Introduction↩︎

Let \(G\) be a real reductive group. The Mackey analogy connects the (tempered) representation theory of \(G\) with the (unitary) representation theory of its Cartan motion group. Since its inception by George Mackey in the 1970s as a tentative statement on the parametrization of representations, it has been studied from various points of view, including the theory of operator algebras and its interplay with canonical topologies on tempered duals [1]–[3].

Among the byproducts of this evolution is some insight into the geometry of the tempered dual \(\widehat{G}_{\mathrm{tempered}}\): the space \(\widehat{G}_{\mathrm{tempered}}\) (which is in general not Hausdorff in the Fell topology) can be assembled from a countable collection of contractible convex cones in finite-dimensional vector spaces. There are at least two distinct, but related, ways of doing this: in terms of lowest \(K\)-types, as in [4], or in terms of the imaginary part of the infinitesimal character, as in [5].

For reductive \(p\)-adic groups, similar phenomena occur in the work of Aubert–Baum–Plymen–Solleveld on the geometry of each Bernstein component of the smooth dual [6], [7], showing that each individual component is usually assembled from elementary pieces of a slightly different kind. In this \(p\)-adic setting, a language gradually emerged to describe the phenomena, using a notion of “stratified equivalence” [8] to relate the Hecke algebras attached to Bernstein blocks to much simpler algebras.

In this paper we use the Mackey embedding recently constructed in [9] to recast the Mackey analogy in the language that was conceived for \(p\)-adic groups. We also prove various properties of the Mackey bijection [3] and of the Mackey embedding [9], in relation with various stratifications of the reduced \(\mathrm{C}^*\)-algebra of \(G\).

Before we describe our results, let us provide some background on the Mackey analogy for real groups, and on the \(p\)-adic phenomena that underlie the notion of stratified equivalence.

The Mackey bijection and the Mackey embedding.

Let \(K\) be a maximal compact subgroup of \(G\). The Cartan motion group \(G_0\) attached to \(G\) and \(K\) is the semidirect product \(K \ltimes (\mathfrak{g}/\mathfrak{k})\), built from the adjoint action of \(K\) on the vector space quotient \(\mathfrak{g}/\mathfrak{k}\) of the Lie algebras of \(G\) and \(K\). The group \(G_0\) has the same dimension as \(G\), but a large abelian normal subgroup; therefore its unitary dual \(\widehat{G}_0\) is easy to describe using Mackey’s “little group” method. The tempered dual \(\widehat{G}_{\mathrm{tempered}}\) is more subtle. But a few years ago, building on suggestions of Mackey [1] and Higson [2], one of us described in [3] a natural bijection \[\widehat{G}_0\longleftrightarrow \widehat{G}_{\mathrm{tempered}}\] using Vogan’s notion of lowest \(K\)-types. This “Mackey bijection” is not a homeomorphism in the Fell topology (unless \(G\) and \(G_0\) are isomorphic), but it is a piecewise homeomorphism: for each finite collection \(E\) of \(K\)-types, it induces a homeomorphism between the subsets consisting of the representations whose set of lowest \(K\)-types is exactly \(E\) [4], [10].

The Mackey bijection has long been known to have connections the Connes–Kasparov isomorphism in operator algebras [2], [4], [11]: it was Alain Connes who pointed out in the 1980s that the Connes–Kasparov isomorphism for real reductive groups can be interpreted as a \(K\)-theoretic reflection of Mackey’s ideas. In recent work of another of us with Higson and Román [9], building on ideas introduced in Román’s thesis, the bijection was promoted to a \(\mathrm{C}^*\)-algebraic embedding \[\alpha\colon \mathrm{C}^*(G_0) \longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\] of the maximal \(\mathrm{C}^*\)-algebra of \(G_0\) into the reduced \(\mathrm{C}^*\)-algebra of \(G\). The Mackey bijection is then recovered by taking an irreducible tempered representation of \(G\), viewed as a representation \(\pi\) of \(\mathrm{C}_{\,\mathrm{r}}^*(G)\); composing it with the embedding \(\alpha\); and, viewing the composition \(\pi\circ\alpha\) as a representation of \(G_0\), extracting the irreducible component that contains the lowest \(K\)-types. One important property of the embedding \(\alpha\) is that it induces the Connes–Kasparov isomorphism in K-theory, giving operator-algebraic substance to the idea that \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\) are similar. In this spirit, it is also known [4] that \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) are assembled from very simple common pieces by Morita equivalences and direct limits; as a result, the homeomorphic pieces of \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\) discussed above are assembled in such a way that the two duals share algebraic-topological invariants.

Bernstein components for \(p\)-adic groups; stratified equivalence.

In the past 20 years, but quite independently of the Mackey analogy, conceptually similar phenomena took on an increasingly important role in certain aspects of the representation theory of reductive \(p\)-adic groups. These appear when one focuses on the geometry of a single connected component \(\mathfrak{s}\) of the smooth dual, the space of equivalence classes of irreducible smooth representations. One can attach to the component \(\mathfrak{s}\) a Hecke algebra \(\mathscr{H}_{\mathfrak{s}}\), a complex algebra whose primitive ideal space identifies with \(\mathfrak{s}\). Under some tameness assumptions and in many cases, including the emblematic case where \(\mathfrak{s}\) is the Iwahori component, the algebra \(\mathscr{H}_{\mathfrak{s}}\) is Morita-equivalent to an affine Hecke algebra \(\widetilde{\mathscr{H}}_{\mathfrak{s}}\), whose structure therefore encapsulates much of the representation theory.

It gradually emerged from the work of Lusztig [12] and of Baum–Nistor [13] for the Iwahori component, and of Aubert–Baum–Plymen–Solleveld in general (see the surveys [6], [7]), that \(\widetilde{\mathscr{H}}_{\mathfrak{s}}\) has a remarkable relation with a simpler algebra \(\widetilde{\mathscr{H}}_0\) that is reminiscent of the \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G_0)\) in the real case. This \(\widetilde{\mathscr{H}}_0\) is the group algebra \(\mathbb{C}[W \ltimes X]\), where \(X\) is the character lattice of a compact torus and \(W\) is a finite group acting on \(X\). The algebras \(\widetilde{\mathscr{H}}_{\mathfrak{s}}\) and \(\widetilde{\mathscr{H}}_0\) share the same underlying vector space, but the algebra structure of \(\widetilde{\mathscr{H}}_{\mathfrak{s}}\) is a deformation of that of \(\widetilde{\mathscr{H}}_0\), and much more complicated. Nevertheless, it turns out that the primitive ideal spaces of \(\mathscr{H}_\mathfrak{s}\) and \(\widetilde{\mathscr{H}}_0\) are in natural bijection, and piecewise homeomorphic.

To discuss such phenomena, Baum–Nistor and Aubert–Baum–Plymen–Solleveld introduce a notion of “stratified equivalence”. At its heart is a notion that we shall call “stratified equivalence morphism” below (see Section 2.2 for details): a stratified equivalence morphism between two algebras \(A\) and \(B\) is an algebra homomorphism which respects certain filtrations \((A_i)\), \((B_i)\) of \(A\) and \(B\), and which on subquotients has the special property that the preimage of any primitive ideal in a subquotient \(B_{i}/B_{i-1}\) is contained in a unique primitive ideal of \(A_i/A_{i-1}\). Such a morphism induces a natural bijection between the primitive ideal spaces of \(A\) and \(B\). If \(A\) and \(B\) are finitely generated and the filtrations are finite, Baum and Nistor show that this bijection is a piecewise homeomorphism; more precisely, that it induces a homeomorphism between the spectra of \(B_{i}/B_{i-1}\) \(A_i/A_{i-1}\) for all \(i\), but not a global homeomorphism in general.

Aubert, Baum, Plymen and Solleveld have highlighted the role of this notion in the \(p\)-adic theory. For instance, if \(\mathfrak{s}\) is the Iwahori component of a reductive \(p\)-adic group, then each of \(\mathscr{H}_\mathfrak{s}\) and \(\widetilde{\mathscr{H}}_0\) can be related, via a stratified equivalence morphism, to a version of Lusztig’s asymptotic Hecke algebra. This plays an important role for the discussion of the geometric structure of \(\mathfrak{s}\) in the work of Aubert, Baum, Plymen and Solleveld; and also in the computation of the \(K\)-theory of \(\mathscr{H}_\mathfrak{s}\), which sheds light on the analytic side of the Baum–Connes conjecture. See [7], [14].

Our results.

