proof, via smooth homology, of the existence of rational families of H-invariant linear forms on G-induced representations, when G/H is a symmetric, reductive, p-adic space, via smooth homology

# Proof, via smooth homology, of the existence of rational families of H-invariant linear forms on G-induced representations, when G/H is a symmetric, reductive, p-adic space, via smooth homology

We fix $F$ a local non archmedean field of characteristic zero, $G$ the points over $F$ of an algebraic reductive group defined over $F$ and $s$ a rational involution of $G$ defined over $F$. We note $H$ the group of fixed points of $G$ under the action of $s$ and $X(G,s)$ the connected component on the neutral element of the set of complex characters of $G$ antiinvariant under the action of $s$. Let $P$ be a $s$-parabolic subgroup of $G$, in other words the intersection $M$ of $P$ with $s(P)$ is an $s$-stable Levi subgroup, we construct from a irreducible, smooth representation $r$ of $M$, a rational family of distributions above the algebraic variety $X(G,s)$, which are $H$-invariant linear forms on tne smooth induced representation $ind(P,G; r )$. Our main trick is the use of homology of groups.