2.1 Real reductive groups

Throughout the paper, we shall use the term real reductive group in the sense that it is used in Vogan’s monograph [Reference VoganVog81, Chapter 0, § 1]. This means, among other things, that a real reductive group G is required to be linear, but it need not be connected, although there are restrictions in that regard. Examples include the group of real points of any connected reductive algebraic group defined over $\mathbb{R}$ , including, for instance, $SL(n,\mathbb{R})$ , $Sp(2n,\mathbb{R})$ , etc.

We shall use the following structure, as discussed in [Reference VoganVog81, Chapter 0, § 1]. Every real reductive group G is equipped with a Cartan involution

\begin{align*}\theta \colon G \longrightarrow G\end{align*}

whose fixed subgroup is precisely K. We shall write the Cartan decomposition of the Lie algebra $\mathfrak{g}$ into $\pm 1$ eigenspaces of $\theta$ as $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{s}$ (we shall not follow the custom of indicating real Lie algebras by a subscript 0; instead, we shall indicate complexifications by a subscript $\mathbb{C}$ ). We can and will equip $\mathfrak{g}$ with an invariant nondegenerate symmetric bilinear form

(2.1) \begin{align} B\colon \mathfrak{g}\times \mathfrak{g} \longrightarrow \mathbb{R}\end{align}

that is positive-definite on $\mathfrak{s}$ and negative-definite on $\mathfrak{k}$ , and that is invariant under $\theta$ . When we speak of inner products on subspaces of $\mathfrak{k}$ or of $\mathfrak{s}$ , we shall always mean the positive-definite inner products on these subspaces obtained from B.

We shall fix throughout a maximal abelian subspace $\mathfrak{a}_{\min}$ of $\mathfrak{s}$ . We shall fix a system of positive restricted roots

(2.2) \begin{align} \Delta^+(\mathfrak{g},\mathfrak{a}_{\min}) \subseteq \Delta(\mathfrak{g},\mathfrak{a}_{\min}) ,\end{align}

along with the associated Iwasawa decomposition $G = \textit{KA}_{\min}N_{\min}$ and minimal parabolic subgroup

(2.3) \begin{align} P_{\min} = M_{\min}A_{\min}N_{\min}\end{align}

(see e.g. [Reference KnappKna02, ChapterVI]). Here, $M_{\min}$ is the centralizer of $\mathfrak{a}_{\min}$ in K.

We shall always denote by $S\subseteq \Delta^{+}(\mathfrak{g},\mathfrak{a}_{\min})$ the set of simple restricted roots. The standard parabolic subgroups of G are the parabolic subgroups that include the chosen minimal parabolic subgroup $P_{\min}$ , and they are in bijection with subsets $I\subseteq S$ , as follows:

(2.4) \begin{align} P_I=M_IA_IN_I, \quad \mathfrak{a}_I = \bigcap \{ {\operatorname{ker}}(\alpha) : \alpha \in I \}.\end{align}

Here, the Levi subgroup $M_IA_I$ is the centralizer of $\mathfrak{a}_I$ in G, and $N_I$ is determined by the requirement that $P_I$ be standard. See, for instance, [Reference KnappKna02, VII.7].

Finally, if $P=\textit{MAN}$ is a standard parabolic subgroup, then we shall always regard the dual vector space $\mathfrak{a}^*$ as a vector subspace of $\mathfrak{a}_{\min}^*$ by means of the map

(2.5) \begin{align} \mathfrak{a}^* \longrightarrow \mathfrak{a}_{\min}^*\end{align}

that takes a linear functional on $\mathfrak{a}$ to its unique extension to a linear functional on $\mathfrak{a}_{\min}$ that vanishes on the orthogonal complement of $\mathfrak{a}$ in $\mathfrak{a}_{\min}$ . The orthogonal complement may also be described as the intersection of $\mathfrak{a}_{\min}$ with $\mathfrak{m}_I$ . When $\mathfrak{a}=\mathfrak{a}_I$ , the image of (2.5) is the space

(2.6) \begin{align} \mathfrak{a}_I^* \cong \{ \gamma \in \mathfrak{a}_{\min}^* : \langle \alpha, \gamma\rangle = 0\,\, \forall \alpha\in I \} .\end{align}

2.2 Imaginary part of the infinitesimal character

Definition 2.1. Denote by $\mathcal{Z}(\mathfrak{g_\mathbb{C}})$ the center of the enveloping algebra of $\mathfrak{g_\mathbb{C}}$ , the complexification of the real Lie algebra $\mathfrak{g}$ . If $\mathfrak{h_\mathbb{C}}$ is any Cartan subalgebra of $\mathfrak{g_\mathbb{C}}$ , then we shall denote by

(2.7) \begin{align} \xi_{\mathfrak{h}}\colon \mathcal{Z}(\mathfrak{g_\mathbb{C}}) \stackrel\cong \longrightarrow \operatorname{Sym}(\mathfrak{h_\mathbb{C}})^{W(\mathfrak{g_\mathbb{C}},\mathfrak{h_\mathbb{C}})}\end{align}

the Harish-Chandra isomorphism (see e.g. [Reference KnappKna02, § V.5]).

On the right-hand side of the Harish-Chandra isomorphism is the Weyl group-invariant part of the symmetric algebra of $\mathfrak{h_\mathbb{C}}$ , or equivalently, the algebra of Weyl group-invariant polynomial functions on $\mathfrak{h_\mathbb{C}}^*$ . Every character of $\operatorname{Sym}(\mathfrak{h_\mathbb{C}})^{W(\mathfrak{g_\mathbb{C}},\mathfrak{h_\mathbb{C}})}$ , or in other words every algebra homomorphism

\begin{align*}\operatorname{Sym}(\mathfrak{h_\mathbb{C}})^{W(\mathfrak{g}_\mathbb{C},\mathfrak{h}_\mathbb{C})} \longrightarrow \mathbb{C} ,\end{align*}

is given by evaluation at some weight $\lambda \in \mathfrak{h}_\mathbb{C}^*$ . Two weights correspond to the same character if and only if they lie in the same $W(\mathfrak{g}_\mathbb{C},\mathfrak{h}_\mathbb{C})$ -orbit.

It follows that every character

\begin{align*}\chi \colon \mathcal{Z}(\mathfrak{g}_\mathbb{C})\longrightarrow \mathbb{C}\end{align*}

may be written as a composition

(2.8) \begin{align} \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \stackrel{\xi_{\mathfrak{h}}}\longrightarrow \operatorname{Sym}(\mathfrak{h}_{\mathbb{C}})^{W(\mathfrak{g}_\mathbb{C},\mathfrak{h}_\mathbb{C})} \stackrel{\mathrm{eval}_{\lambda}\vphantom{(_{(}}}\longrightarrow \mathbb{C}\end{align}

of the Harish-Chandra isomorphism $\xi_{\mathfrak{h}}$ and evaluation at some $\lambda\in \mathfrak{h}_\mathbb{C}^*$ , which is unique up to the action of $W(\mathfrak{g}_\mathbb{C},\mathfrak{h}_\mathbb{C})$ . We shall write such a composition as

\begin{align*}\chi = \xi_{\mathfrak{h},\lambda} \colon \mathcal{Z}(\mathfrak{g}_\mathbb{C})\longrightarrow \mathbb{C} .\end{align*}

Following Vogan [Reference VoganVog00, § 2], we wish to identify some of these characters as real and some as imaginary.

Definition 2.2. Let $\mathfrak{h} = \mathfrak{t} \oplus \mathfrak{a}$ be a $\theta$ -stable Cartan subalgebra of $\mathfrak{g}$ , decomposed into its compact and noncompact parts (i.e. into the $+1$ and $-1$ eigenspaces of $\theta$ , respectively). The real part of its complexification, $\mathfrak{h}_\mathbb{C}$ , is the real vector subspace

\begin{align*}\mathrm{Re}(\mathfrak{h}_\mathbb{C}) = i\mathfrak{t}\oplus \mathfrak{a} \subseteq \mathfrak{h}_\mathbb{C} .\end{align*}

If the group G is realized as a group of real matrices in such a way that the Cartan involution is $\theta(X) = -X^{\mathrm{transp.}}$ , then these are the matrices in $\mathfrak{h}_\mathbb{C}$ with real eigenvalues. A weight $\lambda\in \mathfrak{h}_\mathbb{C}^*$ is real if it is real-valued on $\mathrm{Re}(\mathfrak{h}_\mathbb{C})$ . Similarly, it is imaginary if it is imaginary-valued on $\mathrm{Re}(\mathfrak{h}_\mathbb{C})$ .

Definition 2.4. If $\lambda \in \mathfrak{h}_\mathbb{C}^*$ , then we shall denote by

\begin{align*}\lambda = \mathrm{Re}(\lambda) + \mathrm{Im}(\lambda)\end{align*}

the (unique) decomposition of a weight $\lambda$ as a sum of real and imaginary weights.

From Lemma 2.3, we have the following.

Lemma 2.5 [Reference VoganVog00, Corollary 2.7]. Let $\mathfrak{h}$ be a $\theta$ -stable Cartan subalgebra of $\mathfrak{g}$ . Let $\chi\colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C}$ be any character. If $\chi$ is written in the form

\begin{align*}\chi = \xi_{\mathfrak{h},\lambda}\colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \longrightarrow \mathbb{C}\end{align*}

for some $\lambda \in \mathfrak{h}_\mathbb{C}^*$ , then the characters

\begin{align*}\xi_{\mathfrak{h},\mathrm{Re}(\lambda)}\colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C}\quad \text{and} \quad \xi_{\mathfrak{h},\mathrm{Im}(\lambda)}\colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C}\end{align*}

depend only on $\chi$ , and not on the choices of $\mathfrak{h}$ and $\lambda$ .

Definition 2.6. Given a character $\chi\colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C}$ as in Lemma 2.5, we shall write

\begin{align*}\mathrm{Re}(\chi) =\xi_{\mathfrak{h},\mathrm{Re}(\lambda)}\colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C}\quad \text{and} \quad \mathrm{Im}(\chi) =\xi_{\mathfrak{h},\mathrm{Im}(\lambda)}\colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C} .\end{align*}

These are the real and imaginary parts of $\chi$ .

Now, let $\pi$ be an irreducible unitary representation of G. Recall that, by Schur’s Lemma, $\pi$ has an associated infinitesimal character

\begin{align*}\operatorname{Inf\,Ch} (\pi) : \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C}.\end{align*}

From the above, we can form the real and imaginary parts of the infinitesimal character of $\pi$ :

\begin{align*}\operatorname{Re\,Inf\,Ch} (\pi)\colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C} \quad \text{and} \quad\operatorname{Im\,Inf\,Ch} (\pi) \colon \mathcal{Z}(\mathfrak{g}_\mathbb{C}) \to \mathbb{C}.\end{align*}

We shall need some information about the real and imaginary parts of the infinitesimal character of an irreducible, tempered unitary representation, starting with the following result.

This follows from Harish-Chandra’s classification of the discrete series via Harish-Chandra parameters (which determine the infinitesimal characters of the discrete series). See Remark A.1 in the appendix.

We shall use Theorem 2.7 and the following well-known principle of Harish-Chandra and Langlands [Reference LanglandsLan89] to say something about the infinitesimal character of a general irreducible tempered unitary representation.

