For a representation
$\unicode[STIX]{x1D70B}$
of a connected Lie group
$G$
on a topological vector space
$E$
we defined in [Reference Gimperlein, Krötz and SchlichtkrullGKS11] a vector subspace
$E^{\unicode[STIX]{x1D714}}$
of
$E$
of analytic vectors. Further, we equipped
$E^{\unicode[STIX]{x1D714}}$
with an inductive limit topology. We called a representation
$(\unicode[STIX]{x1D70B},E)$
analytic if
$E=E^{\unicode[STIX]{x1D714}}$
as topological vector spaces.
Some mistakes in the paper have been pointed out by Glöckner (see [Reference GlöcknerGlö13]). For a representation
$(\unicode[STIX]{x1D70B},E)$
and a closed
$G$
-invariant subspace
$F$
of
$E$
we asserted in Lemma 3.6(i) that
$F^{\unicode[STIX]{x1D714}}=E^{\unicode[STIX]{x1D714}}\cap F$
as a topological space. Based on that, we further asserted in Lemma 3.6(ii) that the inclusion
$E^{\unicode[STIX]{x1D714}}/F^{\unicode[STIX]{x1D714}}\rightarrow (E/F)^{\unicode[STIX]{x1D714}}$
is continuous and in Lemma 3.11 that if
$(\unicode[STIX]{x1D70B},E)$
is analytic then so is the restriction to
$F$
. However, there is a gap in the proof of the first assertion, and presently it is not clear to us whether the above statements are then true in this generality (for unitary representations
$(\unicode[STIX]{x1D70B},E)$
they are straightforward). Our proof does give the following weaker version of the two lemmas.
Lemma 1. Let
$(\unicode[STIX]{x1D70B},E)$
be a representation and let
$F\subset E$
be a closed invariant subspace. Then:
-
(i)
$F^{\unicode[STIX]{x1D714}}=E^{\unicode[STIX]{x1D714}}\cap F$
as vector spaces and with continuous inclusion
$F^{\unicode[STIX]{x1D714}}\rightarrow E^{\unicode[STIX]{x1D714}}$
; -
(ii)
$E^{\unicode[STIX]{x1D714}}/E^{\unicode[STIX]{x1D714}}\cap F\subset (E/F)^{\unicode[STIX]{x1D714}}$
continuously; -
(iii) if
$(\unicode[STIX]{x1D70B},E)$
is an analytic representation, then
$\unicode[STIX]{x1D70B}$
induces an analytic representation on
$E/F$
.
Indeed, for (iii) note that if
$E$
is analytic,
$E/F=E^{\unicode[STIX]{x1D714}}/E^{\unicode[STIX]{x1D714}}\cap F\subset (E/F)^{\unicode[STIX]{x1D714}}$
continuously by (ii), and
$(E/F)^{\unicode[STIX]{x1D714}}\subset E/F$
continuously.
Further, we asserted in Proposition 3.7 a general completeness property of the functor which associates
$E^{\unicode[STIX]{x1D714}}$
to
$E$
. However, there is a gap in the proof, which asserts that
$v_{i}\rightarrow v$
in the topology of
$E^{\unicode[STIX]{x1D714}}$
. As statements in this generality are not needed for the main result, we can leave out the proposition (together with Remark 3.8).
Attached to
$G$
we introduced a certain analytic convolution algebra
${\mathcal{A}}(G)$
. A central theme of the paper is the relation of analytic representations of
$G$
to algebra representations of
${\mathcal{A}}(G)$
on
$E$
:
${\mathcal{A}}(G)\times E\rightarrow E$
. In Proposition 4.2(ii), we claimed that the bilinear map
${\mathcal{A}}(G)\times {\mathcal{A}}(G)\rightarrow {\mathcal{A}}(G)$
is continuous. However, the proof shows only separate continuity. For a similar reason, we need to weaken Proposition 4.6 to the following.
