2.1 Pseudodifferential operators on Euclidean space

For our purposes, a convenient general reference for pseudodifferential operators is Chapter 18 of the treatise [Reference Hörmander22] of Hörmander. The pseudodifferential operators of order $m\in \mathbb {Z}$ on $\mathbb {R}^n$ are precisely the operators

(2.1.1) $$ \begin{align} A \colon C_c^\infty (\mathbb{R}^n) \longrightarrow C^\infty (\mathbb{R}^n) \end{align} $$

of the form

(2.1.2) $$ \begin{align} (A\varphi )(x) = (2\pi)^{-n} \int _{\mathbb{R}^n} e^{i\xi\cdot x} a(x,\xi) \hat \varphi (\xi)\, d\xi , \end{align} $$

for which the complete symbol function $a(x,\xi )$ belongs to the symbol class $S^m (\mathbb {R}^n{\times } \mathbb {R}^n)$ defined in [Reference Hörmander22, Def. 18.1.1]. Here $\hat \varphi $ is the Fourier transform of $\varphi $ [Reference Hörmander21, Def. 7.1.1], normalized so that if $a(x,\xi )\equiv 1$ , then $A\varphi = \varphi $ . Among other things, the conditions on the symbol class imply that the integrand is an integrable function. The operator $A$ is continuous for the usual topologies on $C_c^\infty (\mathbb {R}^n)$ and $C^\infty (\mathbb {R}^n)$ [Reference Hörmander22, Thm. 18.1.6].

When the complete symbol function admits an asymptotic expansion in homogeneous functions of strictly decreasing integer order, as detailed in [Reference Hörmander22, Prop. 18.1.3 and Def. 18.1.5], the operator $A$ will be said to be classical. We shall be exclusively concerned with classical operators in what follows.

Every pseudodifferential operator $A$ on $\mathbb {R}^n$ is adjointable, in the sense that there is a linear operator

$$\begin{align*}A^{*} \colon C_c^\infty (\mathbb{R}^n) \longrightarrow C^\infty (\mathbb{R}^n) \end{align*}$$

such that

$$\begin{align*}\langle A\varphi,\psi \rangle_{L^2} = \langle \varphi, A^{*} \psi\rangle_{L^2} \qquad \forall \varphi,\psi\in C_c^\infty (\mathbb{R}^n). \end{align*}$$

If $A$ is a classical pseudodifferential operator, then the adjoint is a classical pseudodifferential operator, too, and it has the same order as $A$ . See [Reference Hörmander22, Thm. 18.1.7].

The composition of properly supported,Footnote ¹ pseudodifferential operators is again pseudodifferential (and classical if the factors are classical), and the order of the composite operator is the sum of the orders of the factors [Reference Hörmander22, Thm. 18.1.8].

2.2 Pseudodifferential operators on smooth manifolds

The space of all classical pseudodifferential operators is invariant under diffeomorphisms [Reference Hörmander22, Thm. 18.1.17]. Because of this, one can transport the definitions of pseudodifferential and classical pseudodifferential operators to any smooth manifold, as in [Reference Hörmander22, Def. 18.1.20].

Hörmander has given a coordinate-free definition of classical pseudodifferential operators that is conceptually helpful [Reference Hörmander20]. If $M$ is a smooth manifold without boundary, then a continuous linear operator

$$\begin{align*}A \colon C_c^\infty (M) \longrightarrow C^\infty (M) \end{align*}$$

is a classical pseudodifferential operator of order $m$ (according to the coordinate-free definition of Hörmander) if and only if for every $\varphi \in C_c^\infty (M) $ and every bounded set of functions $\ell \in C^\infty (M)$ with $d\ell $ nowhere zero throughout $\operatorname { {Supp}}(\varphi )$ , there is a uniform asymptotic expansion