It follows from Theorem 2.8 that every irreducible tempered unitary representation has the same infinitesimal character as some cuspidal principal series representation of the form $\operatorname{Ind}_{P}^G \sigma \otimes \exp(i \nu)$ . But the infinitesimal character of the latter may be readily computed as follows:

(2.9) \begin{align} \bigg \{\!\begin{aligned}\operatorname{Im\,Inf\,Ch} ( \operatorname{Ind}_P^G \sigma\otimes \exp(i \nu) ) & = i \nu \in i \mathfrak{a}^* ,\\\operatorname{Re\,Inf\,Ch} ( \operatorname{Ind}_P^G \sigma\otimes \exp(i \nu) ) & = \operatorname{Inf\,Ch}(\sigma) \in i\mathfrak{t}^* .\end{aligned}\end{align}

For instance, see [Reference KnappKna86, Proposition 8.22]. The infinitesimal character in (2.9) is computed using a $\theta$ -stable Cartan subalgebra of the form $\mathfrak{h} = \mathfrak{t}\oplus \mathfrak{a}$ that is constructed from the parabolic subgroup $P=\textit{MAN}$ in Theorem 2.8 by defining $\mathfrak{t}$ to be the Lie algebra of a maximal torus in $K \cap M$ (which, by another well-known result of Harish-Chandra, is a Cartan subalgebra of $\mathfrak{m}$ ) and, of course, defining $\mathfrak{a}$ to be the Lie algebra of A. The infinitesimal character of the discrete series representation $\sigma$ lies in $i\mathfrak{t}^*$ thanks to Theorem 2.7.

We want to use a single $\theta$ -stable Cartan subalgebra of $\mathfrak{g}$ to describe the infinitesimal characters of all the irreducible tempered unitary representations of G. To this end, we shall use the minimal parabolic subgroup that we fixed in (2.3). Form the ‘minimally compact’ Cartan subalgebra

(2.10) \begin{align} \mathfrak{h}_{\min} = \mathfrak{t}_{\min} \oplus \mathfrak{a}_{\min} ,\end{align}

where $\mathfrak{t}_{\min}$ is the Lie algebra of a maximal torus in $M_{\min}$ . It follows from computations with Cayley transforms (see e.g. [Reference KnappKna02, § VI.7]) that there is an inner automorphism of the complex Lie algebra $\mathfrak{g}_\mathbb{C}$ that conjugates the Cartan subalgebra $\mathfrak{h}_\mathbb{C}$ used in (2.9) to $\mathfrak{h}_{\min,\mathbb{C}}$ , and that conjugates $\mathfrak{a}$ into $\mathfrak{a}_{\min}$ , and $\mathfrak{t}$ to $\mathfrak{t}_{\min} \oplus i (\mathfrak{m}\cap \mathfrak{a}_{\min})$ . This inner automorphism is not unique, but the possible images of $\nu$ lie within a single orbit of the restricted Weyl group.

We arrive at the following result.

Theorem 2.10 [Reference VoganVog00, Corollary 3.5(1)]. The imaginary part of the infinitesimal character of any irreducible tempered unitary representation $\pi$ may be written in the following form, using the $\theta$ -stable Cartan subalgebra $\mathfrak{h}_{\min}=\mathfrak{t}_{\min}\oplus \mathfrak{a}_{\min}$ :

\begin{align*}\operatorname{Im\,Inf\,Ch}(\pi) = [(0,\nu)]\in ( \mathfrak{t}_{\min}^* \oplus i\mathfrak{a}_{\min}^* ) / W(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\min,\mathbb{C}}).\end{align*}

The $W(\mathfrak{g}, \mathfrak{a}_{\min})$ -orbit of $\nu$ in $i \mathfrak{a}_{\min}^*$ is uniquely determined by $\operatorname{Im\,Inf\,Ch}(\pi)$ .

From Lemma 2.9, the inclusion of $i \mathfrak{a}_{\min}^*$ into $ \mathrm{Im}(\mathfrak{h}_{\min,\mathbb{C}})^*$ determines a continuous map

\begin{align*}i \mathfrak{a}_{\min}^*/W(\mathfrak{g}, \mathfrak{a}_{\min})\longrightarrow\mathrm{Im}(\mathfrak{h}_{\min,\mathbb{C}})^* / W(\mathfrak{g}_\mathbb{C},\mathfrak{h}_{\min,\mathbb{C}})\end{align*}

that is a homeomorphism onto its image. From Theorem 2.10, there is a diagram of continuous maps.

(2.11)

We have labeled the bottom map, like the top map, as the imaginary part of the infinitesimal character, and we shall use the bottom version of the imaginary part of the infinitesimal character from now on. According to (2.9), this means, for example, that if $P=\textit{MAN}$ is a standard parabolic subgroup of G, as above, and if $\pi$ is a direct summand of the parabolically induced representation $\operatorname{Ind}_P^G \sigma\otimes \exp(i \nu) $ , as in Theorem 2.8, then

\begin{align*}\operatorname{Im\,Inf\,Ch}(\pi) = [i \nu]\in i \mathfrak{a}_{\min}^*/W(\mathfrak{g}, \mathfrak{a}_{\min}),\end{align*}

where $\mathfrak{a}^*$ is embedded into $\mathfrak{a}_{\min}^*$ using (2.5).

2.3 Stratification of the space of imaginary values of the infinitesimal character

Recall that we fixed a system (2.2) of positive restricted roots and that we are denoting by $S \subseteq \Delta^{+} (\mathfrak{g} ,\mathfrak{a}_{\min})$ the subset of simple roots.

Definition 2.11. We shall write

\begin{align*}\mathfrak{a}_{\mathrm{dom}}^* = \{ \nu\in \mathfrak{a}_{\min}^* : \langle \nu, \alpha \rangle \geqslant 0\,\, \forall \alpha \in S\} .\end{align*}

This is the dominant Weyl chamber in $\mathfrak{a}_{\min}^*$ that is associated to the given choice of system of positive roots.

The natural map

\begin{align*}\mathfrak{a}_{\mathrm{dom}}^*\stackrel \cong \longrightarrow \mathfrak{a}_{\min}^* / W(\mathfrak{g},\mathfrak{a}_{\min})\end{align*}

induced from the inclusion of $\mathfrak{a}_{\mathrm{dom}}^*$ into $\mathfrak{a}_{\min}^*$ is a homeomorphism, and from now on we shall use this homeomorphism to identify imaginary parts of infinitesimal characters with points in $\mathfrak{a}_{\mathrm{dom}}^*$ .

Definition 2.12. For every $I \subseteq S$ , define

\begin{align*}\mathfrak{a}^*_{I,+} = \{ \nu\in \mathfrak{a}_{\mathrm{dom}}^* : \langle \nu,\alpha\rangle = 0 \,\, \forall \alpha \in I\,\, \text{and} \,\,\langle \nu,\beta\rangle \ne 0 \,\, \forall \beta \notin I\} .\end{align*}

Note that $\mathfrak{a}_{I,+}^* \subseteq \mathfrak{a}_I^*$ as in (2.6). Explicitly, we have

(2.12) \begin{align} \mathfrak{a}_{I,+}^* = \mathfrak{a}^*_{\mathrm{dom}}\cap \mathfrak{a}_I^* \setminus \bigcup_{I\subsetneqq J} \mathfrak{a}_J^*.\end{align}

The sets $\mathfrak{a}_{I,+}^*$ are locally closed subsets of $\mathfrak{a}_{\mathrm{dom}}^*$ , and the full chamber $\mathfrak{a}_{\mathrm{dom}}^*$ is the disjoint union of all of them:

(2.13) \begin{align} \mathfrak{a}_{\mathrm{dom}}^* = \bigsqcup _{I \subseteq S} \mathfrak{a}^*_{I,+}.\end{align}

At one extreme, when $I=\emptyset$ , $\mathfrak{a}_{I,+}^*$ is the interior of the dominant chamber; at the other extreme, when $I=S$ , $\mathfrak{a}_{I,+}^*$ is the intersection of $\mathfrak{a}_{\min}$ with the center of $\mathfrak{g} $ (and so, for instance, it is a point if the center of G is compact).

2.4 Classification of representations via the imaginary part of the infinitesimal character

In this section, we shall describe Vogan’s characterization of the fibers of the map $\operatorname{Im\,Inf\,Ch}$ in (2.11) over the various components in (2.13).

1. (i) An irreducible unitary representation $\pi$ of a real reductive group is said to have real infinitesimal character if

   \begin{align*}\operatorname{Im\,Inf\,Ch} (\pi) =0\in i \mathfrak{a}_{\mathrm{dom}}^* .\end{align*}

2. (ii) An irreducible, tempered unitary representation of G is said to be tempiric (terminology of Afgoustidis) if it has real infinitesimal character. We shall denote by $\widehat G_{\mathrm{tempiric}}$ the set of unitary equivalence classes of tempiric representations of G.

Theorem 2.14 [Reference VoganVog00, Corollary 3.5(2)]. Let I be a subset of the set of simple roots $S\subseteq\Delta^+(\mathfrak{g},\mathfrak{a}_{\min})$ , let $\nu\in \mathfrak{a}_{I,+}^*$ , and denote by $P_I$ the associated standard parabolic subgroup of G, as in (2.3). The correspondence

\begin{align*}\sigma \longmapsto \operatorname{Ind}_{P_I}^G \sigma \otimes \exp(i \nu)\end{align*}

determines a bijection from the set of unitary equivalence classes of tempiric representations $\sigma$ of $M_I$ to the set of unitary equivalence classes of irreducible tempered unitary representations $\pi$ of G for which $\operatorname{Im\,Inf\,Ch}(\pi) = i\nu$ .

Theorem 2.14 determines a bijection

(2.14) \begin{align} \widehat G_{\mathrm{tempered}} \cong \bigsqcup _{I\subseteq S} \widehat M_{I,\mathrm{tempiric}} \times i \mathfrak{a}^*_{I,+}.\end{align}

This will be central to what follows.

3.1 The reduced group $C^*$ -algebra and the infinitesimal character

Let G be a real reductive group and denote by $C^*_r (G)$ the reduced group $C^*$ -algebra of G. In this section, we shall use the theory of the imaginary part of the infinitesimal character that we reviewed in the last section to fiber the spectrum $\widehat{G}_{\mathrm{tempered}}$ of $C_r^*(G)$ over $i\mathfrak{a}_{\mathrm{dom}}^*$ . In order to do this, we must prove that the imaginary part of the infinitesimal character defines a continuous map.

Theorem 3.1. The map

\begin{align*}\operatorname{Im\,Inf\,Ch}: \widehat{G}_{\mathrm{tempered}} \to i\mathfrak{a}^*_{\mathrm{dom}}\end{align*}

is continuous.

Definition 3.3. Let G be any Lie group and let $T\in \mathcal{Z}(\mathfrak{g}_\mathbb{C})$ . Define a function

(3.1) \begin{align} \operatorname{Inf\,Ch}_T\colon \widehat{G} \longrightarrow \mathbb{C} \end{align}

on the unitary dual of G by

\begin{align*} \operatorname{Inf\,Ch}_T(\pi)\cdot I = \pi(T) \end{align*}

(this is an identity between operators acting on the smooth vectors in the Hilbert space of the representation $\pi$ ).

We shall use the following simple and well-known fact. We include a proof because we shall use the same argument to prove a less familiar fact later on, in Lemma 5.10.

In this paper, we shall mostly understand the topology on the dual through the bijection between closed sets in the spectrum of a $C^*$ -algebra and closed, two-sided ideals in that $C^*$ -algebra (in which the closed set in the spectrum associated to an ideal is the set of representations that vanish on the ideal). See [Reference DixmierDix77, Chapters 3 and 13] for information about the topology of the unitary dual, and for information about the topology on the spectra of $C^*$ -algebras in general.

But for the proof of the lemma, it will be better to use the following characterization of convergence in the unitary dual in terms of matrix coefficients: a net $\{ \pi_\alpha \}$ of irreducible unitary representations of G converges to an irreducible unitary representation $\pi$ in the dual if and only if there are unit vectors $v_\alpha$ in the Hilbert spaces of the representations $\pi_\alpha$ , and a unit vector v in the Hilbert space of $\pi$ , such that

(3.2) \begin{align} \lim_{\alpha \to \infty} \langle v_\alpha, \pi_\alpha (a) v_\alpha \rangle = \langle v, \pi (a) v \rangle, \quad \forall a \in C_c^\infty (G).\end{align}

See again [Reference DixmierDix77, Chapters 3 and 13].