The striking parallels between the real and \(p\)-adic phenomena described above have been known to the experts for a few years. However, by providing a concrete map at the level of group algebras, the construction of the Mackey embedding \(\alpha\colon \mathrm{C}^*(G_0)\longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) in [9] now makes it possible to study these parallels more formally. In this paper, we shall prove a first result in this direction, and see that it reveals new properties of the Mackey bijection.

In Section 2 we consider two pairs of filtrations for \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\), which we shall denote by \((I_n^0)_{n \in \mathbb{N}}\), \(\left(I_n\right)_{n \in \mathbb{N}}\) and \((\Upsilon^0_k)_{k \in \mathbb{N}}\), \((\Upsilon_{k})_{k \in \mathbb{N}}\) respectively. Both are defined in terms of lowest \(K\)-types. The filtrations \((I_n^0)_{n \in \mathbb{N}}\), \(\left(I_n\right)_{n \in \mathbb{N}}\) of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) are those which appeared in [4] in connection with the Connes–Kasparov isomorphism. The filtrations \((\Upsilon^0_k)_{k \in \mathbb{N}}\) and \((\Upsilon_{k})_{k \in \mathbb{N}}\) are coarser and only keep track of the “height” of lowest \(K\)-types. We prove the following result (see Theorem 8).

::: {#th-a .theorema} Theorem A 1. The Mackey embedding \(\alpha\colon \mathrm{C}^*(G_0)\longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) is a stratified equivalence morphism with respect to the filtrations \(\left(I_n^0\right)_{n \in \mathbb{N}}\) and \(\left(I_n\right)_{n \in \mathbb{N}}\) of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\). It is also a stratified equivalence morphism with respect to the filtrations \(\left(\Upsilon^0_k\right)_{k \in \mathbb{N}}\) and \((\Upsilon_{k})_{k \in \mathbb{N}}\).* :::

The bijection between \(\widehat{G}_{\mathrm{tempered}}\) and \(\widehat{G}_0\) induced by the stratified equivalence property is, in both cases, the Mackey bijection. Once this is proved, the work of Baum and Nistor [13] strongly suggests that the bijection is a piecewise homeomorphism, and induces a homeomorphism between pieces of \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\) corresponding to the spectra of our filtrations’ subquotients. In the case of the filtrations \((I_n^0)_{n \in \mathbb{N}}\), \(\left(I_n\right)_{n \in \mathbb{N}}\) this was already known [4], and amounts to the above statement on the behavior of the Mackey bijection when a fixed set of lowest \(K\)-types is prescribed. But for the coarser filtration, this reveals a stronger property of the Mackey bijection which is, to our knowledge, new. It amounts to the following statement (see Corollary 1).

::: {#th-b .theoremb} Theorem B 1. The Mackey bijection maps each connected component of \(\widehat{G}_{\mathrm{tempered}}\) homeomorphically onto its image in \(\widehat{G}_0\).* :::

Thus the unitary dual \(\widehat{G}_0\) is obtained by gluing together the components of \(\widehat{G}_{\mathrm{tempered}}\) in a rather peculiar manner, that preserves algebraic-topological invariants. Although this result was suggested to us by stratified equivalence, in Section 3 we shall give a direct representation-theoretic proof.

Up to this point our discussion has been focused on two partitions of \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\) according to lowest \(K\)-types, and on the corresponding stratifications of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\). While this is very useful to understand the Mackey bijection, recent work of Bradd, Higson and Yuncken [5] has shown that much further insight on the bijection can be gained by considering another way to partition \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\), according to the “singularity” of the imaginary part of the infinitesimal character. This makes the construction of [3] quite transparent, and in our discussion of the Mackey bijection in Section 1 we will take up notation and ideas from [5].

The partition of \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\) according to the imaginary part of the infinitesimal character also has its own operator-algebraic counterpart. On their way to proving the main results of [5], the authors consider corresponding filtrations of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\). For a given filtration piece, the individual subquotients \(\mathrm{C}^*(G_0; I)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G; I)\) both have spectrum the set of representations whose imaginary part of the infinitesimal character lies on a specific wall in the dominant chamber of all possible imaginary parts. Bradd, Higson and Yuncken point out that these subquotients have a very simple structure up to \(\mathrm{C}^*\)-algebra isomorphism, and that \(\mathrm{C}^*(G_0; I)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G; I)\) are Morita-equivalent. In this way the infinitesimal character leads to a partition of \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\) into yet another set of contractible homeomorphic pieces, assembled differently in both duals.

The proofs of the main results in [5] feature their own versions of the Mackey embedding; these are much simpler than the “global” Mackey embedding \(\alpha\colon \mathrm{C}^*(G_0) \longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) of [9], but they are defined only at the level of the subquotients \(\mathrm{C}^*(G_0; I)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G; I)\) using the very simple structure of these subquotients. Since a large portion of this paper is about the behavior on the Mackey embedding \(\alpha\) on various filtrations of the \(\mathrm{C}^*\)-algebras of \(G_0\) and \(G\), we have taken this opportunity to clarify the relationship between the “global” embedding constructed in [9] and the “local” versions that appear in [5]. This is the theme of Section 4.

Organization of the paper.

Section 1 collects the definitions and results on the tempered dual, and on the Mackey bijection and embedding, that will be used throughout the paper. As mentioned above, the influence of the recent work of Bradd, Higson and Yuncken will be apparent in our choices of notation and presentation.

After that, Sections 2, 3 and 4 can be read more or less independently of one another. Section 2 is about the stratified equivalence property of the Mackey embedding; there we state the appropriate definitions (indicating the slight deviations from those of Baum–Nistor and Aubert–Baum–Plymen–Solleveld), define the filtrations \((\Upsilon_k)_{k \in \mathbb{N}}\) and \(\left(I_n\right)_{n \in \mathbb{N}}\), and prove Theorem A. Section 3 contains the proof of Theorem B. While the result was suggested to us by the inspection of stratified equivalence for the filtrations \((\Upsilon_k)_{k \in \mathbb{N}}\), we give a purely representation-theoretic proof that does not refer to the notions of Section 2. Finally, in Section 4 we discuss the infinitesimal character filtration of Bradd, Higson and Yuncken before proving a comparison theorem between the Mackey embedding \(\alpha\) and the “localized” versions on the subquotients of the infinitesimal character filtration, and collecting various remarks on the (non-)uniqueness properties of the embedding \(\alpha\).

Acknowledgements.

This paper grew out of a discussion that took place on the 28th of February, 2025, during a workshop on tempered representations and \(K\)-theory at Institut Henri Poincaré. We thank all the colleagues who took part in that discussion, in particular Anne-Marie Aubert, Roger Plymen and Maarten Solleveld. We are also indebted to Jacob Bradd, Nigel Higson and Robert Yuncken for several enlightening conversations over the past year; Theorem B was brought on by questions of Bradd and Yuncken.

We extend our gratitude to the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne Université) and its Centre Émile Borel, as well as the project OpART of the Agence Nationale de la Recherche (ANR-23-CE40-0016), for their help and support in relation with the thematic trimester “Representation theory and noncommutative geometry”, and during the preparation of this paper. P. C. is particularly grateful to the Institut Élie Cartan de Lorraine and the Fondation Sciences Mathématiques de Paris for supporting him through visiting professorships.

1 The tempered dual and the Mackey bijection↩︎

1.1 Parameters for the tempered dual of \(G\)↩︎

Throughout the paper, by a real reductive group we shall mean the group of real points of a connected reductive algebraic group defined over \(\mathbb{R}\); and we shall fix throughout a real reductive group \(G\), together with a Cartan involution \(\theta\) of \(G\). We denote by \(K\) the set of fixed-points \(G^\theta\). It is a maximal compact subgroup of \(G\) and we write \(\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}\) for the Cartan decomposition of the Lie algebra \(\mathfrak{g}=\operatorname{Lie}(G)\), where \(\mathfrak{k}\) is the Lie algebra of \(K\). We assume throughout that the Lie algebra \(\mathfrak{g}\) is equipped with a symmmetric, non-degenerate, \(G\)-invariant bilinear form that is positive-definite on \(\mathfrak{p}\) and negative-definite on \(\mathfrak{k}\). This gives positive-definite inner products on each of \(\mathfrak{k}\) and \(\mathfrak{p}\); we shall denote these inner products and their restrictions to various subspaces by \(\left\langle\cdot\,,\,\cdot\right\rangle\), and the corresponding norms by \(\lVert{\cdot}\rVert\), without any further reference. In particular, if \(\mathfrak{a}_1 \subset \mathfrak{a}_2\) are nested subspaces of \(\mathfrak{p}\), any linear functional \(\nu\) on \(\mathfrak{a}_1\) extends uniquely to a linear functional on \(\mathfrak{a}_2\) which is zero on the orthogonal complement of \(\mathfrak{a}_1\) in \(\mathfrak{a}_2\). We shall call this “the” extension of \(\nu\) to \(\mathfrak{a}_2\) without further comment.

