Erratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

1 Erratum to: Math. Z. DOI 10.1007/s00209-016-1629-6

The following changes to the main results of [1] are necessary:

• (1) In Theorem A and Corollary B the following assumption is required: the number of orbits of the complexification $H_{\mathbb {C}}$ on $G_{\mathbb {C}}{/}P_{\mathbb {C}}$ is finite, where P is a minimal parabolic subgroup of G.

• (2) In Theorem C the following additional assumption is required: the number of orbits of $H_{\mathbb {C}}$ on $X_{\mathbb {C}}$ is finite.

Presently we do not know whether these results hold without the additional assumptions.

The source of the mistake is in [2], where the expressions “real algebraic groups” and “real algebraic manifolds” are ambiguous. Moreover, a mistake in [2, Definition 1.1.1] hints to a wrong resolution of this ambiguity, in particular in [2, Theorem D]. This result entered in the proof of Lemma 3.2.1 of [1].

In [2] the terms “real algebraic groups” and “real algebraic manifolds” sometimes mean algebraic groups and manifolds defined over $\mathbb {R}$, and sometimes real points of such. Those two meanings are not equivalent; in particular, the statement that an algebraic group G defined over $\mathbb {R}$ acts on an algebraic manifold X with finitely many orbits implies the statement that $G(\mathbb {R})$ acts on $X(\mathbb {R})$ with finitely many orbits, but is not equivalent to it. Rather, it is equivalent to the stronger statement that $G(\mathbb {C})$ has finitely many orbits on $X(\mathbb {C})$. In particular, in [2, Theorem D] one needs the stronger assumption (only then it follows from the Bernstein–Kashiwara theorem, [2, Thm 3.2.2]), see [3]. We do not know whether this theorem holds under the weaker assumption.

The argument in [1] proves the corrected versions of the main results (see (1, 2) above), after the following revision.

• (a) In Sects. 2 and 3, the expression “real algebraic group” has to be replaced by “algebraic group defined over $\mathbb {R}$” and the expression “real algebraic manifold” has to be replaced by “algebraic manifold defined over $\mathbb {R}$”.

• (b) One has to introduce the following notation: for an algebraic manifold X defined over $\mathbb {R}$, a Zariski closed algebraic submanifold Z and an algebraic bundle ${\mathcal {E}}$ over X denote $\mathcal {S}_{Z}(X,{\mathcal {E}}):=\mathcal {S}_{Z(\mathbb {R})}(X(\mathbb {R}),{\mathcal {E}})$ and $\mathcal {S}_{Z}^*(X,{\mathcal {E}}):=\mathcal {S}_{Z(\mathbb {R})}^*(X(\mathbb {R}),{\mathcal {E}})$, and similarly for the special cases $\mathcal {S}(X), \mathcal {S}(X,{\mathcal {E}}), \mathcal {S}_Z(X),$ and their dual spaces.

• (c) In the proof of Lemma 3.2.1, one has to add that the reason that Proposition 3.1.1 implies the finiteness of the dimension of

  $$\begin{aligned} {\text {H}}_0(\mathfrak {h}, \mathcal {S}_Z(X,{\mathcal {E}})/\mathcal {S}_Z(X,{\mathcal {E}})^i\otimes \chi ) \end{aligned}$$

  is that $Z(\mathbb {R})$ is a finite union of $H(\mathbb {R})$-orbits.

• (d) In Sect. 4, G should be the group of real points of an algebraic reductive group $\mathbf{G}$ defined over $\mathbb {R}$, and H should be the group of real points of an algebraic subgroup $\mathbf{H}\subset \mathbf{G}$. Also, each time that we require H to be a real spherical subgroup we actually need to require the stronger condition that $\mathbf{H}$ has finitely many orbits on $\mathbf{G}/\mathbf{P}$, where $\mathbf{P}$ is a minimal parabolic subgroup of $\mathbf{G}$ defined over $\mathbb {R}$.