Proposition 2. Let
$(\unicode[STIX]{x1D70B},E)$
be an
$F$
-representation. The assignment
$$\begin{eqnarray}(f,v)\mapsto \unicode[STIX]{x1D6F1}(f)v:=\int _{G}f(g)\unicode[STIX]{x1D70B}(g)v\,dg\end{eqnarray}$$
defines a continuous bilinear map
$$\begin{eqnarray}{\mathcal{A}}_{n}(G)\times E\rightarrow E_{n}\end{eqnarray}$$
for every
$n\in \mathbb{N}$
, and a separately continuous map
$$\begin{eqnarray}{\mathcal{A}}(G)\times E\rightarrow E^{\unicode[STIX]{x1D714}}\end{eqnarray}$$
(with convergence of the defining integral in
$E^{\unicode[STIX]{x1D714}}$
). Moreover, if
$(\unicode[STIX]{x1D70B},E)$
is a Banach representation, then the latter bilinear map is continuous.
Proof. The first statement is proved in the article, and thus only the statement for
$\unicode[STIX]{x1D70B}$
a Banach representation remains to be proved. We repeat the first part of the proof, now with
$p$
denoting the fixed norm of
$E$
. The constants
$c,C$
such that
$$\begin{eqnarray}p(\unicode[STIX]{x1D70B}(g)v)\leqslant Ce^{cd(g)}p(v)\quad (g\in G,v\in E)\end{eqnarray}$$
and
$N,C_{1}$
such that
$$\begin{eqnarray}C_{1}:=\int _{G}e^{(c-N)d(g)}\,dg<\infty\end{eqnarray}$$
are then all fixed, and so is
$\unicode[STIX]{x1D716}=1/(CC_{1})$
.
Let
$n\in \mathbb{N}$
and an open
$0$
-neighborhood
$W_{n}\subset E_{n}$
be given. We may assume that
$$\begin{eqnarray}W_{n}=\{v\in E_{n}\mid p(\unicode[STIX]{x1D70B}(K_{n})v)<\unicode[STIX]{x1D716}_{n}\}\end{eqnarray}$$
with
$K_{n}\subset GV_{n}$
compact and
$\unicode[STIX]{x1D716}_{n}>0$
. Let
$$\begin{eqnarray}O_{n}:=\Bigl\{f\in {\mathcal{O}}(V_{n}G)|\sup _{z\in K_{n},g\in G}|f(z^{-1}g)|e^{Nd(g)}<\unicode[STIX]{x1D716}\unicode[STIX]{x1D716}_{n}\Bigr\}\subset {\mathcal{A}}_{n}(G).\end{eqnarray}$$
The computation in the given proof shows that if
$f\in O_{n}$
and
$p(v)<1$
, then
$\unicode[STIX]{x1D6F1}(f)v\in W_{n}$
. The asserted bi-continuity of
${\mathcal{A}}(G)\times E\rightarrow E^{\unicode[STIX]{x1D714}}$
follows.◻
As a consequence, we obtain as in Example 4.10(a), but only for Banach representations
$(\unicode[STIX]{x1D70B},E)$
, that
$E^{\unicode[STIX]{x1D714}}$
is
${\mathcal{A}}(G)$
-tempered. In particular,
${\mathcal{A}}(G)$
need not itself be
${\mathcal{A}}(G)$
-tempered, and we need to replace Lemma 5.1(i) by the following weaker version.
Lemma 3.
$V^{\text{min}}$
is an analytic globalization of
$V$
and it carries an algebra action
$$\begin{eqnarray}(f,v)\mapsto \unicode[STIX]{x1D6F1}(f)v,\quad {\mathcal{A}}(G)\times V^{\text{min}}\rightarrow V^{\text{min}}\end{eqnarray}$$
of
${\mathcal{A}}(G)$
, which is separately continuous.
The main result of the paper, Theorem 5.7, has two statements concerning a Harish-Chandra module
$V$
with a globalization
$E$
:
-
(1) if
$E$
is analytic
${\mathcal{A}}(G)$
-tempered, then
$E=V^{\text{min}}$
; -
(2) if
$E$
is an
$F$
-globalization, then
$E^{\unicode[STIX]{x1D714}}=V^{\text{min}}$
.
The proof, which relied on Lemma 3.11 and Proposition 4.6, respectively, needs to be corrected. The proof of (1) if
$V$
is irreducible needs no modification. For the general case it can be adjusted as follows.