Proof of Lemma 3.4. Let $T \in \mathcal{Z}(\mathfrak{g}_\mathbb{C})$ . Given (3.2) above, we wish to prove that

(3.3) \begin{align} \lim_{\alpha \to \infty}\operatorname{Inf\,Ch}_T (\pi_\alpha) = \operatorname{Inf\,Ch}_T (\pi).\end{align}

Choose $a\in C_c^\infty (G)$ such that $\langle v, \pi(a) v \rangle \ne 0$ and then write

\begin{align*}\operatorname{Inf\,Ch}_T (\pi_\alpha) \langle v_\alpha, \pi_\alpha (a) v_\alpha\rangle = \langle v_\alpha, \pi_\alpha (T)\pi_\alpha (a) v_\alpha\rangle= \langle v_\alpha, \pi_\alpha (Ta) v_\alpha\rangle,\end{align*}

where, in the last inner product, we view T as a differential operator on G. We find that

\begin{align*}\lim_{\alpha \to \infty} \operatorname{Inf\,Ch}_T (\pi_\alpha) \langle v_\alpha, \pi_\alpha (a) v_\alpha\rangle =\langle v, \pi (Ta) v\rangle = \operatorname{Inf\,Ch}_T (\pi) \langle v, \pi (a) v\rangle,\end{align*}

from which (3.3) follows.

Definition 3.5. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ . Define the map

\begin{align*} \operatorname{Inf\,Ch}_{\mathfrak{h}}: \widehat{G}_{\mathrm{tempered}} \longrightarrow \mathfrak{h}^*_{\mathbb{C}}/W(\mathfrak{g}_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}}) \end{align*}

by means of the formula

\begin{align*} \operatorname{Inf\,Ch}_{\mathfrak{h}}(\pi) = [\lambda]\quad \text{when} \quad \operatorname{Inf\,Ch}(\pi) = \xi_{\mathfrak{h},\lambda}. \end{align*}

When $\mathfrak{h} = \mathfrak{h}_{\min}$ , we shall often write simply $\operatorname{Inf\,Ch}$ in place of $\operatorname{Inf\,Ch}_{\mathfrak{h}_{\min}}$ .

Theorem 3.6. Let G be a real reductive group and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ . The map

\begin{align*}\operatorname{Inf\,Ch}_{\mathfrak{h}}: \widehat{G}_{\mathrm{tempered}} \to \mathfrak{h}_{\mathbb{C}}^*/W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}})\end{align*}

is continuous.

Proof of Theorem 3.1. The inclusion

\begin{align*}i\mathfrak{a}^*_{\mathrm{dom}} \xrightarrow{\cong} i\mathfrak{a}^*_{\min}/W(\mathfrak{g},\mathfrak{a}_{\min}) \to \mathrm{Im}(\mathfrak{h}_{\min,\mathbb{C}})^*/ W(\mathfrak{g}_\mathbb{C},\mathfrak{h}_{\min,\mathbb{C}})\end{align*}

is a homeomorphism onto its image. Therefore, it suffices to prove that the composition of $\operatorname{Im\,Inf\,Ch}$ with the above inclusion is continuous. This composition is simply $\operatorname{Inf\,Ch}$ composed with the (continuous) projection

\begin{align*} \mathfrak{h}^*_{\min,\mathbb{C}}/W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\min,\mathbb{C}}) \longrightarrow \mathrm{Im} (\mathfrak{h}^*_{\min,\mathbb{C}})/W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\min,\mathbb{C}}),\end{align*}

so continuity follows from Theorem 3.6.

Remark 3.7. It follows from the Dauns–Hofmann theorem (see [Reference Dauns and HofmannDH68] or [Reference PedersenPed79, § 4.4] for an exposition) that the continuity of $\operatorname{Im\,Inf\,Ch}$ allows us to embed $C_0(i\mathfrak{a}^*_\mathrm{dom})$ into the center of the multiplier algebra of $C_r^*(G)$ . More precisely, there is a unique inclusion of the $C^*$ -algebra $C_0(i \mathfrak{a}_{\mathrm{dom}}^*)$ into the center of the multiplier algebra of $C^*_r(G)$ such that

\begin{align*} \pi(f\cdot a) = f( \operatorname{Im\,Inf\,Ch} (\pi)) \cdot \pi (a) \end{align*}

for every tempered irreducible representation $\pi$ , every $f\in C_0(i \mathfrak{a}_{\mathrm{dom}}^*)$ , and every $a\in C^*_r (G)$ .

3.2 Definition of the subquotients

We shall use the imaginary part of the infinitesimal character to construct a (finite) filtration of $C^*_r(G)$ by ideals, whose subquotients have a particularly simple structure.

Definition 3.8. For every $I\subseteq S $ , define

\begin{align*}U_I = \{ i\nu\in i\mathfrak{a}_{\mathrm{dom}}^* :\langle \nu,\beta\rangle \ne 0 \,\, \forall \beta \notin I \} = \bigsqcup _{J\subseteq I} i \mathfrak{a}^*_{J,+} .\end{align*}

Thanks to the continuity of $\operatorname{Im\,Inf\,Ch}$ , the tempered representations for which the imaginary part of the infinitesimal character lies in $U_I$ form an open subset of $\widehat{G}_{\mathrm{tempered}}$ . We shall denote by

\begin{align*} C^*_r(G;U_I)\triangleleft C^*_r (G) \end{align*}

the $C^*$ -algebra ideal in the reduced group $C^*$ -algebra associated to this open subset of $\widehat{G}_{\mathrm{tempered}}$ . That is, $C^*_r(G,U_I)$ is the mutual kernel of all the irreducible representations $\pi$ of $C^*_r(G)$ for which $\operatorname{Im\,Inf\,Ch}(\pi)$ lies in the complement of $U_I$ .

It follows from the formula (2.12) that $i \mathfrak{a}_{I,+}^*$ is a closed subset of $U_I$ . We shall denote by

\begin{align*} \pi_I \colon C^*_r(G;U_I)\longrightarrow C^*_r (G; I) \end{align*}

the quotient associated to this closed subset. That is, $C^*_r (G; I)$ is the quotient of $C^*_r(G;U_I)$ by the ideal consisting of all elements that vanish in every representation $\pi$ with $\operatorname{Im\,Inf\,Ch}(\pi)\in U_I \setminus i\mathfrak{a}^*_{I,+}$ .

By elementary $C^*$ -algebra theory [Reference DixmierDix77, Chapter 2], every irreducible representation $\pi$ of $C^*_r (G)$ for which $\operatorname{Im\,Inf\,Ch}(\pi) \in U_I$ restricts to an irreducible representation of $C^*_r (G ; U_I)$ , and moreover, every irreducible representation of $C^*_r (G ; U_I)$ extends uniquely to an irreducible representation $\pi$ of $C^*_r (G)$ for which $\operatorname{Im\,Inf\,Ch}(\pi) \in U_I$ . Furthermore, an irreducible representation $\pi$ of $C^*_r (G ; U_I)$ factors through the quotient $C^*_r(G;I)$ if and only if $\operatorname{Im\,Inf\,Ch}(\pi)\in i \mathfrak{a}^*_{\min,I}$ . More precisely, we have the following.

Lemma 3.9. The procedure that associates to each irreducible representation $\pi'$ of $C^*_r(G;I)$ the unique extension to $C^*_r (G)$ of the composition of $\pi'$ with the quotient map $C^*_r (G;U_I)\to C^*_r (G;I)$ in Definition 3.8 determines a homeomorphism

\begin{align*}\widehat{C^*_r(G;I)} \stackrel \cong \longrightarrow \{ [\pi]\in \widehat G_{\mathrm{tempered}} : \operatorname{Im\,Inf\,Ch}(\pi) \in i\mathfrak{a}^*_{I,+} \}.\end{align*}

3.3 Discrete series representations and tempiric representations

We shall use the following substantial results of Harish-Chandra about the discrete series representations of real reductive groups (in the statement, by a K-type of a representation $\pi$ of G, we mean an irreducible constituent of $\pi\vert_K$ ).

Theorem 3.10 has the following consequence.

Proof. By Theorem 2.8 and (2.9), for every tempiric representation $\pi$ of G there is a standard parabolic subgroup $P = \textit{MAN}$ of G and a discrete series representation $(\sigma, H_\sigma)$ of M such that $\pi$ embeds as a subrepresentation of $\operatorname{Ind}_P^G \sigma \otimes 1$ . By Frobenius reciprocity, for a given irreducible representation $(\gamma, V_\gamma)$ of K,

\begin{align*}\operatorname{Hom}_K(V_\gamma, \operatorname{Ind}_P^G \sigma \otimes 1) \cong \operatorname{Hom}_{M \cap K}(V_\gamma, H_\sigma).\end{align*}

Therefore, $V_\gamma$ occurs in $\operatorname{Ind}_P^G \sigma \otimes 1$ only if an $(M\cap K)$ -type occurring in $V_\gamma$ also occurs in $H_\sigma$ . By Theorem 3.10, there are finitely many discrete series representations $\sigma$ of M which contain one of the (finitely many) $(M\cap K)$ -types occurring in $V_\gamma$ . We conclude that $V_\gamma$ is contained in finitely many representations of the form $\operatorname{Ind}_P^G \sigma \otimes 1$ , and in particular it is contained in finitely many tempiric representations.

3.4 Structure of the subquotients

In this section, we shall fix a subset $I{\subseteq} S$ and determine $C^*_r (G;I)$ up to isomorphism. For this and later purposes, we shall need to recall some details about the compact model of a parabolically induced representation. For a full account, see [Reference KnappKna86, § VII.1].

Let $P = \textit{MAN}$ be a parabolic subgroup of G, let $(\sigma, H_\sigma)$ be a unitary representation of M, and let $\nu \in \mathfrak{a}^*$ . The unitary representation

\begin{align*}\pi_{\sigma, \nu} = \operatorname{Ind}_{P}^G \sigma \otimes \exp(i\nu)\end{align*}

may be realized on the Hilbert space

\begin{align*}\operatorname{Ind}_{P}^G H_\sigma = L^2(K, H_\sigma)^{M \cap K},\end{align*}

where $M \cap K$ acts on $L^2(K, H_\sigma)$ by $(m \cdot \psi)(k) = \sigma(m)\psi(km)$ . The unitary operator

\begin{align*}\pi_{\sigma, \nu}(g): L^2(K, H_\sigma)^{M \cap K} \longrightarrow L^2(K, H_\sigma)^{M \cap K}\end{align*}

is defined by

(3.4) \begin{align} (\pi_{\sigma, \nu}(g)\psi)(k) = e^{-(i\nu+\rho)(\log a(g^{-1}k))} \sigma(m(g^{-1}k))^{-1} \psi(k(g^{-1}k)),\end{align}

where $\rho(H) = \tfrac{1}{2}\operatorname{tr} (\operatorname{ad}(H)\vert_{\mathfrak{n}})$ for $H \in \mathfrak{a}$ , and where a factorization

\begin{align*}g = k(g)m(g)a(g)n(g)\in \textit{KMAN} = G\end{align*}

is used; the factorization is not unique, but the operator $\pi_{\sigma, \nu}(g)$ is nevertheless well-defined.