(2.2.1) $$ \begin{align} e^{-i t \ell } \cdot A(e^{i t\ell }\varphi ) \sim \sum_{k=0}^\infty \alpha_k(\varphi ,\ell ) t ^{m-k}\qquad\text{as}\quad t \to +\infty, \end{align} $$

with $\alpha _k(\varphi ,\ell )\in C^\infty (M)$ , meaning that for every $N$ , the set of all functions

(2.2.2) $$ \begin{align} t ^{N-m}\cdot \left ( e^{-i t \ell } \cdot A(e^{i t\ell }\varphi ) - \sum_{k=0}^{N-1} \alpha_k(\varphi ,\ell ) t ^{m-k}\right ) \in C^\infty(M), \end{align} $$

for all $\ell $ and all $t \ge 1$ , is a bounded set in $C^\infty (M)$ . See [Reference Hörmander20, Sec. 2] for further details and discussion.

Hörmander proves that when $M {=} \mathbb {R}^n$ , the compactly supported operators of the above type, meaning those for which there exists $\varphi \in C_c^\infty (M)$ with

(2.2.3) $$ \begin{align} A \varphi\psi = A\psi = \varphi A\psi\qquad \forall \psi \in C^\infty _c(M) , \end{align} $$

are precisely the compactly supported pseudodifferential operators of the type considered in the previous section. The complete symbol function is characterized by

$$\begin{align*}a(x,\xi) = \bigl ( e^{-i \xi } \cdot A( e^{i\xi}\varphi )\bigr ) (x) \end{align*}$$

where $\xi \in \mathbb {R}^n$ is regarded as a linear function on $\mathbb {R}^n$ using the standard dot product on $\mathbb {R}^n$ , and $\varphi $ is as in (2.2.3) above.

It follows that the pseudodifferential operators in Hörmander’s coordinate-free definition are precisely the operators with the following property: for every $\varphi \in C^\infty _c(M)$ and every diffeomorphism from a neighborhood of the support of $\varphi $ to an open subset of $\mathbb {R}^n$ , the operator $\varphi A \varphi $ transfers, via that diffeomorphism, to a (compactly supported) pseudodifferential operator on $\mathbb {R}^n$ in the sense of the previous section. These are the operators that we shall be considering from this point onward; they coincide with the operators on manifolds defined in [Reference Hörmander22, Def. 18.1.20].

We shall work in slightly greater generality. If $M$ is a smooth manifold, and if $E$ and $F$ are smooth vector bundles over $M$ , then using local coordinates on $M$ and local frames for $E$ and $F$ , one can define pseudodifferential and classical pseudodifferential operators

(2.2.4) $$ \begin{align} A \colon C_c^\infty (M;E)\longrightarrow C^\infty (M;F) \end{align} $$

that act between spaces of smooth sections of $E$ and $F$ ; this is done in [Reference Hörmander22, Def. 18.1.32]. The operators are characterized by the property that for any local frames of $E_1$ and $E_2$ over an open set $U\subseteq M$ , the entries of the matrix of operators determined by $\varphi A \varphi $ and the frames are classical pseudodifferential operators on $U$ .

2.3 The principal symbol

Let $A$ be a pseudodifferential operator of order $m$ on a smooth manifold $M$ . The value of the leading coefficient function

$$\begin{align*}\alpha_0(\varphi ,\ell)\colon M \longrightarrow \mathbb{C} \end{align*}$$

in the asymptotic expansion (2.2.1) at a point $x\in M$ depends only on the value of $\varphi $ at $x$ and the value of the differential $d\ell $ at $x$ . So from $\alpha _0$ we obtain a function

$$\begin{align*}a_0 \colon T^{*}M\setminus M \longrightarrow \mathbb{C} , \end{align*}$$

that is characterized by

$$\begin{align*}a_0(x,\xi)\varphi (x) = \alpha_0(\varphi ,\ell)(x)\quad\text{if } (d\ell)(x) = \xi. \end{align*}$$

This is the principal symbol of $A$ . It is a smooth function on $T^{*}M\setminus M$ that is homogeneous of degree $m$ on each cotangent fiber. See [Reference Hörmander20, Lemmas 2.2 & 2.3] or [Reference Hörmander22, pp. 82-83].

