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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:
22E35 Analysis on \(p\)-adic Lie groups
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