We also fix a maximal abelian subspace \(\mathfrak{a}_{\min}\subset \mathfrak{p}\) and a system of positive restricted roots \[\Delta^+(\mathfrak{g},\mathfrak{a}_{\min})\subset\Delta(\mathfrak{g},\mathfrak{a}_{\min}),\] which will remain fixed throughout the paper. They determine an Iwasawa decomposition \[G=KA_{\min}N_{\min}\] where \(A_{\min}=\exp(\mathfrak{a}_{\min})\) and \(N_{\min}=\exp(\mathfrak{\mathfrak{n}_{\min}})\), with \(\mathfrak{n}_{\min}\) defined to be the direct sum of root spaces \(\bigoplus\limits_{\alpha\in\Delta^+(\mathfrak{g},\mathfrak{a}_{\min})}\mathfrak{g}_\alpha\). Finally, we denote by \(M_{\min}\) the centralizer of \(\mathfrak{a}_{\min}\) in \(K\); then \[P_{\min}= M_{\min}A_{\min}N_{\min}\] is a minimal parabolic subgroup of \(G\). Having fixed these elements of notation, let us now introduce parameters that will allow us to describe the tempered representation theory of \(G\).

1.1.1 Imaginary part of the infinitesimal character; tempiric representations↩︎

Let \(S\subset\Delta^+(\mathfrak{g},\mathfrak{a}_{\min})\) denote the set of simple restricted roots. As in [5], subsets of \(S\) will play a central role in the parametrization of the tempered dual of \(G\) that we use below.

A parabolic subgroup of \(G\) is a closed subgroup containing a conjugate of \(P_{\min}\). Such a group \(P\) is called standard if \(P_{\min}\subset P\), and it is well known that standard parabolic subgroups of \(G\) are in natural one-to-one correspondence with subsets of \(S\), as follows. Any subset \(I\subset S\) determines a unique standard parabolic subgroup \(P_I\) with Langlands decomposition \[P_I=M_IA_IN_I\] where \(A_I = \exp(\mathfrak{a}_I)\) for \(\mathfrak{a}_I=\bigcap_{\alpha\in I}\operatorname{Ker}\alpha\), where the Levi component \(L_I=M_IA_I\) of \(P_I\) is the centralizer in \(G\) of \(\mathfrak{a}_I\), where \(M_I\) is the subgroup of \(L_I\) generated by all compact subgroups of \(L_I\), and where the unipotent radical \(N_I\) is \(\exp(\mathfrak{n}_I)\) with \(\mathfrak{n}_I=\bigoplus_{\alpha\in S\setminus I}\mathfrak{g}_\alpha\). See for instance [15] for further details.

We set \[\mathfrak{a}_{\operatorname{dom}}^\ast=\left\lbrace\, \nu\in\mathfrak{a}_{\min}^*\::\:\left\langle\nu\,,\,\alpha\right\rangle\geq0\,,\,\forall \alpha\in S\, \right\rbrace;\] this is the dominant Weyl chamber in \(\mathfrak{a}_{\min}^*\) associated with the given choice of positive roots.

Still following [5] (see Section 2.3 there), for \(I\subset S\), we further define \[\mathfrak{a}_{I, +}^\ast=\left\lbrace\, \nu\in\mathfrak{a}_{\operatorname{dom}}^\ast\::\:\left\langle\nu\,,\,\alpha\right\rangle=0\,,\,\forall \alpha\in I\quad\text{and}\quad\left\langle\nu\,,\,\beta\right\rangle\neq0\,,\,\forall \beta\in S\setminus I\, \right\rbrace.\] We shall need the following properties of the sets \(\mathfrak{a}_{I, +}^\ast\).

(a) For fixed \(I \subset S\), the set \(\mathfrak{a}_{I, +}^\ast\) is contained in \(\mathfrak{a}_I^*\), and we have \[\mathfrak{a}_{I, +}^\ast=\mathfrak{a}_{\operatorname{dom}}^\ast\cap \mathfrak{a}_I^*\setminus\bigcup_{I\subsetneq J} \mathfrak{a}_J^*.\]

(b) The set \(\mathfrak{a}^*_\varnothing\) is the interior of \(\mathfrak{a}_{\operatorname{dom}}^\ast\), whereas \(\mathfrak{a}^*_S\) identifies with the intersection of \(\mathfrak{a}_{\min}\) with the center of \(\mathfrak{g}\).

(c) The sets \(\mathfrak{a}_{I, +}^\ast\) are locally closed in \(\mathfrak{a}_{\operatorname{dom}}^\ast\), and give a partition of \(\mathfrak{a}_{\operatorname{dom}}^\ast\): \[\label{eq-partition-chambre}\mathfrak{a}_{\operatorname{dom}}^\ast=\bigsqcup_{I\subset S}\mathfrak{a}_{I, +}^\ast.\tag{2}\]

Let us now turn to representations. If \(\pi\) is an irreducible unitary representation of \(G\), we shall denote the imaginary part of its infinitesimal character by \[\operatorname{Im\,inf\,ch}(\pi)\in\mathfrak{a}_{\min}^*/W(\mathfrak{g},\mathfrak{a}_{\min}).\] See [5] for details regarding this notion. Although the basic ideas probably date back to Harish-Chandra, the present formulation draws on work of Vogan [16], [17].

For our purposes it will be convenient to view the imaginary part of the infinitesimal character as an element of the chamber \(\mathfrak{a}_{\operatorname{dom}}^\ast\), using the fact that the latter is a fundamental domain for the action of \(W\) on \(\mathfrak{a}_{\min}^\ast\): the inclusion of \(\mathfrak{a}_{\operatorname{dom}}^\ast\) into \(\mathfrak{a}_{\min}^\ast\) induces a bijection

\[\label{eq-chambre-dom} \mathfrak{a}_{\operatorname{dom}}^\ast\overset{\sim}{\longleftrightarrow} \mathfrak{a}_{\min}^\ast/W.\tag{3}\]

Therefore it makes sense to use the decomposition 2 to partition the unitary dual of \(G\) according to which subset \(\mathfrak{a}_{I, +}^\ast\) contains the imaginary part of the infinitesimal character. This is one of the main ingredients of [5].

The following extreme case is crucial to discuss the Mackey bijection.

In Section 1.1.3 below we shall outline a description of the tempered dual \(\widehat{G}_{\mathrm{tempered}}\) in terms of tempiric representations of Levi subgroups. This relies on the following fact, again drawing on work of Vogan, which provides a classification of tempered irreducible representations in terms of the imaginary part of their infinitesimal character. In the statement we shall freely use notation related to parabolic induction; some of the basics will be recalled in Section 1.1.3.

See [5] for a sketch of proof of this result, due to Vogan in a greater generality, and for further comments.

1.1.2 Lowest \(K\)-types↩︎

Alongside the imaginary part of the infinitesimal character, another way to sort unitary representations of \(G\) is to study their \(K\)-types. Let us summarize here the elements of Vogan’s theory of lowest \(K\)-types that we shall use later. See [18], and e.g. [5] or [9], for additional details and references.

Norm-bounded subsets of \(\widehat{K}\) are finite. It follows that every admissible representation has at least one lowest \(K\)-type, and at most finitely many. While the height function \(\lVert{\cdot}\rVert_{\widehat{K}}\) in Definition 2 depends on the chosen inner product and on the chosen set of positive roots, the lowest \(K\)-types of \(\pi\) depend only on \(\pi\), not on the other choices.

1.1.3 Parametrization of the tempered dual↩︎

If \(P\) is a parabolic subgroup of \(G\) with Langlands decomposition \(P=MAN\), and if we are given a tempered representation \(\sigma\) of \(M_I\) and an element \(\nu\) of \(\mathfrak{a}^+\), then the pair \((\sigma,\nu)\) determines a tempered representation \(\sigma\otimes e^{i\nu}\) of the Levi component \(L=MA\), which can be extended as \(\sigma\otimes e^{i\nu}\otimes 1_N\) to \(P\) and unitarily induced to \(G\). The representation \(\operatorname{Ind}_P^G\sigma\otimes e^{i\nu}\otimes 1_N\) obtained in this way is said to be parabolically induced from \((\sigma,\nu)\).