It follows from Hörmander’s coordinate-free definition of classical pseudodifferential operators that the principal symbol may be characterized in terms of the operator $A$ by the formula

$$\begin{align*}a_0(x,\xi)\varphi (x) = \lim_{t\to +\infty} t^{-m} e^{-i\ell(x)} (A e^{it \ell}\varphi) (x) \quad \text{when }(d\ell)(x)=\xi, \end{align*}$$

and that the limit is uniform over bounded families of smooth functions $\ell $ .

In the case of a pseudodifferential operator

$$\begin{align*}A \colon C_c^\infty (M;E) \longrightarrow C^\infty (M;F) , \end{align*}$$

acting on bundle sections, the principal symbol of $A$ is a smooth section of the pullback of the bundle $\operatorname { {Hom}} (E,F)$ to $T^{*}M\setminus M$ that is homogeneous of order $m$ in each fiber [Reference Hörmander22, p. 92]. It is characterized by the formula

(2.3.1) $$ \begin{align} a_0(x,\xi)s(x) = \lim_{t\to +\infty} t^{-m}e^{-i\ell(x)}( A e^{it \ell}s)(x) \quad \text{when }(d\ell)(x)=\xi, \end{align} $$

for all smooth, compactly supported sections $s$ . Once again the limit is uniform over bounded families of smooth functions $\ell $ .

For various purposes below, it will be convenient to use the following notion of compactification of a finite-dimensional real vector space, which adds to the vector space a “sphere at infinity.”

Definition 2.3.2. If $\mathfrak {v}$ is any finite-dimensional vector space, then we shall denote by $\overline {\mathfrak {v}}$ the topological compactification of $\mathfrak {v}$ (meaning a compact Hausdorff space including $\mathfrak {v}$ as a dense open subset) for which the continuous map

$$\begin{align*}\mathfrak{v} \rightarrow \mathfrak{v} , \qquad v \mapsto (1{+}\|v\|) ^{-1} v \end{align*}$$

extends to a homeomorphism from $\overline {\mathfrak {v}} $ to the closed unit ball in $\mathfrak {v}$ . Here $\|\,\cdot \, \|$ is any norm on $\mathfrak {v}$ (the compactification is independent of the choice of norm).

The action of $GL(\mathfrak {v})$ on $\mathfrak {v}$ extends to a continuous action on the compactification above. With this, if $\mathfrak {V}$ is any real vector bundle, then we may define a space $\overline {\mathfrak {V}}$ by applying the compactification construction fiberwise (compare [Reference Atiyah3, Sec. 1.2]). We shall write the added boundary as $\partial \mathfrak {V}$ below.

The principal symbol of an order zero pseudodifferential operator on a smooth manifold $M$ , being homogeneous of order zero on each fiber of $T^{*}M\setminus M$ , extends to a continuous function (or section) on $\overline {T^{*}M} \setminus M$ , and is determined by the restriction of this extension to $\partial T^{*}M$ :

$$\begin{align*}a_0 \colon \partial T^{*}M \longrightarrow \mathbb{C}. \end{align*}$$

If the fiberwise compactification is constructed using a Riemannian metric on $M$ , then $\partial T^{*}M$ identifies with the unit sphere bundle in $T^{*}M$ and therefore carries a canonical smooth structure, independent of the choice of metric. The principal symbol is then a smooth function. Moreover, it is evident from the definitions that if the principal symbol of a classical pseudodifferential operator of order $0$ is identically zero, then $ A$ is in fact of order ${-}1$ . We therefore obtain a short exact sequence of algebras and algebra morphisms

$$\begin{align*}0 \longrightarrow \Psi^{-1}_c (M) \longrightarrow \Psi^0_c(M) \longrightarrow C^\infty _c(\partial T^{*}M) \longrightarrow 0 , \end{align*}$$

where $\Psi _c^{-1}(M)$ and $\Psi ^0_c(M)$ are the algebras of compactly supported order $-1$ and order $0$ classical pseudodifferential operators, respectively.