Like in the paper, it suffices to consider an exact sequence of Harish-Chandra modules
$0\rightarrow V_{1}\rightarrow V\rightarrow V_{2}\rightarrow 0$
, where both
$V_{1}$
and
$V_{2}$
have unique analytic
${\mathcal{A}}(G)$
-tempered globalizations. We show that the same holds for
$V$
.
Let
$E_{1}$
be the closure of
$V_{1}$
in
$E$
and
$E_{2}=E/E_{1}$
. By Lemma 1(iii),
$E_{2}$
is an analytic
${\mathcal{A}}(G)$
-tempered globalization of
$V_{2}$
, so that by assumption
$E_{2}=V_{2}^{\text{min}}={\mathcal{A}}(G)V_{2}$
as topological vector spaces.
In a first step we prove that
$E_{1}=V_{1}^{\text{min}}={\mathcal{A}}(G)V_{1}$
as vector spaces. For that, we note first that
$E_{1}$
is
${\mathcal{A}}(G)$
-tempered and that
$V_{1}^{\text{min}}\subset E_{1}$
continuously. Next, by Proposition 5.3 (which holds for any
${\mathcal{A}}(G)$
-tempered representation), we may embed
$E_{1}\subset F_{1}$
continuously into a Banach globalization of
$F_{1}$
of
$V_{1}$
. Moreover, the proof shows that the embedding is compatible with the action by
${\mathcal{A}}(G)$
. It follows that
$E_{1}^{\unicode[STIX]{x1D714}}\subset F_{1}^{\unicode[STIX]{x1D714}}$
continuously and as
${\mathcal{A}}(G)$
-modules. Further, note that since
$E$
is analytic, from Lemma 1(i), we also obtain
$E_{1}^{\unicode[STIX]{x1D714}}=E^{\unicode[STIX]{x1D714}}\cap E_{1}=E_{1}$
as vector spaces. Hence,
$V_{1}^{\text{min}}\subset E_{1}\subset F_{1}^{\unicode[STIX]{x1D714}}$
. By assumption,
$V_{1}$
has a unique
${\mathcal{A}}(G)$
-tempered globalization and hence
$F_{1}^{\unicode[STIX]{x1D714}}\simeq V_{1}^{\text{min}}$
. Therefore,
$V_{1}^{\text{min}}\subset E_{1}\subset F_{1}^{\unicode[STIX]{x1D714}}\simeq V_{1}^{\text{min}}$
. As these maps respect the structure as
${\mathcal{A}}(G)$
-modules, the inclusion is also surjective:
$V_{1}^{\text{min}}=E_{1}$
.
Being an inductive limit,
$E_{1}=F_{1}^{\unicode[STIX]{x1D714}}$
is an ultrabornological space, and
$V_{1}^{\text{min}}$
is webbed (see the reference in the proof of Proposition 4.6). We conclude from the open mapping theorem that
$V_{1}^{\text{min}}=E_{1}$
also as topological vector spaces.
With Lemma 5.2, we now have a diagram of topological vector spaces
where the vertical arrow in the middle signifies the continuous inclusion
$V^{\text{min}}={\mathcal{A}}(G)V\subset E$
, and where the rows are exact. The five lemma implies
$V^{\text{min}}=E$
as a vector space, and as in the article we conclude from [DS79] that this is then a topological identity.
Finally, for (2) we recall from Corollary 3.5 that
$(E^{\infty })^{\unicode[STIX]{x1D714}}=E^{\unicode[STIX]{x1D714}}$
. The Casselman–Wallach smooth globalization theorem asserts the existence of a Banach globalization
$F$
of
$V$
such that
$F^{\infty }=E^{\infty }$
and therefore
$F^{\unicode[STIX]{x1D714}}=E^{\unicode[STIX]{x1D714}}$
. In particular,
$E^{\unicode[STIX]{x1D714}}$
is
${\mathcal{A}}(G)$
-tempered by Proposition 2. Now (1) applies.
Acknowledgement
The authors wish to thank Helge Glöckner for pointing out the discussed mistakes.