These representations can be realized in various Hilbert spaces – see for instance [20]. We shall use the “compact picture”, which has the advantage that the carrier Hilbert space is independent of the parameter \(\nu\). It is obtained as the set of \(K \cap L\)-equivariant vectors in the space of square-integrable, \(H_\sigma\)-valued functions on \(K\). The action on \(G\) on this space is defined by means of the Iwasawa decomposition, but we shall not need the precise formula. For brevity’s sake, as in [9] we shall denote this space by \[\label{eq-def-Htau} \operatorname{Ind}H_\sigma = \operatorname{L^2} (K, H_\pi)^{K \cap L}.\tag{4}\]

In order to describe the tempered dual of \(G\) in a language suited to our purposes, it will be useful to consider parabolically induced representations labeled by subsets of roots, as in Section 1.1.1.

The following statement amounts to a classification of the irreducible tempered representations of \(G\):

The three assertions in the theorem encapsulate everything that was used in [3] to prove that the construction there provides a bijection between \(\widehat{G}_{\mathrm{tempered}}\) and the unitary dual of \(G_0\) ; see the discussion in [3] for reformulations of the relevant parts of the literature concerning each assertion. The version of the classification formulated in Theorem 3 is implicitly used throughout [5].

1.2 The unitary dual of \(G_0\) and the Mackey bijection↩︎

1.2.1 Unitary dual of the Cartan motion group↩︎

1.2.1.1 Construction of unitary representations of \(G_0\).

In the rest of the paper, we define the Cartan motion group \(G_0\) for \(G\) (and \(K\)) to be the semidirect product \(K \ltimes \mathfrak{p}\) attached to the adjoint action of \(K\) on \(\mathfrak{p}\). Let us recall Mackey’s recipe for constructing unitary representations of \(G_0\), and his classification of irreducible unitary representations.

The main tool is the coadjoint action of \(K\) on the dual vector space \(\mathfrak{p}^\ast\). Suppose given a pair \((\chi, \mu)\) where \(\chi\) is an element of \(\mathfrak{p}^\ast\) and \(\mu\) is a unitary representation of the stabilizer \(K_\chi\), acting on a vector space \(H_\mu\). To such data we attach the unitary representation \[\varrho_{\chi, \mu}=\operatorname{Ind}_{K_\chi \ltimes \mathfrak{p}}^{G_0}(\mu \otimes \chi)\] of \(G_0\), realized in its compact picture, namely the closed subspace of \(\operatorname{L^2}(K, H_\mu)\) defined by \[\operatorname{Ind}H_\mu=\left\{\;f \in \operatorname{L^2}(K, H_\mu) \;: \;f(km)=\mu(m)k \;\text{ for all m \in K_\chi and k \in K}\;\right\},\] with the action of \(G_0\) defined by \(\varrho_{\chi, \mu}(k, v) f = u \longmapsto e^{i \langle k \cdot \chi, u\rangle} f(k^{-1} u)\).

1.2.1.2 The spectral extended quotient \(\mathfrak{p}^\ast\!/\!\!/K\) and its topology.

Let \(\mathscr{D}\) be the set of all pairs \((\chi, \mu)\) where \(\chi\) is an element of \(\mathfrak{p}^\ast\) and \(\mu\) is an irreducible unitary representation of the stabilizer \(K_\chi\).

Define an equivalence relation on \(\mathscr{D}\) by: \((\chi, \mu) \sim (\chi', \mu')\) if and only if there exists \(k \in K\) such that, first, we have \(\chi' = \chi \circ \operatorname{ad}(k)\), and second, the representation \(\mu'\) of \(K_{\chi'}\) on \(H_{\mu'}\) is equivalent to the representation \(u \longmapsto \mu(k^{-1}uk)\) of \(K_{\chi'}\) on \(H_\mu\).

Let \(\mathfrak{p}^\ast \!/\!\!/K\) be the quotient \(\mathscr{D}/\!\!\sim\): it is the spectral extended quotient of Baum and Connes, used by Aubert, Baum, Plymen and Solleveld under various names (see e.g. [6]).

For our purposes, it is important to recall that every \(K\)-orbit in \(\mathfrak{p}\) meets \(\mathfrak{a}_{\min}\), and that the intersection is a single \(W\)-orbit; see [15]. Since \(\mathfrak{a}_{\min}^\ast/W\) identifies with \(\mathfrak{a}_{\operatorname{dom}}^\ast\) as in 3 , it follows that every \(K\)-orbit in \(\mathfrak{p}^\ast\) meets \(\mathfrak{a}_{\operatorname{dom}}^\ast\) in a unique point. If \(\mathscr{D}_{\mathfrak{a}_{\operatorname{dom}}^\ast}\) denotes the subset of \(\mathscr{D}\) consisting of all pairs \((\chi, \mu)\) with \(\chi \in \mathfrak{a}_{\operatorname{dom}}^\ast\), we deduce that the inclusion \(\mathscr{D}_{\mathfrak{a}_{\operatorname{dom}}^\ast} \hookrightarrow \mathscr{D}\) induces a bijection between \(\mathscr{D}_{\mathfrak{a}_{\operatorname{dom}}^\ast}\) and \(\mathfrak{p}^\ast \!/\!\!/K\).

We shall use this to define a topology on \(\mathfrak{p}^\ast \!/\!\!/K\), building on the following observation.

As an immediate consequence of the Lemma, for every \((\chi, \mu) \in \mathscr{D}_{\mathfrak{a}_{\operatorname{dom}}^\ast}\), there exists \(\varepsilon_{\chi, \mu}>0\) with the property that if \(\chi' \in \mathfrak{a}_{\operatorname{dom}}^\ast\) satisfies \(\lVert \chi-\chi'\rVert < \varepsilon_{\chi, \mu}\) then the stabilizer \(K_{\chi'}\) contains \(K_\chi\). We equip \(\mathscr{D}_{\mathfrak{a}_{\operatorname{dom}}^\ast}\) with the topology which is generated, in the sense explained e.g. in [21], by the sets \[\mathscr{O}_{\chi, \mu}^{\varepsilon} = \left\{\;(\chi', \mu) \in \mathscr{D}_{\mathfrak{a}_{\operatorname{dom}}^\ast} \;: \;\lVert \chi-\chi'\rVert < \varepsilon \text{ and } \operatorname{Hom}_{K_{\chi'}}(\mu', \mu) \neq 0\;\right\}\] for \(0 < \varepsilon < \varepsilon_{\chi, \mu}\). Transferring this topology using the bijection \(\mathscr{D}_{\mathfrak{a}_{\operatorname{dom}}^\ast} \leftrightarrow \mathfrak{p}^\ast \!/\!\!/K\) above, we obtain a topology on the spectral extended quotient \(\mathfrak{p}^\ast \!/\!\!/K\).

Classification of the unitary irreducible representations.

The following statement identifies the unitary dual of \(G_0\), equipped with the Fell topology, and the spectral extended quotient \(\mathfrak{p}^\ast\!/\!\!/K\), equipped with the above topology.

Part (a)-(b) are due to Mackey [22]; part (c) is due to Baggett [23], although we have phrased it a little differently—our formulation of (c) can be extracted e.g. from the survey of Echeterhoff and Emerson [24] on the dual spaces of crossed products in an operator-algebraic context.

Reformulation using the imaginary part of the infinitesimal character.

We shall now rephrase the classification in Theorem 4 in a manner parallel to that in Section 1.1.3.

First, let us define an analogue, for irreducible representations of \(G_0\), of the imaginary part of the infinitesimal character. The projection \((\chi, \mu) \longmapsto \chi\), from \(\mathscr{D}\) to \(\mathfrak{p}^\ast\), descends to a surjection \[\operatorname{pr}\colon \mathfrak{p}^\ast \!/\!\!/K\longrightarrow \mathfrak{p}^\ast\!/K\] from the spectral extended quotient \(\mathfrak{p}^\ast \!/\!\!/K\) to the ordinary quotient \(\mathfrak{p}^\ast\!/K\). Now, as mentioned in the discussion leading up to Lemma 1 above, every \(K\)-orbit in \(\mathfrak{p}^\ast\) meets \(\mathfrak{a}_{\min}^\ast\) in a single \(W\)-orbit, and therefore meets \(\mathfrak{a}_{\operatorname{dom}}^\ast\) in a single point. Combining these observations, we obtain a canonical surjection \[\mathfrak{p}^\ast \!/\!\!/K \overset{\operatorname{pr}}{\longrightarrow} \mathfrak{p}^\ast\!/K \simeq \mathfrak{a}_{\min}^\ast/W \overset{\simeq}{\longrightarrow} \mathfrak{a}_{\operatorname{dom}}^\ast,\] where the final bijection is given by 3 . Therefore Theorem 4 determines a canonical surjection \[\operatorname{Im\,inf\,ch}\colon \widehat{G}_0\longrightarrow \mathfrak{a}_{\operatorname{dom}}^\ast.\] Given an irreducible representation \(\varrho\) of \(G_0\), we say that the corresponding element of \(\mathfrak{a}_{\operatorname{dom}}^\ast\) is the imaginary part of the infinitesimal character of \(\varrho\). For the connection with the usual notion of infinitesimal character, and for remarks on the absence of a “real part of the infinitesimal character” in contrast to the situation for reductive groups, see [5].