Theorem 3.14. For every $ \sigma\in \widehat M_{I,\mathrm{tempiric}}$ , there is a unique surjective $C^*$ -algebra morphism

\begin{align*} \pi_\sigma \colon C^*_r (G;I)\longrightarrow C_0 (\mathfrak{a}^*_{I,+}, \mathfrak{K} (\operatorname{Ind}_{P_I}^G H_\sigma)) \end{align*}

such that

\begin{align*}\pi_\sigma(f)(\nu) = \pi_{\sigma,\nu}(f)\end{align*}

for every $\nu\in \mathfrak{a}^*_{I,+}$ and every $f\in C^*_r (G;I)$ .

Proof. Of course, we would like to define $\pi_\sigma$ by the formula

\begin{align*}\pi_\sigma(f)(\nu) = \pi_{\sigma,\nu}(f)\end{align*}

that appears in the statement of the theorem. That we may do so follows from the considerations in [Reference Clare, Crisp and HigsonCCH16, Lemma 4.10], which show that $\pi_{\sigma,\nu}(f)$ is a continuous function of $\nu$ , and indeed a $C_0$ -function of $\nu$ (this is essentially the Riemann–Lebesgue lemma; cf. the proof of Lemma 5.19 below, which deals with the same issue for a different group). It remains to prove that $\pi_\sigma$ is surjective; this follows from Dixmier’s Stone–Weierstrass theorem [Reference DixmierDix77, Theorem 11.1.8].

Theorem 3.15. There is a unique $C^*$ -algebra isomorphism

\begin{align*} C^*_r (G;I)\stackrel \cong\longrightarrow \bigoplus _{\sigma\in \widehat M_{I,\mathrm{tempiric}}} C_0(\mathfrak{a}^*_{I,+}, \mathfrak{K} (\operatorname{Ind}_{P_I}^G H_\sigma) ) \end{align*}

whose composition with the projection onto the $\sigma$ -summand in the direct sum is $\pi_\sigma$ .

4.1 Irreducible unitary representations of the Cartan motion group

In what follows, we shall use the Cartan decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{s}$ to identify the quotient vector space $\mathfrak{g}/\mathfrak{k}$ with $\mathfrak{s}$ , and so write the Cartan motion group as

(4.1) \begin{align}G_0 = K\ltimes \mathfrak{s}.\end{align}

Let $\xi$ be a linear functional on the real vector space $\mathfrak{s}$ , and let $K_\xi$ be its isotropy subgroup in K for the coadjoint action. Mackey [Reference MackeyMac49] proved that if $\tau$ is any irreducible unitary representation of $K_\xi$ , then the induced unitary representation

(4.2) \begin{align}\rho_{\tau,\xi}= \operatorname{Ind}_{K_\xi\ltimes \mathfrak{s}}^{K\ltimes \mathfrak{s}} \tau\otimes \exp (i \xi)\end{align}

is an irreducible unitary representation of $G_0$ . He further proved that every irreducible unitary representation of $G_0$ is equivalent to a representation of this form, and that the only equivalences among these representations are the obvious ones coming from K-conjugacies, so that from (4.2) we obtain a bijection

(4.3) \begin{align}\widehat G_0 \cong \bigg( \bigsqcup_{\xi\in \mathfrak{s}^*} \widehat K_\xi \!\bigg ) / K .\end{align}

For a review of these results from the $C^*$ -algebraic point of view of this paper, see [Reference HigsonHig08, § 3.1].

The bijection in (4.3) may be adapted to our current purposes by using the inclusion of $\mathfrak{a}_{\min}$ into $\mathfrak{s}$ , along with the following terminology and conventions (for our later purposes, we shall give more details here than are immediately needed). We identify $\mathfrak{a}_{\min}^*$ with a subspace of $\mathfrak{s}^*$ by extending each $\nu\in\mathfrak{a}_{\min}^*$ by zero on the orthogonal complement of $\mathfrak{a}_{\min}$ in $\mathfrak{s}$ to obtain an element of $\mathfrak{s}^*$ . Having done so, every functional in $\mathfrak{s}^*$ may be conjugated by K to a functional in $\mathfrak{a}_{\min}^*$ , and moreover to a unique functional in $\mathfrak{a}_{\mathrm{dom}}^*$ . Indeed, the inclusion of $\mathfrak{a}^*$ into $\mathfrak{s}^*$ induces a homeomorphism

\begin{align*} \mathfrak{a}^*_{\min} / W(\mathfrak{g},\mathfrak{a}_{\min}) \stackrel \cong \longrightarrow \mathfrak{s}^*/ K .\end{align*}

Definition 4.1. Let I be a subset of the set of simple roots in $\Delta ^+ (\mathfrak{g},\mathfrak{a}_{\min})$ and let $\nu\in \mathfrak{a}_{I,+}^*$ (notation from Definition 2.12). Let $K_I$ be the centralizer of $\mathfrak{a}_I$ in K (this may also be described as the isotropy group of the single element $\nu$ , as well as the intersection $K\cap M_I$ ), and let $\sigma$ be a unitary representation of $K_I$ . We shall denote by $\rho_{\sigma,\nu}$ the unitary representation

\begin{align*}\rho_{\sigma, \nu} = \operatorname{Ind}_{K_I\ltimes \mathfrak{s}}^{K\ltimes \mathfrak{s}} \sigma\otimes \exp (i \nu)\end{align*}

of $G_0 = K\ltimes \mathfrak{s}$ . Here, $\nu$ is extended to a linear functional on $\mathfrak{s}$ that is zero on the orthogonal complement of $\mathfrak{a}_I$ in $\mathfrak{s}$ . We shall realize the representation as follows. We let $K_I$ act on $L^2(K, H_\sigma)$ by $(k_I \cdot f)(k) = \sigma(k_I)f(kk_I)$ . The Hilbert space of the representation is $\operatorname{Ind}_{K_I}^K H_\sigma = L^2(K,H_\sigma)^{K_I}$ and the action of $G_0$ on an element $f \in L^2(K,H_\sigma)^{K_I}$ is determined by

\begin{align*}\begin{cases}(k\cdot f )(k_1) = f (k^{-1}k_1), & \forall k,k_1\in K,\\(X \cdot f)(k_1) = e^{i \nu (\operatorname{Ad}_{k_1}^{-1}(X))}f(k_1), & \forall X \in \mathfrak{s}\,\,\forall k_1\in K.\end{cases}\end{align*}

With this notation established, the bijection (4.3) can be recast as follows.

Theorem 4.3. The representations $\rho_{\tau, \nu}$ of $G_0$ , as I ranges over the finite subsets of the set S of simple restricted roots, as $\tau $ ranges over the unitary dual of $K_I$ , and as $\nu$ ranges over $\mathfrak{a}^*_{I,+}$ , are irreducible and pairwise inequivalent. They exhaust the unitary dual of $G_0$ , and determine a bijection

\begin{align*} \widehat G_0 \cong \bigsqcup _{I\subseteq S} \widehat K_I \times \mathfrak{a}^*_{I,+}.\end{align*}

4.2 Imaginary part of the infinitesimal character for the motion group

The vector subspace $\mathfrak{s}\subseteq\mathfrak{g}$ , considered as an abelian Lie algebra, is a Lie subalgebra of the Lie algebra of the Cartan motion group $G_0$ , and there is an inclusion

\begin{align*} \operatorname{Sym}(\mathfrak{s}_{\mathbb{C}})^K\longrightarrow \mathcal{Z}(\mathfrak{g}_{0,\mathbb{C}})^K .\end{align*}

Often, but not always, this is an isomorphism. In any case, each $T\in \operatorname{Sym}(\mathfrak{s}_{\mathbb{C}})^K$ determines a continuous map

(4.4) \begin{align} \operatorname{Inf\,Ch}_T\colon \widehat G_0 \longrightarrow \mathbb{C}\end{align}

by the formula $\operatorname{Inf\,Ch}_T(\pi) \cdot I = \pi (T)$ (action on smooth vectors).

The map (4.4) is easy to compute: identifying the element $T\in \operatorname{Sym}(\mathfrak{s}_{\mathbb{C}})^K$ with a K-invariant polynomial function on $\mathfrak{s}^*$ , we have

\begin{align*}\operatorname{Inf\,Ch}_T(\rho_{\tau,\nu}) = T( i \nu).\end{align*}

Lemma 4.5. The map

(4.5) \begin{align}&\operatorname{Inf\,Ch} \colon \widehat G_0 \longrightarrow i\mathfrak{s}^*/K,\nonumber\\&\operatorname{Inf\,Ch} (\rho_{\tau, \nu}) = i\nu \in i\mathfrak{s}^* /K\end{align}

is continuous.

Remark 4.6. As in Remark 3.7, the Dauns–Hofmann theorem allows us to embed $C_0(i\mathfrak{s}^*/K)$ into the center of the multiplier algebra of $C^*(G_0)$ . However, the Dauns–Hofmann theorem is not necessary as Fourier theory [Reference HigsonHig08, Theorem 3.2] provides an isomorphism

\begin{align*}C^* (G_0)\cong C(i\mathfrak{s}^*, \mathfrak{K}(L^2(K)))^K\end{align*}

under which the action of $C_0(i\mathfrak{s}^*/K)$ is simply pointwise multiplication.

Since the inclusion of $i \mathfrak{a}_{\min}^*$ into $i \mathfrak{s}^*$ as imaginary-valued linear functionals vanishing on the orthogonal complement of $\mathfrak{a}_{\min}$ induces a homeomorphism

\begin{align*}i \mathfrak{a}^*_{\min} / W(\mathfrak{g},\mathfrak{a}_{\min}) \stackrel \cong \longrightarrow i \mathfrak{s}^*/ K ,\end{align*}

we can view $\operatorname{Inf\,Ch}(\pi)$ as an element of $i\mathfrak{a}_{\min}^*/ W(\mathfrak{g},\mathfrak{a}_{\min})$ , and so obtain from (4.5) a continuous map

(4.6) \begin{align} \operatorname{Inf\,Ch} \colon \widehat G_0 \longrightarrow i\mathfrak{a}_{\min}^*/ W(\mathfrak{g},\mathfrak{a}_{\min}).\end{align}

A comparison between this and the map $\operatorname{Im\,Inf\,Ch}$ that we defined in the reductive case, see (2.11), makes it reasonable to also use the notation $\operatorname{Im\,Inf\,Ch}$ for the map in (4.6), which we shall do from now on. In other words, we are effectively declaring the real part of the infinitesimal character to be always zero on $\widehat{G}_0$ . Thus,

(4.7) \begin{align} \operatorname{Im\,Inf\,Ch}(\rho_{\tau,\nu}):=i\nu \in i\mathfrak{a}_{\min}^*/ W(\mathfrak{g},\mathfrak{a}_{\min}) \cong i\mathfrak{a}^*_{\mathrm{dom}}.\end{align}

4.3 Subquotients of the motion group $C^*$ -algebra

Using the continuous map $\operatorname{Im\,Inf\,Ch}$ just defined on $\widehat{G}_0$ , we can now repeat the constructions of § 3.2 more or less word for word, with G replaced by $G_0$ .

Definition 4.7. We shall denote by

\begin{align*} C^*(G_0;U_I)\triangleleft C^* (G_0) \end{align*}

the $C^*$ -algebra ideal in $C^* (G_0)$ associated to the open subset of the unitary dual consisting of representations whose infinitesimal character belongs to the open subset $U_I\subseteq i\mathfrak{a}_{\mathrm{dom}}^*$ in Definition 3.8. We shall denote by $C^* (G_0; I)$ the quotient $C^*$ -algebra associated to the closed subset $i\mathfrak{a}^*_{\min,I}\subseteq U_I$ , and by

\begin{align*} \rho_I\colon C^*(G_0;U_I)\longrightarrow C^* (G_0; I) \end{align*}

the quotient mapping.

