Corrigendum: Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules

Abstract

We correct the proof of the main result of the paper, Theorem 5.7. Our corrected proof relies on weaker versions of a number of intermediate results from the paper. The original, more general, versions of these statements are not known to be true.

For a representation $\unicode[STIX]{x1D70B}$ of a connected Lie group $G$ on a topological vector space $E$ we defined in [GKS11] 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 [Glö13]). 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:

1. (i) $F^{\unicode[STIX]{x1D714}}=E^{\unicode[STIX]{x1D714}}\cap F$ as vector spaces and with continuous inclusion $F^{\unicode[STIX]{x1D714}}\rightarrow E^{\unicode[STIX]{x1D714}}$ ;

2. (ii) $E^{\unicode[STIX]{x1D714}}/E^{\unicode[STIX]{x1D714}}\cap F\subset (E/F)^{\unicode[STIX]{x1D714}}$ continuously;

3. (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. (1) if $E$ is analytic ${\mathcal{A}}(G)$ -tempered, then $E=V^{\text{min}}$ ;

2. (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.