As in Section 1.1.3 this leads to a partition of \(\widehat{G}_0\) according to the singularity of the imaginary part of the infinitesimal character. Recall the partition 2 of \(\mathfrak{a}_{\operatorname{dom}}^\ast\) into subsets \(\mathfrak{a}_{I, +}^\ast\) indexed by subsets \(I \subset S\) of the set of simple roots. Given a subset \(I\) of \(S\), a unitary representation \(\mu\) of the compact group \(K_I = M_I \cap K\), and an element \(\nu\) of \(\mathfrak{a}_{I, +}^\ast\), we shall denote by \[\label{eq-def-rho-imunu} \varrho_{I, \mu, \nu} \colon G_0 \longrightarrow \operatorname{End}\left(\operatorname{Ind}\; H_\mu\right)\tag{6}\] the representation introduced as \(\varrho_{\nu^\ast, \mu}\) above, for the unique element \(\nu^\ast\) of \(\mathfrak{p}^\ast\) which extends \(\nu\) and is zero on the orthogonal complement of \(\mathfrak{a}_{\min}\).

For later purposes, we mention that in 6 the unitary representation \(\mu\) of \(K_I\) is allowed to be reducible. But the case where \(\mu\) is irreducible is of course important, and we shall often use the following reformulation of parts (a) and (b) in Theorem 4.

1.2.2 The Mackey bijection↩︎

With the above preparations, it is easy to define the Mackey bijection \(\widehat{G}_0\longrightarrow \widehat{G}_{\mathrm{tempered}}\).

It follows from Theorem 5 that every unitary irreducible representation of \(G_0\) is equivalent to a representation \(\varrho_{I, \mu, \nu}\) for an essentially unique triple \((I, \mu, \nu)\). On the other hand, it follows from Theorem 2 that every irreducible representation \(\mu\) of \(K_I\) arises as the unique lowest \(K_I\)-type of a tempiric representation \(\tau(\mu)\) of \(M_I\), and the equivalence class of \(\tau(\mu)\) is determined by that of \(\mu\).

The following theorem, originally established in [3] with a slightly different formulation, is now an immediate consequence of Theorems 3 and 5:

It is clear that \(\mathscr{M}\) preserves the imaginary part of the infinitesimal character, in the sense that \(\operatorname{Im\,inf\,ch}(\mathscr{M}(\pi))=\operatorname{Im\,inf\,ch}(\pi)\) for every \(\pi \in \widehat{G}_0\). It is less obvious, but no less true, that it preserves the lowest \(K\)-types of representations: as in Section 1.1.2, to any element \(\pi\) of \(\widehat{G}_0\) we associate the finite subset \(\operatorname{LKT}_{G_0}(\pi) \subset \widehat{K}\) of lowest \(K\)-types of \(\pi\); then [3] says that \[\label{eq-mackey-preserves-lkt}\operatorname{LKT}_G\big(\mathscr{M}(\pi)\big)=\operatorname{LKT}_{G_0}(\pi)\tag{7}\] for every \(\pi \in \widehat{G}_0\).

1.3 The Mackey embedding of \(\mathrm{C}^*\)-algebras↩︎

For this section we need to interpret the (tempered) representations of \(G\) and \(G_0\) as representations of certain \(\mathrm{C}^*\)-algebras. We shall refer to Dixmier [25] and Bekka, de la Harpe and Valette [26] for the details of the connection. For our purposes it will suffice to say that every unitary (resp. tempered) representation \(\pi\) of \(G_0\) (resp. \(G\)), acting on a complex Hilbert space \(H\), can be “integrated” to a nondegenerate representation of the universal \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G_0)\) (resp. of the reduced \(\mathrm{C}^*\)-algebra \(\mathrm{C}_{\,\mathrm{r}}^*(G)\)), acting on the same Hilbert space \(H\). We shall still denote by \(\pi\) the corresponding \(\mathrm{C}^*\)-morphism from \(\mathrm{C}^*(G_0)\) (resp. \(\mathrm{C}_{\,\mathrm{r}}^*(G)\)) to the \(\mathrm{C}^*\)-algebra \(\operatorname{\mathfrak{B}}(H)\) of all continuous linear endomorphisms of \(H\). This procedure preserves irreducibility, unitary equivalence, and produces all nondegenerate representations of \(\mathrm{C}^*(G_0)\) or \(\mathrm{C}_{\,\mathrm{r}}^*(G)\). In this way the spectra of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) (i.e. the sets of equivalence classes of irreducible representations) are identified with \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\), respectively. We shall often make such identifications without further comment.

Perhaps the main result of [9] is the construction of a natural embedding of \(\mathrm{C}^*(G_0)\) into \(\mathrm{C}_{\,\mathrm{r}}^*(G)\), as follows.

(i) $\varrho_{I, \tau, \nu}$ contains $\varrho_{I, \mu, \nu}$ with
    multiplicity one,

(ii) the set of lowest $K$-types in $\varrho_{I, \tau, \nu}$ is
     exactly $\operatorname{LKT}\left(\pi_{I, \tau, \nu}\right)$,
     and

(iii) these lowest $K$-types all appear in the irreducible summand
      $\varrho_{I, \mu, \nu}$ of $\varrho_{I, \tau, \nu}$.

For all this see [@Mackey_bijection] and [@CHR].

Throughout the rest of the paper we fix an embedding \(\alpha\colon \mathrm{C}^*(G_0)\longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) with the property in Theorem 7. In Section 4 we shall moreover assume that it is obtained from the construction in [9], yielding continuous families \(\left(\mathscr{B}_{\tau, \nu}\right)_{\nu \in \mathfrak{a}_I^\ast}\) of intertwining operators as in Remark 1(c). See Section 4.2, as well as Section 4.3 for related remarks on the (non-)uniqueness of an embedding \(\alpha\) that satisfies 8 .

2 The Mackey embedding as a stratified equivalence↩︎

2.1 Basics on the Jacobson and Fell topologies↩︎

In Section 1.3 we recalled that the duals \(\widehat{G}_{\mathrm{tempered}}\) and \(\widehat{G}_0\) can be identified with the spectra of the reduced \(\mathrm{C}^*\)-algebra \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) and the universal \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G_0)\), respectively.

In this section we collect basic facts on the interplay between the structure of a \(\mathrm{C}^*\)-algebra \(A\) and the Jacobson topology on its spectrum. We will use these when \(A\) is either \(\mathrm{C}^*(G_0)\) or \(\mathrm{C}_{\,\mathrm{r}}^*(G)\), in order to view the locally closed subsets of \(\widehat{G}_0\) or \(\widehat{G}_{\mathrm{tempered}}\) as the spectra of certain subquotients of \(\mathrm{C}^*(G_0)\) or \(\mathrm{C}_{\,\mathrm{r}}^*(G)\). As an application we shall outline a proof that containment of a given \(K\)-type defines an open condition in the unitary dual of \(G\) or \(G_0\). Readers who already know these facts can safely skip this section.

2.1.1 Primitive ideals and the Fell topology↩︎

Let \(A\) be a \(\mathrm{C}^*\)-algebra. By the spectrum of \(A\) we shall mean the set of equivalence classes of irreducible representations of \(A\), as in [25]; we denote it by \(\widehat{A}\) or \(\operatorname{Spec}(A)\). By a primitive ideal of \(A\), we shall mean the annihilator \(\{ f \in A \;: \;\pi(f)=0\}\) of an irreducible representation \(\pi\) of \(A\); this annihilator depends only on the equivalence class of the representation \(\pi\). We shall denote by \(\operatorname{Prim}(A)\) the set of all primitive ideals, as in [25], and equip it with the Jacobson topology [25]. We shall identify \(\operatorname{Prim}(A)\) with \(\widehat{A}\) as in [25].

By definition of the Jacobson topology, the closed subsets of \(\widehat{A}\) then correspond to closed (two-sided) ideals in \(A\): if \(X \subset \widehat{A}\) is closed, then \[J_X = \{\;f \in A \;: \;\forall \pi \in X, \pi(f)=0\;\}\] is a closed ideal in \(A\), and every closed ideal arises in this way.