2.4 The pseudodifferential extension of C*-algebras

It will be important for our purposes that compactly supported classical pseudodifferential operators of order $0$ extend to bounded Hilbert space operators:

Theorem 2.4.1 (See for instance [Reference Hörmander22, Thm. 18.1.1])

If $A$ is a compactly supported classical pseudodifferential operator of order $0$ on a smooth manifold $M$ , then $A$ is $L^2$ -bounded: for any choice of smooth measure on $M$ , there is a constant $C\ge 0$ such that

$$\begin{align*}\| A \varphi \|_{L^2(M)} \le C \| \varphi \|_{L^2(M)} \qquad \forall \varphi \in C_c^\infty (M). \end{align*}$$

We shall also require the following slightly more specialized results, related to ${C}^{*}$ -algebra theory.

Theorem 2.4.4 [Reference Seeley43, Lem. 7.2]; see also [Reference Kohn and Nirenberg31, Thm. 5]

If $A$ is a compactly supported, classical pseudodifferential operator of order zero on a smooth manifold $M$ , then

$$\begin{align*}\| A\|_{L^2(M)\to L^2 (M)} \ge \sup \bigl \{\, |a_0(m,\xi)| : m\in M,\,\, \xi\in T^{*}_mM \setminus \{ 0\}\,\bigr \} \end{align*}$$

(the $L^2$ -operator norm of $A$ is of course the best possible constant in Theorem 2.4.1; the supremum above is finite because $a_0$ is homogeneous in $\xi $ of order $0$ ).

It follows from Theorem 2.4.4 that the principal symbol gives rise to a continuous homomorphism of Banach algebras, and in fact ${C}^{*}$ -algebrasFootnote ²

$$\begin{align*}\overline {\Psi^0_c(M)} \longrightarrow C_0(\partial T^{*}M), \end{align*}$$

where the overbar denotes the ${C}^{*}$ -algebra completion in the operator norm.

The following result, or rather the equivariant version that we shall derive from it in Section 3.5, will play a crucial role in the paper.

Theorem 2.4.5 [Reference Seeley43, Thm. 14.1]; see also [Reference Kohn and Nirenberg31, Cor. 6.1]

The above extended principal symbol map fits into a short exact sequence of ${C}^{*}$ -algebras

$$\begin{align*}0 \longrightarrow \mathfrak{K}(L^2 (M)) \longrightarrow \overline{\Psi^0_c(M)} \longrightarrow C_0(\partial T^{*}M) \longrightarrow 0. \end{align*}$$

3.1 Sections of equivariant vector bundles

Let $E$ be a smooth, right- $G$ -equivariant, hermitian vector bundle over $M$ , and denote by $C^\infty (M;E)$ and $C_c^\infty (M;E)$ its spaces of smooth sections, and smooth, compactly supported sections, respectively. Both spaces carry right actions of the group $G$ that are defined by

$$\begin{align*}(s \cdot g) (m) = s(mg^{-1})\cdot g \qquad \forall g \in G,\,\,\, \forall m \in M. \end{align*}$$

The right-hand side of the formula involves the action

$$\begin{align*}E_{mg^{-1}} \to E_m, \quad e\mapsto e \cdot g \end{align*}$$

of the group element $G$ between the fibers of $E$ .

Now fix a $G$ -invariant smooth measure on $M$ and a left-invariant Haar measure on $G$ . Denote by $C_c^\infty (G)$ the space of smooth, compactly supported complex functions on $G$ , equipped with the standard convolution multiplication and the involution that is used in groupoid ${C}^{*}$ -algebra theory [Reference Renault39, Sec. II.1]:

(3.1.1) $$ \begin{align} (f_1*f_2)(g) = \int _G f_1(gg_1)f_2(g_1^{-1} ) \, dg_1\quad\text{and}\quad f^{*}(g) = \overline{f(g^{-1})}. \end{align} $$