Theorem 4.8. For every $ \tau\in \widehat K_I$ , there is a unique surjective $C^*$ -algebra morphism

\begin{align*} \rho_\tau \colon C^* (G_0;I)\longrightarrow C_0(\mathfrak{a}^*_{I,+}, \mathfrak{K} (\operatorname{Ind}_{K_I}^K H_{\tau})) \end{align*}

such that

\begin{align*}\rho_\tau(\rho_I(a))(\nu) = \rho_{\tau,\nu}(a)\end{align*}

for every $\nu\in \mathfrak{a}^*_{I,+}$ and every $a\in C^* (G_0;U_I)$ .

Theorem 4.9. There is a unique $C^*$ -algebra isomorphism

\begin{align*} C^* (G_0;I)\stackrel \cong\longrightarrow \bigoplus _{\tau\in \widehat K_{I}} C_0 (\mathfrak{a}^*_{I,+}, \mathfrak{K} (\operatorname{Ind}_{K_I}^K H_\tau) ) \end{align*}

whose composition with the projection onto the $\tau$ -summand in the direct sum is the morphism $\rho_\tau$ from Theorem 4.8.

5.1 Deformation families of Lie groups

Let G be a Lie group and let K be a compact subgroup of G (or, for nearly everything that follows, any closed subgroup). The deformation to the normal cone construction, applied to $K\subseteq G$ , yields a smooth family of Lie groups $ \{G_t\}_{t\in[0,\infty)}$ with

(5.1) \begin{align} G_t = \begin{cases} G, & t \gt 0, \\ K \ltimes (\mathfrak{g/k}), & t=0. \end{cases}\end{align}

See [Reference HigsonHig08, § 6.2] for this particular instance of the deformation to the normal cone construction. We shall use the following features.

1. (i) The disjoint union $\boldsymbol{{G}}= \bigsqcup \{ G_t : t \geqslant 0\}$ is equipped with a smooth manifold (with boundary) structure for which the map

   \begin{align*} \boldsymbol{{G}} \to [0, \infty) ,\quad G_t\ni g\mapsto t\in [0, \infty)\end{align*}
   is a submersion.

2. (ii) The fiberwise multiplication, inverse, and unit maps $\boldsymbol{{G}}\times_{[0, \infty)} \boldsymbol{{G}}\to \boldsymbol{{G}}$ , $\boldsymbol{{G}}\to\boldsymbol{{G}}$ , and $[0, \infty)\to \boldsymbol{{G}}$ are smooth maps.

3. (iii) If V is a finite-dimensional real vector space, and if $E\colon V\to G$ is a smooth map for which $E(0)=e$ , and for which the map

   \begin{align*} K \times V \to G,\quad (k,v) \mapsto k\cdot E(v) \end{align*}
   is a diffeomorphism, then the map
   \begin{align*} K \times V \times [0, \infty) \to \boldsymbol{{G}}; \quad (k,v,t) \mapsto \begin{cases} k\cdot E(tv)\in G_t, & t \gt 0, \\ (k, dE(v)) \in G_0, & t=0 \end{cases} \end{align*}
   is a diffeomorphism, where $dE\colon V\to \mathfrak{g}/\mathfrak{k}$ , is the composition of the derivative of E at $0\in V$ with the projection $\mathfrak{g}\to \mathfrak{g}/\mathfrak{k}$ (for this, we use $G_0 = K \ltimes \mathfrak{g} / \mathfrak{k}$ ).

Equip the groups $G_t$ with Haar measures $\mu_t$ , as follows. Fix a Haar measure $\mu=\mu_1$ on $G= G_t$ , and then define

(5.2) \begin{align} \mu_t = t^{-d} \mu \quad \text{for }t \gt 0,\end{align}

where $d = \dim (G/K)$ . As for $\mu_0$ , the measure $\mu$ with which we started determines a Lebesgue measure on $\mathfrak{g}$ , which may be written as the product of a Lebesgue measure $\mu_{\mathfrak{k}}$ on $\mathfrak{k}$ and a Lebesgue measure $\mu_{\mathfrak{s}}$ on $\mathfrak{s}$ (the summands in the Cartan decomposition $\mathfrak{g} = \mathfrak{k}\oplus \mathfrak{s}$ ). The measure $\mu_{\mathfrak{k}}$ determines a Haar measure $\mu_K$ on K, and we define $\mu_0$ to be the product of $\mu_K$ and $\mu_{\mathfrak{s}}$ on $G \cong K\ltimes \mathfrak{s}$ .

The measures $\mu_t$ vary smoothly with t in the sense that if f is a smooth and compactly supported function on $\boldsymbol{{G}}$ , then $\int_{G_t} f(g_t) \, d \mu_t g_t$ is a smooth function of t. Moreover, the operations of fiberwise convolution and adjoint define a $*$ -algebra structure on the space $C_c^\infty (\boldsymbol{{G}})$ of all smooth and compactly supported functions on $\boldsymbol{{G}}$ .

Definition 5.2. We shall denote by $C^*_r(\boldsymbol{{G}})$ the $C^*$ -algebra completion of the $*$ -algebra $C_c^\infty (\boldsymbol{{G}})$ in the norm

\begin{align*} \| f\| = \sup \{ \| f_t\|_{C^*_r (G_t)} : t\geqslant 0 \} , \end{align*}

where $f_t$ is the restriction of f to $G_t\subseteq \boldsymbol{{G}}$ .

The spectrum of $C^*_r (\boldsymbol{{G}})$ is the disjoint union

\begin{align*} \widehat{\boldsymbol{{G}}}_{\mathrm{tempered}} : = \widehat{C^*_r(\boldsymbol{{G}})} = \bigsqcup_{t\geqslant 0} \widehat{G}_{t,\mathrm{tempered}} .\end{align*}

We shall equip it with the usual hull-kernel topology it carries as the spectrum of a $C^*$ -algebra. But, for the most part, we shall only need the following simpler notion of convergence among representations of the groups $G_t$ .

Example 5.6. Let $\pi:G \to U(H)$ be a unitary representation. Define

\begin{align*}\begin{cases} \pi_t (g)= \pi(g), & t \gt 0, \\ \pi_0 (k,X) = \pi(k), & t =0.\end{cases}\end{align*}

Then, $\{\pi_t\}_{t\in[0,\infty)}$ is a continuous family of representations.

Example 5.7. Let $P=\textit{MAN}$ be a parabolic subgroup of a real reductive group. Form the deformation to the normal cone $\boldsymbol{{P}}$ associated to P and the compact subgroup $K\cap M$ . Let $\sigma$ be an irreducible unitary representation of M and let $\nu \colon \mathfrak{a} \to \mathbb{R}$ be a linear functional. Define unitary representations $\lambda_t $ of $P_t$ on the Hilbert space of $\sigma$ by

\begin{align*} \begin{cases} \lambda_t (man) = e^{i t^{-1} \nu (\log (a))}\cdot \sigma(m), & t\gt 0, \\ \lambda_0(k,X) = e^{i \nu(X)} \cdot \sigma(k), & t = 0, \end{cases}\end{align*}

where $\nu$ is transferred to a linear functional on $ \mathfrak{p}/(\mathfrak{k}\cap \mathfrak{m})$ by setting it equal to zero on the image of $\mathfrak{m}\oplus \mathfrak{n}$ . This is a continuous family of representations of $\boldsymbol{{P}}$ .

The following proposition is a special case of a more general principle that unitary induction takes continuous families to continuous families.

Proposition 5.8. Let $P=\textit{MAN}$ be a parabolic subgroup of a real reductive group G. Let $\sigma$ be a unitary representation of M and let $\nu \in \mathfrak{a}^*$ . The representations

\begin{align*}\pi_t = \begin{cases} \pi_{\sigma, t^{-1} \nu}, & t\gt 0, \\\rho_{\sigma,\nu}, & t =0\end{cases}\end{align*}

form a continuous family of unitary representations of $\boldsymbol{{G}}$ .

Proof. Note that $\boldsymbol{{P}}$ embeds as a smooth closed submanifold of $\boldsymbol{{G}}$ . For $g\in \boldsymbol{{G}}$ , we have a (non-unique) decomposition

\begin{align*}g= \kappa(g)\cdot \varphi(g)\end{align*}

with $\kappa(g)\in K$ and $ \varphi(g)\in \boldsymbol{{P}}$ . It is not usually possible to choose $\kappa$ and $\varphi$ to be globally smooth functions, but we can at least choose them to be smooth in a neighborhood of any point $g\in\boldsymbol{{G}}$ .

Now, for $\xi\in L^2 (K,H_\sigma)^{M\cap K}$ , we may write

\begin{align*} (\pi(g)\xi)(k') = \delta^{-1/2}(\varphi (g^{-1}k')) \cdot \lambda(\varphi(g^{-1}k'))^{-1} \xi(\kappa(g^{-1}k')),\end{align*}

where $\delta$ is the modular function for P (a necessary ingredient in unitary induction) and where $\lambda$ is as in Example 5.7. This, plus the formula above, shows that $g\mapsto \pi(g)\xi$ is continuous from $\boldsymbol{{G}}$ to $L^2(K,H_\sigma)$ , as claimed.

5.2 The imaginary part of the infinitesimal character for the Mackey deformation

We turn now to infinitesimal characters in the context of the deformation to the normal cone family. The fundamental observation underlying all of this is the following lemma. For the groups of interest to us, it is a consequence of condition (iii) on $\boldsymbol{{G}}$ in § 5.1.

Let us apply this to elements in the enveloping algebra of G. If $T\in \mathcal{U}(\mathfrak{g}_{\mathbb{C}})$ , and if T has order k for the usual increasing filtration on the enveloping algebra (so that T is a sum of products of k or fewer elements from $\mathfrak{g}$ ), then the family of right-invariant differential operators

\begin{align*}T_t = t^k T \colon C_c^\infty (G_t) \longrightarrow C_c^\infty(G_t)\quad (t\gt0)\end{align*}

extends to a smooth family of right-invariant differential operators defined on all the fibers of $\boldsymbol{{G}}$ . We obtain from this a families version of Lemma 3.4; we shall formulate it for real reductive groups and the tempered dual only because we have not defined the full unitary dual of the family $\boldsymbol{{G}}$ .

Lemma 5.10. Let G be a real reductive group and let $T\in\mathcal{Z}(\mathfrak{g}_{\mathbb{C}})$ . If1pt T has order k, and if $T_t = t^k T$ for $t\ne 0$ , then the function

\begin{align*} \widehat G_t \ni \pi_t \longmapsto \operatorname{Inf\,Ch}_{T_t}(\pi_t)\in \mathbb{C}\quad (t\gt0) \end{align*}

extends at $t=0$ to a continuous function on $\widehat {\boldsymbol{{G}}}_{\mathrm{tempered}}$ .