Given a closed ideal \(J\) in \(A\), every irreducible representation of \(J\) extends uniquely to an irreducible representation of \(A\), and this determines a homeomorphism between \(\widehat{J}\) and the open subset \(\widehat{A}^J\) of representations that do not vanish on \(J\): see [25]. In this way, if \(J \subset J' \subset A\) are nested closed ideals in \(A\), then the spectra of \(J\) and \(J'\) correspond to nested open subsets in the spectrum of \(A\), and we shall identify the spectrum of \(J'/J\) with the locally closed subset of \(\widehat{A}\) given by the set-theoretic difference of these two open subsets. If \(\pi\) is a representation of \(A\) whose equivalence class belongs to that subset, we shall also use the letter \(\pi\) to denote its restriction to \(J'\), or the representation of \(J'/J\) obtained from the latter by passing to the quotient.

2.1.2 Containment of \(K\)-types as an open condition↩︎

Recall that if \(G\) is a locally compact group, the unitary dual \(\widehat{G}\) identifies with the spectrum of the universal \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G)\); this identifies the Fell topology on \(\widehat{G}\) with the Jacobson topology on the spectrum of \(\mathrm{C}^*(G)\). As an illustration of the ideas in Section 2.1.1, we indicate an application to unitary group representations which is widely known, but perhaps not that easy to locate in the literature, and important for our later purposes.

2.2 Stratified equivalence morphisms for topological \(\mathbb{C}\)-algebras↩︎

This subsection collects definitions that, up to minor variation, arose from the study of certain Hecke algebras attached to Bernstein components of the smooth dual of \(p\)-adic groups: for this see the work of Lusztig [12], Baum–Nistor [13], and Aubert–Baum–Plymen–Solleveld (especially [8]).

Now suppose \(A\) is a topological \(\mathbb{C}\)-algebra. By a filtration of \(A\) we shall mean an increasing family \(\mathcal{J}=(J_n)_{n \in \mathbb{N}}\) of closed ideals, with \(J_0=\{0\}\), such that the union \(\bigcup_{n \in \mathbb{N}} J_n\) is dense in \(A\). If there exists \(n \in \mathbb{N}\) such that \(J_n=A\), we shall say the filtration \(\mathcal{J}\) is finite, and call number of pieces of \(\mathcal{J}\) the smallest integer \(n\geq 1\) such that \(J_n=A\).

In the rest of this paper, we shall use the notions of spectrum-preserving and stratified equivalence morphisms in the context of \(\mathrm{C}^*\)-algebra morphisms. These are of course also homomorphisms of topological \(\mathbb{C}\)-algebras, but in the \(\mathrm{C}^*\)-algebraic context the notion of primitive ideal is that of Section 2.1, and therefore slightly different from the more algebraic version used in Definition 4. However, using the \(\mathrm{C}^*\)-algebraic notion of primitive ideal, Definitions 4 and 5 apply verbatim, and yield the notions of spectrum-preserving and stratified equivalence morphisms which we shall use in the rest of this paper.

2.3 Two stratifications of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) by lowest \(K\)-types↩︎

2.3.1 Stratification by the height of the lowest \(K\)-types↩︎

Recall that on each of \(\widehat{G}_{\mathrm{tempered}}\) and \(\widehat{G}_0\), we defined a “height of the lowest \(K\)-types” function \[\begin{align} \pi & \longmapsto \lVert{\operatorname{LKT}(\pi)}\rVert_{\widehat{K}} \end{align}\] in Section 1.1.2. The height functions attached to \(\widehat{G}_{\mathrm{tempered}}\) and \(\widehat{G}_0\) have the same image in \(\mathbb{R}_{\geq 0}\). It is a discrete, locally finite set \(\Lambda\) of real numbers, and we shall write \[\Lambda = \{ R_1, \ldots, R_k, \ldots \}\] where \((R_k)_{k \geq 1}\) is a strictly increasing sequence of nonnegative real numbers. To simplify our later notation, we set \(R_0=-1\) (any negative real number would do).

We shall define filtrations of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) according to the height of lowest \(K\)-types. For any nonnegative integer \(k\), let \({\mathscr{V}}_k\) be following the subset of \(\widehat{G}_{\mathrm{tempered}}\): \[\label{eq-def-vk} \mathscr{V}_k=\left\{\;\pi \in \widehat{G}_{\mathrm{tempered}}\;: \;\lVert{\operatorname{LKT}(\pi)}\rVert_{\widehat{K}} > R_k\;\right\}.\tag{9}\] Consider its annihilator \({\Upsilon}_k\) in \(\mathrm{C}_{\,\mathrm{r}}^*(G)\): \[{\Upsilon}_k = \left\{\;f \in \mathrm{C}_{\,\mathrm{r}}^*(G)\;: \;\forall \pi \in {\mathscr{V}}_{k}, \; \pi(f)=0 \;\right\}.\]

Similarly, on the \(G_0\)-side let \({\mathscr{V}}_k^0\) be the subset of \(\widehat{G}_0\) which comprises the irreducible representations \(\pi\) such that \(\lVert{\operatorname{LKT}(\pi)}\rVert_{\widehat{K}} > R_k\), and let \({\Upsilon}^0_k\) be the annihilator of \(\mathcal{V}_k^0\) in \(\mathrm{C}^*(G_0)\).

2.3.2 Stratification by the possible sets of lowest \(K\)-types↩︎

In this section, we define another pair of filtrations of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) which keep track of the precise sets of lowest \(K\)-types of representations, not just the height of the lowest \(K\)-types. As a result, these filtrations will be finer than the ones in Proposition 1.

Let \(\mathscr{E}\) be the collection of subsets of \(\widehat{K}\) that arise as \(\operatorname{LKT}(\pi)\) for at least one irreducible tempered representation \(\pi\) of \(G\). In the notation of § 1.1.2, on any given element \(E\) of \(\mathscr{E}\), the function \(\lambda \longmapsto \lVert{\lambda}\rVert_{\widehat{K}}\) is constant; in other words the image \(\lVert{E}\rVert_{\widehat{K}}\) consists of a single number.

The set \(\mathscr{E}\) is countable; we shall fix a labelling \[\mathscr{E} = \{E_1, E_2, \ldots \}\] satisfying the following conditions:

(i) If \(\lVert{E_p}\rVert_{\widehat{K}} < \lVert{E_q}\rVert_{\widehat{K}}\), then \(p<q\);

(ii) If \(\lVert{E_p}\rVert_{\widehat{K}}=\lVert{E_q}\rVert_{\widehat{K}}\) and \(\operatorname{Card}(E_p)>\operatorname{Card}(E_q)\), then \(p<q\).

For any positive integer \(n\), let \(\widetilde{\mathscr{W}}_n\) be the following subset of \(\widehat{G}_{\mathrm{tempered}}\): \[\widetilde{\mathscr{W}}_n = \left\{\;\pi \in \widehat{G}_{\mathrm{tempered}}\;: \;\pi_{|K} \not\supset E_n\;\right\}.\] Consider its annihilator \(J_n\) in \(\mathrm{C}_{\,\mathrm{r}}^*(G)\): \[J_n = \left\{\;f \in \mathrm{C}_{\,\mathrm{r}}^*(G)\;: \;\forall \pi \in \widetilde{\mathscr{W}}_{n}, \; \pi(f)=0\;\right\}.\] This an ideal of \(\mathrm{C}_{\,\mathrm{r}}^*(G)\): it is the ideal denoted by \(\mathbf{J}[n]\) in [4]. Accordingly \(\widetilde{\mathscr{W}}_n\) is a closed subset of \(\widehat{G}_{\mathrm{tempered}}\). We set \(J_{0}=0\) and \(\widetilde{\mathscr{W}}_0 = \widehat{G}_{\mathrm{tempered}}\).

