Abstract
Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle \({\mathcal {E}}\) on X. Let \(\mathfrak {h}\) be the Lie algebra of H. Let \(\mathcal {S}(X,{\mathcal {E}})\) be the space of Schwartz sections of \({\mathcal {E}}\). We prove that \(\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})\) is a closed subspace of \(\mathcal {S}(X,{\mathcal {E}})\) of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let \(\pi \) be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants \(V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}\) is an isomorphism if and only if any linear \(\mathfrak {h}\)-invariant functional on V is continuous in the topology induced from \(\pi \). The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.
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Acknowledgments
We would like to thank Eitan Sayag and Joseph Bernstein for fruitful discussions, and the anonymous referee for useful remarks.
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A. Aizenbud was partially supported by ISF grant 687/13 and a Minerva foundation grant. D. Gourevitch was partially supported by ISF grant 756/12 and ERC Starting Grant StG 637912. B. Krötz was supported by ERC Advanced Investigators Grant HARG 268105.
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Aizenbud, A., Gourevitch, D., Krötz, B. et al. Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture. Math. Z. 283, 979–992 (2016). https://doi.org/10.1007/s00209-016-1629-6
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DOI: https://doi.org/10.1007/s00209-016-1629-6