Theorem 5.11. Let G be a real reductive group and let $\mathfrak{h}_{\mathbb{C}}$ be a Cartan subalgebra of $\mathfrak{g}_{\mathbb{C}}$ . The family of maps

\begin{align*} t \cdot \operatorname{Inf\,Ch}: \widehat G _{t,\mathrm{tempered}} \longrightarrow \mathfrak{h}_{\mathbb{C}}^* / W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}})\quad (t\gt 0) \end{align*}

extends to a continuous map

\begin{align*} \widehat{\boldsymbol{{G}}}_{\mathrm{tempered}} \longrightarrow \mathfrak{h}_{\mathbb{C}}^* / W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}}). \end{align*}

Specifically, taking $\mathfrak{h}=\mathfrak{h}_{\mathrm{min}}$ , the continuous extension to $t=0$ is given by the infinitesimal character of $G_0$ , from (4.7),

\begin{align*} \operatorname{Inf\,Ch}:\widehat{G}_0 \to i\mathfrak{a}^*_{\mathrm{min}}/W(\mathfrak{g},\mathfrak{a}_{\mathrm{min}}) \subseteq \mathfrak{h}_{\min,\mathbb{C}}^* / W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}}). \end{align*}

Proof. According to Chevalley’s theorem [Reference ChevalleyChe55], there are homogeneous, Weyl group-invariant polynomial functions $p_1,\ldots, p_r $ on $\mathfrak{h}^*_{\mathbb{C}}$ , say of degrees $k_1,\dots, k_r$ , that freely generate the algebra of all Weyl group-invariant polynomial functions on $\mathfrak{h}^*_{\mathbb{C}}$ . The function

\begin{align*}\mathfrak{h}^*_{\mathbb{C}}/W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}}) \longrightarrow \mathbb{C}^r, \quad \lambda\longmapsto (p_1(\lambda), \dots, p_r (\lambda))\end{align*}

is then a homeomorphism (indeed, an isomorphism of affine varieties). The compositions of this homeomorphism with the maps in the statement of the theorem are the maps

\begin{align*} \widehat G_t \ni \pi_t \longmapsto ( \operatorname{Inf\,Ch}_{T_{1,t}}(\pi_t), \dots, \operatorname{Inf\,Ch}_{T_{r,t}}(\pi_t)),\end{align*}

where $T_j \in \mathcal{Z}(\mathfrak{g}_{\mathbb{C}})$ corresponds to $p_j\in \mathbb{C}[\mathfrak{h}_{\mathbb{C}}^*]$ under the Harish-Chandra isomorphism, and where $T_{j,t}= t^{k_j} T_j$ . Since $T_j$ has order $k_j$ , it follows from Lemma 5.10 that the composition gives a continuous map from $\widehat{\boldsymbol{{G}}}_{\mathrm{tempered}}$ to $\mathbb{C}^r$ . This proves the existence of the continuous extension.

In order to determine the value of this extension at $t = 0$ , we make an explicit computation using the minimal principal series of G. We have seen in (2.9) that for the minimal parabolic subgroup $P= P_{\min}$ of G,

\begin{align*}\operatorname{Inf\,Ch} (\pi_{\sigma, t^{-1}\nu}) = \operatorname{Inf\,Ch}(\sigma)\oplus i t^{-1}\nu\in i \mathfrak{t}^*_{\min} \oplus i \mathfrak{a}^*_{\min} ,\end{align*}

so that

(5.3) \begin{align}t \cdot \operatorname{Inf\,Ch} (\pi_{\sigma, t^{-1}\nu}) = t \cdot \operatorname{Inf\,Ch}(\sigma)\oplus i \nu\in i\mathfrak{t}^*_{\min} \oplus i \mathfrak{a}^*_{\min} .\end{align}

The representation $\pi_{\sigma,\nu}$ will be irreducible if $\nu$ is fixed by no nontrivial element of $W(\mathfrak{g},\mathfrak{a}_{\min})$ , but in any case it always has an infinitesimal character. According to Proposition 5.8, the representations $\pi_{\sigma, t^{-1}\nu}$ of $G_t$ extend to a continuous family of representations with limit $\rho_{\sigma,\nu}$ on $G_0$ . The continuous extension is therefore given on $\widehat G_0$ by

\begin{align*} \rho_{\tau,\nu} \longmapsto 0\oplus i\nu \in i\mathfrak{t}^*_{\min} \oplus i \mathfrak{a}^*_{\min},\end{align*}

for every irreducible constituent $\rho_{\tau,\nu}$ of $\rho_{\sigma,\nu}$ Since every $\rho_{\tau,\nu}$ occurs in some $\rho_{\sigma, \nu}$ , this completely determines the continuous extension to $t=0$ , as claimed.

The imaginary part of the infinitesimal character always belongs to the image of the inclusion of $i\mathfrak{a}^*_{\mathrm{dom}}$ into $\mathfrak{h}^*_{\mathbb{C}}/W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}})$ ; see Theorem 3.1. Therefore, composing the map of Theorem 5.11 with the projection

\begin{align*} \mathfrak{h}^*_{\mathbb{C}}/W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}}) \longrightarrow \mathrm{Im} (\mathfrak{h}^*_{\mathbb{C}})/W(\mathfrak{g}_{\mathbb{C}},\mathfrak{h}_{\mathbb{C}}),\end{align*}

we obtain the following.

Theorem 5.12. Let G be a real reductive group. There is a continuous map

\begin{align*}\widehat{\boldsymbol{{G}}}_{\mathrm{tempered}} \longrightarrow i \mathfrak{a}_{\mathrm{dom}}^*\end{align*}

such that

\begin{align*}\widehat G_{t,\mathrm{tempered}} \ni \pi_t \longmapsto t\cdot \operatorname{Im\,Inf\,Ch}({\pi_t}) \quad (t\gt0)\end{align*}

and

\begin{align*}\widehat G_{0,\mathrm{tempered}} \ni \pi_0 \longmapsto \operatorname{Im\,Inf\,Ch}({\pi_0}) \quad (t=0) .\end{align*}

Thus, for $t\gt0$ , the scaled imaginary part of the infinitesimal character of the irreducible representation $\pi_t = \pi_{\sigma,t^{-1}\nu}$ of $G_t = G$ is $i \nu$ , while the scaled imaginary part of the infinitesimal character of the irreducible representation $\rho_{\tau,\nu}$ of $G_0$ is $i \nu$ .

Remark 5.14. Similarly to G and $G_0$ , by the Dauns–Hofmann theorem there is a unique inclusion of $C_0(i \mathfrak{a}^*_{\mathrm{dom}})$ into the center of the multiplier algebra of $C_r^*(\boldsymbol{{G}})$ such that

\begin{align*} \pi_t ( (f\cdot a)(t)) = f( t \cdot \operatorname{Im\,Inf\,Ch} (\pi_t )) \cdot \pi_t (a) \end{align*}

for all $t\gt0$ and

\begin{align*} \pi_0 ( (f\cdot a)(0)) = f( \operatorname{Im\,Inf\,Ch} (\pi_0 )) \cdot \pi_0 (a) \end{align*}

for all irreducible tempered unitary representations $\pi_t$ of $G_t$ and $\pi_0$ of $G_0$ , all $f\in C_0(i \mathfrak{a}_{\mathrm{dom}}^*)$ , and all $a\in C^* _r(\boldsymbol{{G}})$ .

5.3 Subquotients of the $C^*$ -algebra of continuous sections

Definition 5.15. We shall denote by

\begin{align*} C^*_r(\boldsymbol{{G}} ;U_I)\triangleleft C^*_r (\boldsymbol{{G}} ) \end{align*}

the $C^*$ -algebra ideal associated to the open subset of the tempered dual consisting of representations whose scaled infinitesimal character belongs to the open subset $U_I\subseteq i\mathfrak{a}_{\mathrm{dom}}^*$ . We shall denote by

\begin{align*} C^*_r(\boldsymbol{{G}} ;U_I)\to C^*_r (\boldsymbol{{G}} ; I) \end{align*}

the quotient of this ideal associated to the closed subset $i\mathfrak{a}^*_{I,+}\subseteq U_I$ .

Remark 5.17. We may also ask whether or not the field of quotients $\{ A_t/J_t\}_{t\in T}$ becomes a continuous field of $C^*$ -algebras, if one deems the continuous sections to be those that lift to continuous sections of the original field. This is not necessarily so, but the field is at least upper semicontinuous, in the sense that for every liftable section s, and every $t_0\in T$ ,

\begin{align*}\limsup_{t\to t_0} \|s(t)\|_{A_t/J_t} \leqslant \| s(t_0)\|_{A_{t_0}/J_{t_0}}.\end{align*}

In fact, the fields of quotients introduced in this section are continuous fields, and not just upper-semicontinuous fields. We shall not need this fact, but it follows from a minor supplementary argument; see Remark 5.20 below.

5.4 Structure of the subquotients

We shall now determine the structure of the subquotients $C^*_r (\boldsymbol{{G}}, I)$ . The formulation of the result and the method of proof are borrowed from [Reference Higson and RománHR20]; see also [Reference Clare, Higson and RománCHR26].

Lemma 5.18. Let G be a real reductive group and let I be a subset of the set S of simple roots in $\Delta^+(\mathfrak{g}, \mathfrak{a}_{\min})$ . There is a unique continuous action of the multiplicative group $\mathbb{R}^{\times}_{+}$ on $C^*_r (G;I)$ by $C^*$ -algebra automorphisms,

\begin{align*} \alpha_t\colon C_r^* (G,I) \longrightarrow C^*_r (G, I)\quad (t\gt 0), \end{align*}

such that

\begin{align*} \pi_{\sigma, \nu}(\alpha_t (f)) = \pi_{\sigma, t^{-1}\nu} (f) \end{align*}

for all $t\gt0$ , all $f\in C^*_r (G;I)$ , all $\sigma\in \widehat M_{I,\mathrm{tempiric}}$ , and all $\nu \in \mathfrak{a}^*_{I,+}$ .

Proof. The uniqueness assertion is clear, since the representations $\pi_{\sigma,\nu}$ separate the points of $C^*_r (G; I)$ . Existence follows from the isomorphism

(5.4) \begin{align} C^*_r (G;I)\stackrel \cong\longrightarrow \bigoplus _{\sigma\in \widehat M_{I,\text{tempiric}}} C_0(\mathfrak{a}^*_{I,+}, \mathfrak{K} (\operatorname{Ind}_{P_I}^G H_\sigma) )\end{align}

in Theorem 3.15, using which we may define $\alpha_t$ by requiring it to correspond with the automorphisms

\begin{align*}\alpha_{\sigma,t} \colon C_0(\mathfrak{a}^*_{I,+}, \mathfrak{K} (\operatorname{Ind}_{P_I}^G H_\sigma) )\longrightarrow C_0 (\mathfrak{a}^*_{I,+}, \mathfrak{K} (\operatorname{Ind}_{P_I}^G H_\sigma))\end{align*}

of the summands on the right-hand side above, defined by

\begin{align*} \alpha_{\sigma,t} (h)(\nu) = h(t^{-1}\nu). \end{align*}

Lemma 5.19. Let G be a real reductive group and let I be a subset of the set S of simple roots in $\Delta^+(\mathfrak{g}, \mathfrak{a}_{\min})$ . There is a unique $C^*$ -algebra homomorphism

\begin{align*} \alpha_0\colon C^*_r (G_0;I) \longrightarrow C^*_r (G; I) \end{align*}

such that

\begin{align*} \pi_{\sigma, \nu} (\alpha_0(f_0)) = \rho_{\sigma,\nu}(f_0) \end{align*}

for all $f_0\in C^*(G_0;I)$ , all $\sigma\in \widehat M_{I,\mathrm{tempiric}}$ , and all $\nu\in \mathfrak{a}^*_{I,+}$ .

Proof. As with Lemma 5.18, the uniqueness assertion is clear, and to prove existence, we can again use the isomorphism (5.4), using which it suffices to construct a suitable $C^*$ -homomorphism

(5.5) \begin{align} \alpha_0\colon C^*_r (G_0;I) \longrightarrow \bigoplus _{\sigma\in \widehat M_{I,\mathrm{tempiric}}} C_0(\mathfrak{a}^*_{I,+}, \mathfrak{K} (\operatorname{Ind}_{P_I}^G H_\sigma)).\end{align}

We define

(5.6) \begin{align} \alpha_0\colon f_0\longmapsto \oplus_\sigma [ \nu \mapsto \rho_{\sigma,\nu}(f_0)].\end{align}

Considering an individual summand, if $f_0$ is a smooth and compactly supported function on $G_0$ , and if $\varphi,\psi\in L^2(K, H_\sigma)^{K_I}$ , then

(5.7) \begin{align}\langle \varphi , \rho_{\sigma,\nu} (f_0) \psi \rangle = \int_K\int_K\int_{\mathfrak{s}} f_0 (k'k^{-1}, \operatorname{Ad}_k X) e^{-i \nu(X)} \langle \varphi(k'), \psi(k) \rangle \, dk'\, dk \, dX ,\end{align}

and the Riemann–Lebesgue lemma shows that the above is a $C_0$ -function on $\nu$ . If, moreover, $f_0$ is K-bi-finite, then the operator $\rho_{\sigma,\nu} (f_0)$ is identically zero on a subspace of finite codimension in $L^2 (K,H_\sigma)^{K_I}$ , and on the finite-dimensional orthogonal complement it is unitarily equivalent to a finite matrix with the above matrix coefficients (5.7) as entries. Therefore, $\rho_{\sigma,\nu} (f_0)$ is a compact operator-valued $C_0$ function of $\nu$ , for every $\sigma$ . But, in addition, K-bi-finiteness implies that only finitely many summands on the right of (5.6) are nonzero.