For any nonnegative integer \(n\), we now set \[{I}_n = J_0 + \cdots + J_n;\] this is the ideal denoted by \(\mathbf{I}[n]\) in [4]. In representation-theoretic terms, set \[\label{eq-def-wn}\mathscr{W}_n= \bigcap \limits_{j \leq n}\widetilde{\mathscr{W}}_j.\tag{10}\] This is the closed set of those \(\pi \in \widehat{G}_{\mathrm{tempered}}\) such that \(\pi_{|K}\) does not contain any of the sets \(E_j\) for \(j \leq n\); or equivalently, such that \(\operatorname{LKT}(\pi)\) does not contain any of the \(E_j\), \(j \leq n\). Then \(I_n\) is the annihilator of \(\mathscr{W}_n\): \[I_n = \left\{\;f \in \mathrm{C}_{\,\mathrm{r}}^*(G)\;: \;\forall \pi \in \mathscr{W}_n, \; \pi(f)=0\;\right\}.\] The spectrum of \(I_n\) is the open subset of \(\widehat{G}_{\mathrm{tempered}}\) whose elements are the irreducible tempered representations \(\pi\) such that \(\operatorname{LKT}(\pi)\) contains \(E_j\) for some \(j \leq n\). Therefore the union of the spectra of all ideals \(I_n\) is all of \(\widehat{G}_{\mathrm{tempered}}\), and as a result the union of all ideals \(I_n\) is dense in \(\mathrm{C}_{\,\mathrm{r}}^*(G)\). Since it is clear that \(I_n \subset I_{n+1}\) for all \(n\), we see that \(\left(I_n\right)_{n \in \mathbb{N}}\) is an increasing filtration of \(\mathrm{C}_{\,\mathrm{r}}^*(G)\). The subquotients of this filtration satisfy \[\operatorname{Spec}(I_n/I_{n-1}) = \operatorname{LKT}_G^{-1}(E_n), \quad \forall n \geq 1.\] Switching to the Cartan motion group, define ideals in \(\mathrm{C}^*(G_0)\) by setting \(J_0^0=0\) and \[J^0_n = \left\{\;f \in \mathrm{C}^*(G_0)\;: \quad \forall \pi \in \widehat{G}_0\:,\: \left( \pi_{|K} \not\supset E_n \right) \implies \pi(f)=0\;\right\}\] for \(n \geq 1\); then set \[I_n^0 =J_0^0+\cdots + J_n^0\] for \(n \geq 0\). This is an ideal of \(\mathrm{C}^*(G_0)\), and we define \(\mathscr{W}_n^0\) to be the closed subset of \(\widehat{G}_0\) with annihilator \(I_n^0\). The union of the sets \(\mathcal{W}_n^0\) is all of \(\widehat{G}_0\), and \((I_{n}^0)_{n \in \mathbb{N}}\) is an increasing filtration of \(\mathrm{C}^*(G_0)\).

2.4 Proof of the stratified equivalence property↩︎

We shall prove that the Mackey embedding \(\alpha\colon \mathrm{C}^*(G_0) \longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) is a stratified equivalence morphisms for the two sets of filtrations of \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) defined in Sections 2.3.1 and 2.3.2. The proofs are very similar, but neither of the results for one set of filtrations implies the result for the other set of filtrations. Therefore we shall give parallel statements and proofs.

The first step is to prove that the Mackey embedding respects the two sets of filtrations:

This will be an easy consequence of the following more precise statement about the composition of any irreducible tempered representation \(\pi\) of \(G\) (viewed as a representation of \(\mathrm{C}_{\,\mathrm{r}}^*(G)\)) with the Mackey embedding. The statement uses the notation of Section 2.3, and in particular the families \((\mathcal{V}_k)_{k \in \mathbb{N}}\) and \((\mathcal{W}_n)_{n \in \mathbb{N}}\) of closed subsets of \(\widehat{G}_{\mathrm{tempered}}\) defined in 9 and 10 .

We have now proved that the Mackey embedding \(\alpha\colon \mathrm{C}^*(G_0) \longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) preserves the filtrations of Sections 2.3.1 and 2.3.2, and turn to the proof that it induces spectrum-preserving morphisms on subquotients. For \(k, n \geq 1\), write \[\begin{align} \alpha_k^\Upsilon\colon \Upsilon_k^0/\Upsilon_{k-1}^0 & \longrightarrow \Upsilon_k/\Upsilon_{k-1}\\ \intertext{and} \alpha_n\colon I_n^0/I_{n-1}^0 & \longrightarrow I_n/I_{n-1} \end{align}\] for the \(\mathrm{C}^*\)-morphisms induced by \(\alpha\).

Recall that the spectra of \(\Upsilon_{k}^0/\Upsilon_{k-1}^0\) and \(\Upsilon_{k}/\Upsilon_{k-1}\) can be identified with the subsets of \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\) comprising the representations whose lowest \(K\)-types have height exactly \(R_k\). Since the Mackey bijection preserves lowest \(K\)-types, it induces a bijection betwen these spectra. The same applies to the spectra of \(I^0_n/I^0_{n-1}\) and \(I_{n}/I_{n-1}\), which can be identified with the subsets of \(\widehat{G}_0\) and \(\widehat{G}_{\mathrm{tempered}}\) comprising the representations whose set of lowest \(K\)-types is exactly \(E_n\).

The key fact to prove the proposition is:

Let us now prove Proposition 1. We shall prove (b); part (a) is obtained by the exact same argument.

The first thing to be checked is that in the notation of Lemma 4, the only primitive ideal of \(\Upsilon_k^0/\Upsilon_{k-1}^0\) that contains (\(\alpha^\Upsilon_k)^{-1}(V)\) is \(V_0\). Let \(W_0\) be a primitive ideal in \(\Upsilon_k^0/\Upsilon_{k-1}^0\) that contains the ideal \((\alpha^\Upsilon_k)^{-1}(V)\). By Lemma 4 this means \(V_0 \subset W_0\). But the primitive ideals of \(\Upsilon^0_k/\Upsilon_{k-1}^0\) are all maximal among closed ideals: this follows from [25], because the \(\mathrm{C}^*\)-algebra \(\Upsilon^0_k/\Upsilon_{k-1}^0\) is liminal in the sense of [25]. To see that it is liminal, recall that \(\Upsilon^0_k/\Upsilon_{k-1}^0\) is a subquotient of \(\mathrm{C}^*(G_0)\), and every subquotient of a liminal \(\mathrm{C}^*\)-algebra is again liminal [25]; since \(\mathrm{C}^*(G_0)\) is liminal — which can be seen e.g. by combining [2] and [27] — this implies our statement. Therefore the inclusion \(V_0 \subset W_0\) can hold only if \(V_0 = W_0\).

The final thing to be checked is that the map \(\pi \longmapsto \varrho\), from the spectrum of \(\Upsilon_{k}/\Upsilon_{k-1}\) to the spectrum of \(\Upsilon^0_{k}/\Upsilon^0_{k-1}\), is bijective. But since this map can be identified with the restriction of the Mackey bijection to the subsets of \(\widehat{G}_{\mathrm{tempered}}\) and \(\widehat{G}_0\) where the lowest \(K\)-types of representations have height \(R_k\), the result follows from the fact that the Mackey bijection preserves lowest \(K\)-types. This concludes the proof.0◻
We have obtained:

In both (a) and (b) the bijection \(\widehat{G}_{\mathrm{tempered}}\longrightarrow \widehat{G}_0\) attached to \(\alpha\) by Remark 1 is, of course, the Mackey bijection: this follows immediately from Lemma 4.

3 Topological properties of the Mackey bijection↩︎

In this section, we prove that for all \(k \geq 1\), the Mackey bijection induces a homeomorphism between the spectra of \(\Upsilon_k^0/\Upsilon_{k-1}^0\) and that of \(\Upsilon_k/\Upsilon_{k-1}\), and similarly that for all \(n \geq 1\), it induces a homeomorphism between the spectra of \(I_n^0/I_{n-1}^0\) and \(I_n/I_{n-1}\).

If \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) were finitely generated as \(\mathbb{C}\)-algebras, this would be an immediate consequence of the work of Baum and Nistor [13], as in Remark 1. However, the algebras \(\mathrm{C}^*(G_0)\) and \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) are not finitely generated, so Baum and Nistor’s “automatic homeomorphism” theorem does not apply in our context. We do not know whether a similar property holds without the finite-type hypothesis in a \(\mathrm{C}^*\)-algebraic framework.

Nevertheless, it is already known that the Mackey bijection does induce a homeomorphism between the the spectrum of \(I_n^0/I_{n-1}^0\) and that of \(I_n/I_{n-1}\): see [4].

The corresponding statement about the spectra of \(\Upsilon_k^0/\Upsilon_{k-1}^0\) and \(\Upsilon_k/\Upsilon_{k-1}\) is strictly stronger than [4]. It gives further insight into the relationship between the geometries of \(\widehat{G}_{\mathrm{tempered}}\) and \(\widehat{G}_0\), and to the best of our knowledge it was not known. We shall give a purely representation-theoretic proof here.

The theorem has the following easy consequence, which we highlighted as Theorem B in the introduction.

Let us also point out that Theorem 9 implies the corresponding statement for the spectra of \(I_n^0/I_{n-1}^0\) and \(I_n/I_{n-1}\), which was already known [4].

