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$$H$$-invariant distribution vectors of induced representations for a $$p$$-adic reductive symmetric space $$G/H$$. (Vecteurs distributions $$H$$ invariants de représentations induites, pour un espace symétrique réductif $$p$$-adique $$G/H$$.) (French. English summary) Zbl 1151.22012
Let $$F$$ be a nonarchimedean local field of characteristic zero, and let $$G$$ be the group of $$F$$-points of a reductive algebraic linear group $$\mathbf G$$ defined over $$F$$. Let $$\sigma$$ be an involution of $$\mathbf G$$, which is defined over $$F$$, and let $$H$$ denote the group of $$F$$-points of an open subgroup, defined over $$F$$, of the group of fixed points by $$\sigma$$. The article starts the study of harmonic analysis on the reductive symmetric space $$G/H$$, in analogy with those on real symmetric spaces. The first step, achieved in the paper, is the construction of families of $$H$$-invariant linear forms on generalized principal series. The main tool is smooth homology. For a given smooth module $$V$$, let $$V_G=J_G(V)$$ be the space of $$G$$-coinvariants (that is the quotient of $$V$$ by the subspace spanned by the vectors $$gv-v$$, $$v\in V$$, $$g\in G$$). Then the functor $$J_G$$ is right exact, and $$H_*(G,V)$$ – the smooth homology of $$V$$ – is defined to be the homology of the complex $\cdots\to J_G(P_1)\to J_G(P_0)\to 0$ obtained by applying $$J_G$$ to a projective resolution $\cdots\to P_2\to P_1\to P_0\to V\to 0$ of $$V$$. $$H_*(G,V)$$ does not depend on the choice of the projective resolution of $$V$$. Moreover, $$H_0(G,V)$$ is canonically isomorphic to $$J_G(V)$$.

##### MSC:
 2.2e+36 Analysis on $$p$$-adic Lie groups
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