It follows that the formula (5.6) does indeed define an element of the right-hand side in (5.5), not just for a smooth, compactly supported, and K-bi-finite function $f_0$ on $G_0$ , but, by an approximation argument, for any $f_0\in C^*_r (G_0)$ , since the functions we have considered up to now are dense in $C^*_r (G_0)$ . It is evident that the homomorphism $\alpha_0$ so defined has the required property.

In view of the descriptions of $C^*_r(G;I)$ and $C^*_r (G_0;I)$ given in Theorems 3.15 and 4.9, respectively, the embedding $\alpha_0$ in Lemma 5.19 determines the embedding of $C^*$ -algebras that is represented by the dotted arrow in the following commuting diagram.

We shall give a full account of this embedding in Theorem 6.12.

Theorem 5.21. Let G be a real reductive group and let I be a subset of the set S of simple roots in $\Delta^+(\mathfrak{g}, \mathfrak{a}_{\min})$ . If $\{ f_t\}$ is a continuous section of the continuous field of $C^*$ -algebras $\{ C^*_r (G_t; U_I) \}$ , then

\begin{align*}\lim_{t\searrow 0} \alpha_t( \pi_I(f_t)) = \alpha_0 (\rho_I(f_0)) ,\end{align*}

where the limit is taken in the norm topology of $C^*_r(G,I)$ , and where $\pi_I:C^*_r(G,U_I) \to C^*_r (G, I)$ and $\rho_I:C^*_r(G_0,U_I) \to C^*_r (G_0, I)$ are the quotient maps that were introduced in Definitions 3.8 and 4.7.

Remark 5.22. If we apply the representation $\pi_{\sigma,\nu}$ to both sides of the limit formula in the theorem, and use the characteristic properties of $\alpha_t$ and $\alpha_0$ , then we obtain the formula

\begin{align*}\lim_{t\searrow 0} \pi_{\sigma,t^{-1}\nu}(f_t) = \rho_{\sigma,\nu}(f_0) .\end{align*}

This is a consequence of Example 5.7 and Proposition 5.8. In effect, Theorem 5.21 promotes this pointwise limit formula (valid for $\sigma$ and one value of $\nu$ at a time) to a uniform limit formula over all $\sigma$ and $\nu$ . This is a technical matter: the main conceptual content of the theorem is already present in Proposition 5.8.

Proof of Theorem 5.21. Let $P_I = \textit{MAN}$ be the Langlands decomposition of $P_I$ . Let $\{ f_t\}$ be a continuous section of the continuous field of $C^*$ -algebras $\{ C^*_r (G_t; U_I) \}$ . The same argument that we used in the proof of Lemma 5.19 shows that it suffices to prove that for every tempiric representation $\sigma$ , and for every pair $\varphi, \psi$ of smooth functions in $ L^2 (K,H_\sigma)^{K\cap M}$ , the limit formula

\begin{align*}\lim_{t\searrow 0} \langle \varphi , \pi_{\sigma,t^{-1}\nu}(f_t)\psi \rangle = \langle \varphi , \rho_{\sigma,\nu}(f_0)\psi \rangle\end{align*}

holds uniformly in $\nu$ .

It follows from the formula (3.4) for $\pi_{\sigma, \nu}(g)$ , plus a change of variables, that if $t\gt0$ , and if $f_t$ is a smooth and compactly supported function on $G_t=G$ , then

(5.8) \begin{align}(\pi_{\sigma,t^{-1}\nu}(f_t)\psi)(\ell) = t^{-d} \int _G f_t(\ell g^{-1}) e^{-i t^{-1}\nu(\log(a (g))}e^{-\rho (\log (a (g))} \sigma(m(g))^{-1}\psi(k(g)) \, dg,\end{align}

where $d = \dim (G/K)$ . Here, we are using the smooth family of Haar measures $t^{-d}dg$ on $G_t=G$ as in (5.2).

Let us now fix an Iwasawa decomposition $M = K'A'N'$ , with $K' = K\cap M$ and $A'\subseteq A_{\min}$ . The integral in (5.8) may then be written as a multiple integral

\begin{align*}& t^{-d}\int_K\int_{A'} \int_{N'}\int_A\int_N f_t(\ell(ka'n'an)^{-1}) e^{-i t^{-1}\nu(\log(a))} e^{-\rho (\log (a))} \\&\quad \times \sigma(a'n')^{-1}\psi(k) e^{+2(\rho' (\log (a'))+\rho(\log (a)))} \, dk \, da'\, dn'\, da\, dn ,\end{align*}

for suitable choices of Haar measure on the factor groups. See, for instance, [Reference KnappKna86, § V.6] for this change-of-variables formula. It follows that the inner product $ \langle \varphi , \pi_{\sigma,t^{-1}\nu} (f_t) \psi \rangle $ is equal to

(5.9) \begin{align} & t^{-d} \int _K \int_K\int_{A'}\int_{N'}\int_A\int_N f_t (\ell(ka'n'an)^{-1}) e^{-it^{-1} \nu(\log(a))}\nonumber \\&\quad \times \langle \varphi (\ell),\sigma(a'n')^{-1}\psi(k) \rangle e^{2\rho' (\log (a'))+ \rho(\log (a))} \, d\ell \, dk \, da'\, dn'\, da\, dn .\end{align}

Let us make some adjustments to (5.9). The exponential maps for the groups A’, N’, A, and N are measure-preserving diffeomorphisms (for suitable multiples of Lebesgue measure on the Lie algebras). So, we may switch to Lie algebra coordinates in the integrals over these groups. Then, after scaling all four Lie algebra coordinates by a factor of t, we obtain the reformulation of the integral in (5.9):

(5.10) \begin{align} & \int _K \int_K \int_{\mathfrak{a}'}\int _{\mathfrak{n}'} \int_{\mathfrak{a}}\int_{\mathfrak{n}} f_t (\ell(k\exp(tX')\exp(tY')\exp(tX)\exp(tY))^{-1} )\nonumber\\& \quad \times e^{-i \nu(X)} \langle \varphi (\ell),\sigma(\exp(tX')\exp(tY'))^{-1}\psi(k) \rangle e^{+ t (2\rho' (X') + \rho (X))}d\ell \, dk\, dX'\, dY'\, dX\, dY .\end{align}

The point about this reformulation is that the expression

\begin{align*}k\exp(tX')\exp(tY')\exp(tX)\exp(tY) \in G_t\end{align*}

that is included in the integrand above determines a diffeomorphism

(5.11) \begin{align} K \times \mathfrak{a}'\times \mathfrak{n}'\times \mathfrak{a}\times \mathfrak{n} \times \mathbb{R} \stackrel \cong \longrightarrow\boldsymbol{{G}}\end{align}

for which

\begin{align*}(k,X',Y',X,Y,0) \longmapsto (k, X'{+}Y'{+}X{+}Y)\in G_0 = K \ltimes (\mathfrak{g}/\mathfrak{k}).\end{align*}

So, if the functions $f_t$ on the individual $G_t$ ( $t\gt0$ ) assemble and extend to a smooth, compactly supported function on $\boldsymbol{{G}}$ , then the integrand in (5.10) is uniformly compactly supported on $K\times \mathfrak{a}'\times \mathfrak{n}'\times \mathfrak{a}\times \mathfrak{n} $ as $t\gt 0$ varies, and moreover, as t tends to 0, it converges uniformly to

\begin{align*} f_0 (\ell(k, X'{+}Y'{+}X{+}Y)^{-1}) e^{-i \nu(X)} \langle \varphi (\ell), \psi(k) \rangle\end{align*}

(note that extending the functional $\nu$ , defined on $\mathfrak{a}$ , to a linear functional on $\mathfrak{g}/\mathfrak{k}$ that vanishes on $\mathfrak{a'}$ , $\mathfrak{n'}$ , and $\mathfrak{n}$ is the same as extending it to a linear functional on $\mathfrak{s}$ that vanishes on the orthogonal complement of $\mathfrak{a}$ , and then identifying $\mathfrak{s}$ with $\mathfrak{g}/\mathfrak{k}$ ). This gives convergence of the integral in (5.10) to

(5.12) \begin{align} \int_K\int_K\int_{\mathfrak{a}'+\mathfrak{n}'+\mathfrak{a}+\mathfrak{n}} f_0 (\ell(k,Z)^{-1}) e^{-i \nu(Z)} \langle \varphi(\ell), \psi(k) \rangle \, d\ell\, dk \, dZ ,\end{align}

which, as we have seen in (5.7), is $\langle\varphi, \rho_{\sigma, \nu} (f_0)\psi\rangle$ , and the convergence is uniform in $\nu$ , as required.

With this rather technical computation completed, we can readily determine the structure of the $C^*$ -algebra $C^*_r (\boldsymbol{{G}}; I)$ in a way that will be useful for the K-theory computations of the next section.

Definition 5.23. Let $\alpha:A_0 \rightarrow A$ be a $C^*$ -algebra homomorphism. The mapping cone $C^*$ -algebra is

\begin{align*} \operatorname{Cone}(\alpha) = \{(f_0,F)\in A_0\oplus C_0([0,\infty), A) : \alpha(f) = F(0) \} . \end{align*}

Theorem 5.24 (Cf. [Reference Higson and RománHR20, Theorem 5.2.2]). Let $\alpha_0\colon C^*_r (G_0;I)\to C^*_r (G;I)$ be the $C^*$ -algebra morphism from Lemma 5.19. The formula

\begin{align*} \alpha_I(f) = ( f_0, t \mapsto \alpha_t(f_t) )\quad (f\in C^*_r (\boldsymbol{{G}} ; I) ) \end{align*}

defines a $C^*$ -algebra isomorphism

\begin{align*} \alpha_I \colon C^*_r (\boldsymbol{{G}} ;I) \stackrel \cong \longrightarrow \operatorname{Cone}(\alpha_0) . \end{align*}

6.1 Vogan’s theorem on minimal K-types

Definition 6.1 (See [Reference VoganVog81, Definition 5.4.18]). Let G be a real reductive group with maximal compact subgroup K. If $\tau$ is an irreducible unitary representation of K, then its norm is

\begin{align*}\| \tau\| = \| \mu + 2 \rho_K\| ,\end{align*}

where $\mu$ is a highest weight for some choice of maximal torus $T\subseteq K$ and some choice of a system of positive roots in $\Delta (\mathfrak{k}_{\mathbb{C}},\mathfrak{t}_{\mathbb{C}})$ ; the choices do not affect the norm.

The simplicity of these definitions belies their importance. We shall be concerned in this paper with the following fundamental result about minimal K-types, due to Vogan, which, as we shall see, completely determines the tempered dual.