4 Localizations of the Mackey embedding↩︎

The definition of the Mackey embedding \(\alpha\colon \mathrm{C}^*(G_0) \longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) in [9] is somewhat intricate, and necessitates rather delicate properties of intertwining operators. In contrast, the recent work of Bradd, Higson and Yuncken features apparently much simpler \(\mathrm{C}^*\)-morphisms that are analogues of the Mackey embedding, but are defined only on certain subquotients of \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) and \(\mathrm{C}^*(G_0)\)—for filtrations of these two algebras that are defined in terms of the imaginary part of the infinitesimal character. These “local” versions of the embedding are key ingredients for the proofs of the main results of [5].

In this section we compare the global Mackey embedding of [9] and the local versions in [5], and reflect on the (non-)uniqueness of an embedding of \(\mathrm{C}^*(G_0)\) into \(\mathrm{C}_{\,\mathrm{r}}^*(G)\) satisfying the spectral property 8 .

4.1 The infinitesimal character filtration; localized embeddings↩︎

4.1.1 The stratification and subquotients for \(G\)↩︎

For \(I \subset S\), define an open subset \({U}_{I}\) of \(\mathfrak{a}_{\operatorname{dom}}^\ast\) as in [5], by \[{U}_{I}= \bigcup_{J \subset I} \mathfrak{a}_{I, +}^\ast.\] Because the map \(\operatorname{Im\,inf\,ch}\colon \widehat{G}_{\mathrm{tempered}}\longrightarrow \mathfrak{a}_{\operatorname{dom}}^\ast\) is continuous [5], the set \(\operatorname{Im\,inf\,ch}^{-1}(U_I)\) is open in \(\widehat{G}_{\mathrm{tempered}}\).

In fact, the following observation of Bradd, Higson and Yuncken shows that for each \(I \subset S\), the subquotient \(\mathrm{C}_{\,\mathrm{r}}^*(G; I)\) has a very simple structure, and that its spectrum can be identified with \(\widehat{M}_{I, \mathrm{tempiric}}\times \mathfrak{a}_{I, +}^\ast\) as a topological space, not just as a set.

In the statement, the notation \(\operatorname{\mathfrak{K}}(\operatorname{Ind}\, H_\tau)\) means the \(\mathrm{C}^*\)-algebra of compact operators on the Hilbert space \(\operatorname{Ind}\, H_\tau\) from 4 , the notation \(\mathscr{C}_0\left(\mathfrak{a}_{I, +}^\ast, \operatorname{\mathfrak{K}}\left(\operatorname{Ind}H_\tau\right)\right)\) means the \(\mathrm{C}^*\)-algebra of continuous functions \(\mathfrak{a}_{I, +}^\ast\longrightarrow \operatorname{\mathfrak{K}}\left(\operatorname{Ind}H_\tau\right)\) that vanish at infinity, and the \(\oplus\) sign means the \(\mathrm{C}^*\)-algebraic direct sum.

4.1.2 The analogues for \(G_0\)↩︎

For every subset \(I\) of \(S\), the ideal \(\mathrm{C}_{\,\mathrm{r}}^*(G; {U}_{I})\) and the subquotient \(\mathrm{C}_{\,\mathrm{r}}^*(G; I)\) have counterparts in the \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G_0)\). The infinitesimal character map \(\operatorname{Im\,inf\,ch}\colon \widehat{G}_0\longrightarrow \mathfrak{a}_{\operatorname{dom}}^\ast\) is, again, continuous [5]. Thus we may consider the ideal \(\mathrm{C}^*(G_0; {U}_{I}) \subset \mathrm{C}^*(G)\) defined as the annihilator of \(\widehat{G}_0\setminus \operatorname{Im\,inf\,ch}^{-1}({U}_{I})\), and the quotient \(\mathrm{C}^*(G_0; I)\) of \(\mathrm{C}^*(G_0; {U}_{I})\) by the ideal of all elements that vanish in every representation \(\varrho\) with \(\operatorname{Im\,inf\,ch}(\varrho) \in {U}_{I}\setminus \mathfrak{a}_{I, +}^\ast\). Thus the spectrum of \(\mathrm{C}^*(G_0; {U}_{I})\) may be identified with the set of elements \(\varrho \in \widehat{G}_0\) such that \(\operatorname{Im\,inf\,ch}(\varrho) \in {U}_{I}\), and the spectrum of \(\mathrm{C}^*(G_0; I)\) with the set of all representations \(\varrho_{I, \mu, \nu}\) for \(\mu \in \widehat{K}_{I}\) and \(\nu \in \mathfrak{a}_{I, +}^\ast\). We shall take up the same notational conventions as in Remark 1 concerning the identification of \(\varrho_{I, \mu, \nu}\) as a representation of \(G\), of \(\mathrm{C}^*(G_0; {U}_{I})\), or of \(\mathrm{C}^*(G_0; I)\).

Again, the subquotients \(\mathrm{C}^*(G_0; I)\) have a very simple structure: there is a unique \(\mathrm{C}^*\)-algebra isomorphism \[\label{eq-isom-cstarGzero-i} \mathscr{F}^0_I\colon \mathrm{C}^*(G_0; I) \longrightarrow \;\bigoplus \limits_{\mu \in \widehat{K}_{I}}\mathscr{C}_0\left(\mathfrak{a}_{I, +}^\ast, \operatorname{\mathfrak{K}}\left(\operatorname{Ind}H_\mu\right)\right)\tag{22}\] such that, for any element \(f\) in \(\mathrm{C}^*(G_0; I)\), the \(\mu\)-component of the Fourier transform \(\mathscr{F}_I^0(f)\) is the function \(\nu \longmapsto \varrho_{I, \mu, \nu}\). For all this, see [5].

4.1.3 Local version of the Mackey embedding↩︎

4.2 The compatibility between the global and local embeddings↩︎

Henceforth we shall assume that the Mackey embedding \(\alpha\colon \mathrm{C}^*(G_0)\longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) is constructed as in [9]. Therefore it comes with continuous families of intertwining operators, as in Remark 1: for each \(I \subset S\) and every tempiric representation \(\tau\) of \(M_I\), the construction in \(\cite{CHR}\) provides a continuous family \((\mathcal{B}_{\tau,\nu})_{\nu \in \mathfrak{a}_I^\ast}\) of unitary automorphisms of \(\operatorname{Ind}H_\tau\) such that \[\label{eq-spectral-CHR-refined} \pi_{I, \tau, \nu} \left( \alpha (g)\right) = \mathscr{B}_{\tau, \nu}\;\varrho_{I, \tau, \nu}(g) \;\mathscr{B}_{\tau, \nu}^{-1} \qquad \forall g \in \mathrm{C}^*(G_0).\tag{23}\]

For each \(I \subset S\), we may use these families to construct an automorphism of \(\mathrm{C}_{\,\mathrm{r}}^*(G; I)\), as follows. Calling in the Fourier transform \(\mathscr{F}_I\) in Lemma 5, we define \(\mathscr{B}_I\colon \mathrm{C}_{\,\mathrm{r}}^*(G; I) \longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G; I)\) to be the unique map with the following property: for every \(\varphi \in \mathrm{C}_{\,\mathrm{r}}^*(G; I)\) and \(\tau \in \widehat{M}_{I, \mathrm{tempiric}}\), if we denote by \(\widehat{\varphi}_\tau\) the \(\tau\)-summand of \(\mathscr{F}_I(\varphi)\), then the \(\tau\)-summand of \(\mathscr{F}_I(\mathcal{B}_{I}(\varphi))\) is the function \[\begin{align} \mathfrak{a}_{I, +}^\ast& \longrightarrow \operatorname{End}\left(\operatorname{Ind}\,H_\tau\right) \\ \nu & \longmapsto \mathscr{B}_{\tau, \nu} \widehat{\varphi}_\tau(\nu) \mathscr{B}_{\tau, \nu}^{-1}. \end{align}\] Because the operators \(\mathcal{B}_{\tau, \nu}\) are unitary and depend continuously on \(\nu\), the map \(\mathscr{B}_I\) is well-defined, and is a \(\mathrm{C}^*\)-algebra automorphism.

Our next result compares the Mackey embedding \(\alpha\colon \mathrm{C}^*(G_0) \longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G)\) with the collection of canonical embeddings \(\alpha^I\colon \mathrm{C}^*(G_0; I) \longrightarrow \mathrm{C}_{\,\mathrm{r}}^*(G; I)\), as \(I\) ranges over the subsets of \(S\).

4.3 A converse statement and further remarks↩︎

The following simple lemma shows that the compatibility properties (a) and (b) in Proposition 1, concerning the behavior of the Mackey embedding \(\alpha\) on the ideals and subquotients of the infinitesimal character filtration, are enough to characterize the embedding \(\alpha\).

References↩︎