Theorem 6.4 (See [Reference VoganVog07, Theorem 1.2]). Let G be a real reductive group with maximal compact subgroup K. Every tempiric representation has a unique minimal K-type, which occurs in the representation with multiplicity one. The map that associates to a tempiric representation its minimal K-type is a bijection

\begin{align*}\widehat G _{\mathrm{tempiric}}\stackrel \cong \longrightarrow \widehat K .\end{align*}

Among other things, this theorem immediately gives a parametrization of the tempered dual of G in terms of irreducible representations of compact subgroups, for we have, from (2.14) and Vogan’s bijection,

\begin{align*} \widehat G_{\mathrm{tempered}} \cong \bigsqcup _{I\subseteq S} \widehat M_{I,\mathrm{tempiric}} \times i \mathfrak{a}^*_{I,+} \cong \bigsqcup _{I\subseteq S} \widehat K_{I} \times i \mathfrak{a}^*_{I,+} \cong \widehat{G}_{0}.\end{align*}

This is the Mackey bijection of Afgoustidis [Reference AfgoustidisAfg21].

In the next section, we shall prove that the Connes–Kasparov isomorphism in operator K-theory is equivalent to a K-theoretic version of Vogan’s theorem that involves the following concepts.

Definition 6.6. Denote by

\begin{align*} \mathrm{mult} \colon R(K) \longrightarrow R_{\mathrm{tempiric}} (G) \end{align*}

the homomorphism of abelian groups defined by

\begin{align*} [\tau]\longmapsto \sum_{\sigma\,\,\mathrm{tempiric}} \operatorname{mult} (\tau : \sigma)\cdot [\sigma] \quad \forall \tau \in \widehat K , \end{align*}

where $\operatorname{mult} (\tau : \sigma)$ is the multiplicity with which $\tau$ occurs in the restriction of the tempiric representation $\sigma$ to K.

It is a simple matter of solving linear equations to prove, using Vogan’s minimal K-type theorem (Theorem 6.4), that the multiplicity homomorphism in Definition 6.6 is an isomorphism. The main goal of this paper is to show that it is also possible to prove that $\operatorname{mult}$ is an isomorphism without using Vogan’s minimal K-type theorem but instead using the Connes–Kasparov isomorphism.

This is very close to a new proof of Vogan’s minimal K-type theorem. For instance, if we were to assume that no $\tau\in \widehat K$ may occur as a minimal K-type of two inequivalent tempiric representations, then elementary linear algebra would allow us to deduce Vogan’s theorem from the isomorphism of the multiplicity map.

6.2 The Connes–Kasparov isomorphism

We turn, finally, to the Connes–Kasparov isomorphism. In view of the aims of this paper, it is certainly important to note that Lafforgue [Reference LafforgueLaf02b] has given a proof of the Connes–Kasparov isomorphism using techniques from K-theory and index theory, rather than representation theory. On the other hand, representation-theoretic treatments of the Connes–Kasparov isomorphism can be found in [Reference WassermannWas87, Reference HigsonHig08, Reference AfgoustidisAfg19, Reference Clare, Higson and SongCHS24, Reference BraddBra25].

We shall use the following formulation of the Connes–Kasparov isomorphism, which is due to Connes (see [Reference ConnesCon94, Proposition 9, p. 141]).

and hence (using Bott periodicity) a six-term exact sequence in K-theory.

The assertion that the K-theory of $C^*_r (\boldsymbol{{G}} )$ vanishes is therefore equivalent to the assertion that the vertical maps in the six-term sequence are isomorphisms, and it is these that Connes considers in [Reference ConnesCon94].

6.3 K-theory of the mapping cone

The main theorem of this paper is the following consequence of the Connes–Kasparov isomorphism for real reductive groups. In fact, this statement is equivalent to the Connes–Kasparov isomorphism for real reductive groups.

Theorem 6.11. Let G be a real reductive group and let K be a maximal compact subgroup of G. The morphism of abelian groups

\begin{align*} \operatorname{mult}: R(K) & \longrightarrow R_{\mathrm{tempiric}} (G),\\ [\tau] &\longmapsto \sum_{\sigma} \operatorname{mult} (\tau : \sigma)\cdot [\sigma], \quad \forall \tau \in \widehat K, \end{align*}

is an isomorphism.

To prove the theorem, we shall relate the morphism in the statement to the induced maps in K-theory associated to the $C^*$ -algebra homomorphisms

\begin{align*}\alpha_0 \colon C^*_r (G_0;I)\longrightarrow C^*_r (G;I)\quad (I \subseteq S)\end{align*}

from Lemma 5.19. The point is that this is a morphism between $C^*$ -algebras whose K-theory groups are easily determined in representation-theoretic terms. First, from the isomorphism

\begin{align*} C^*_r(G_0; I) \stackrel\cong \longrightarrow \bigoplus_{\tau \in \widehat{K}_I} C_0(\mathfrak a^*_{I,+}, \mathfrak{K}(\operatorname{Ind}_{K_I}^K H_\tau))\end{align*}

in Theorem 4.9, we obtain isomorphisms

\begin{align*}\begin{aligned}K_* ( C^*_r(G_0; I) )& \stackrel\cong \longrightarrow \bigoplus_{\tau \in \widehat{K}_I} K_* ( C_0(\mathfrak a^*_{I,+}, \mathfrak{K}(\operatorname{Ind}_{K_I}^K H_\tau)) ) \\& \,\, \cong\,\, \bigoplus_{\tau \in \widehat{K}_I} K_* ( C_0(\mathfrak a^*_{I,+})) \\& \,\, \cong\,\, \bigoplus_{\tau \in \widehat{K}_I} \mathbb{Z} ,\end{aligned}\end{align*}

using first the stability property of K-theory and then the Bott periodicity theorem, which implies that the K-theory of $C_0(\mathfrak a^*_{I,+})$ is the infinite cyclic group $\mathbb{Z}$ . We fix a choice of generator for each I. Second, using Theorem 3.15 in a similar fashion, we have isomorphisms

\begin{align*}K_* ( C^*_r(G; I) ) \stackrel\cong \longrightarrow \bigoplus_{\sigma \in \widehat{M}_{I,\mathrm{tempiric}}} \mathbb{Z} .\end{align*}

Theorem 6.12. There is a commutative diagram as below in which the vertical isomorphisms are the ones described above, and the bottom morphism is the multiplicity map for the group $M_I$ :

\begin{align*}\operatorname{mult}_I:[\tau]\mapsto \sum_{\sigma\in \widehat M_{I,\text{tempiric}}} \operatorname{mult}(\tau:\sigma)[\sigma] .\end{align*}

For this, we shall calculate the map

\begin{align*}C_0(\mathfrak a^*_{I,+}, \mathfrak{K}(\operatorname{Ind}_{K_I}^K H_\tau)) \longrightarrow C_0(\mathfrak a^*_{I,+}, \mathfrak{K}(\operatorname{Ind}_{P_I}^G H_\sigma))\end{align*}

which induces the middle arrow in K-theory. Let $F \in C_0(\mathfrak{a}^*_{I, +}, \mathfrak{K}(\operatorname{Ind}_{K_I}^K H_\tau))$ , and let $f_0$ be the element of $C_r^*(G_0; I)$ such that $\rho_{\tau, \nu}(f) = F(\nu)$ and $\rho_{\tau', \nu}(f) = 0$ when $[\tau'] \neq [\tau]$ . By definition, the composition

\begin{align*}C_r^*(G_0; I) \to C_r^*(G;I) \to C_0(\mathfrak{a}^*_{I,+}, \mathfrak{K}(\operatorname{Ind}_{P_I}^G H_\sigma)) \end{align*}

sends $f_0$ to the function $\rho_{\sigma}(f_0)$ defined by $\rho_\sigma(f_0)(\nu) = \rho_{\sigma, \nu}(f_0)$ .

Now, we have the decomposition

\begin{align*}L^2 (K,H_\sigma) &= \bigoplus_{\tau'\in\widehat{K}_I }\operatorname{mult}(\tau': \sigma)\cdot L^2 (K,H_{\tau'}),\\\rho_{\sigma,\nu}(f_0) &= \bigoplus_{\tau'\in\widehat{K}_I }\operatorname{mult}(\tau': \sigma)\cdot \rho_{\tau',\nu}(f_0),\end{align*}

where $n \cdot T$ denotes the direct sum of T with itself n times. It follows that the middle arrow in the above diagram is induced by the map

\begin{align*}F \mapsto \operatorname{mult}(\tau: \sigma) \cdot F,\end{align*}

sending a matrix to the direct sum of that matrix with itself $\operatorname{mult}(\tau: \sigma)$ times. In terms of K-theory, this corresponds to multiplication by $\operatorname{mult}(\tau: \sigma)$ , once appropriate generators corresponding to $[\tau]$ and $[\sigma]$ are fixed.

We now turn to the proof of Theorem 6.11. We will prove it simultaneously with the following theorem, by using the Connes–Kasparov isomorphism and induction on the real rank of the group G (note that $G = K$ when the real rank is 0). We recall that the real rank of G is the dimension of $\mathfrak{a}_{\min}$ .

Proof of Theorems 6.11 and 6.13. Assume that Theorem 6.11 holds for groups with real rank strictly less than r, and suppose that G has real rank r. First, suppose that $I \subset S$ is a proper subset of S. As the group $M_I$ has lower real rank, the morphism

\begin{align*}R(K_I) \to R_{\mathrm{tempiric}}(M_I)\end{align*}

is an isomorphism by the inductive hypothesis. By Theorem 6.12, this means that the $C^*$ -homomorphism $\alpha_{0}: C^*(G_0;I) \to C^*_r(G;I)$ induces an isomorphism on K-theory, so by Theorem 5.24, we obtain that $K_*(C^*_r(\boldsymbol{{G}};I)) = 0$ for $I\subset S$ proper.

Now, for each $0 \leqslant k \leqslant r$ , define

\begin{align*}U_k = \bigcup_{|I| = k} U_I,\end{align*}

where $|I|$ denotes the cardinality of I. Define $C_r^*(\boldsymbol{{G}}; U_k)$ to be the $C^*$ -algebra corresponding to the open set $U_k$ . Then, $U_k \setminus U_{k-1}$ is the disjoint union of closed subsets

\begin{align*}U_k \setminus U_{k-1} = \bigsqcup_{|I| = k} i\mathfrak{a}^*_{I,+},\end{align*}

and so we have the exact sequence

\begin{align*}0 \to C_r^*(\boldsymbol{{G}};U_{k-1}) \to C_r^*(\boldsymbol{{G}} ; U_k) \to \bigoplus_{|I| = k} C_r^*(\boldsymbol{{G}}; I) \to 0.\end{align*}

It therefore follows from the six-term exact sequence in K-theory, and from induction on k, that $K_*(C_r^*(\boldsymbol{{G}}; U_k)) = 0$ for $k \lt r$ .

Now consider $I = S$ . In this case, we note that

\begin{align*}U_r \setminus U_{r-1} = i \mathfrak{a}^*_{S,+},\end{align*}

and $C_r^*(\boldsymbol{{G}}; U_r) = C_r^*(\boldsymbol{{G}})$ , so we have an exact sequence

\begin{align*}0 \to C_r^*(\boldsymbol{{G}}; U_{r-1}) \to C_r^*(\boldsymbol{{G}}) \to C_r^*(\boldsymbol{{G}}; S) \to 0.\end{align*}

As both $C_r^*(\boldsymbol{{G}}; U_{r-1})$ and $C_r^*(\boldsymbol{{G}})$ vanish in K-theory (this is where we use Connes–Kasparov isomorphism), so must $C_r^*(\boldsymbol{{G}}; S)$ , which proves Theorem 6.13 for G. By Theorem 6.12, this also proves Theorem 6.11 for G.

We have used the Connes–Kasparov isomorphism to prove Theorem 6.11. Conversely, by arguing exactly as in the above proof, we can prove that $C_r^*(\boldsymbol{{G}})$ has vanishing K-theory if we are given Theorem 6.11. That is, we have the following.