The spaces $C^\infty (M;E)$ and $C_c^\infty (M;E)$ carry right module actions of the algebra $C_c^\infty (G)$ defined by the formula

(3.1.2) $$ \begin{align} (s* f) (m) = \int _G ( sg_1^{-1}) (m) f(g_1^{-1}) \, dg_1 \quad \forall s \in C^\infty (M;E)\,\,\,\forall f \in C_c^\infty (G). \end{align} $$

Here the scalar $f(g)$ has been written on the right-hand side of the vector $(sg)(m)$ in an effort to make the formula easier to parse. When $M{=}G$ and when $E$ is trivial, so that $C_c^\infty (M,E) = C_c^\infty (G)$ , (3.1.2) agrees with (3.1.1).

The space $C^\infty (M;E)$ also carries the obvious $L^2$ -inner product, and indeed a $C_c^\infty (G)$ -valued inner product

(3.1.3) $$ \begin{align} \langle\,\,\,\,,\,\,\,\rangle_{C^\infty _c (G)} : C_c^\infty (M;E) \times C_c^\infty (M;E) \longrightarrow C_c^\infty (G) \end{align} $$

that is defined by

$$\begin{align*}\langle s_1, s_2 \rangle _{C^\infty _c (G)} : g \longmapsto \langle s_1, s_2 g^{-1} \rangle_{L^2(M;E)}. \end{align*}$$

When $C_c^\infty (M,E) = C_c^\infty (G)$ , this is simply $s_1^{*}*s_2\in C_c^\infty (G)$ , where the involution and product are as in (3.1.1). In general, the inner product is complex-sesquilinear and satisfies

$$\begin{align*}\begin{aligned} \langle s_1, s_2* f \rangle _{C^\infty _c (G)} & = \langle s_1, s_2 \rangle _{C^\infty _c (G)}* f\\ \langle s_1, s_2 \rangle _{C^\infty _c (G)} & = \langle s_2, s_1 \rangle _{C^\infty _c (G)}^{*} \end{aligned} \end{align*}$$

for all $s_1,s_2\in C_c^\infty (M;E)$ and all $f\in C_c^\infty (G)$ .

We shall use the realization of the reduced ${C}^{*}$ -algebra of $G$ from groupoid ${C}^{*}$ -algebra theory; see [Reference Renault39, Sec. II.1] again. The $*$ -algebra $C^\infty _c (G)$ is represented faithfully by left-convolution (using the left Haar measure) as bounded operators on the Hilbert space $L^2 (G,dg^{-1})$ (this is a $*$ -representation), and ${C}^{*}_r(G)$ is by definition the completion in the operator norm. When $G$ is unimodular, which is our main interest, this is the usual definition of the reduced ${C}^{*}$ -algebra [Reference Pedersen38, Sec. 7.2]. In general, pointwise multiplication by $\Delta (g)^{-1/2}$ , where $\Delta $ is the modular function of $G$ , as in [Reference Pedersen38, Sec. 7.1], is an algebra automorphism of $C_c^\infty (G)$ that extends to a $*$ -isomorphism from ${C}^{*}_r(G)$ as we have defined it to the standard version of the reduced ${C}^{*}$ -algebra.

The inner product (3.1.3) has the additional properties that

$$\begin{align*}\langle s,s\rangle _{C^\infty_c (G)} \ge 0 \quad \text{and} \quad \langle s,s\rangle _{C^\infty_c (G)} = 0 \,\, \Rightarrow \,\, s =0 , \end{align*}$$

where the inequality indicates positivity in the ${C}^{*}$ -algebra ${C}^{*}_r (G)$ . The formula

$$\begin{align*}\| s\|^2 = \| \langle s,s\rangle_{C^\infty _c (G)} \|_{{C}^{*}_r (G)} \end{align*}$$

defines a norm on the complex vector space $C_c^\infty (M;E)$ . We shall denote by ${C}^{*}_r(M;E)$ the Banach-space completion of $C^\infty _c (M;E)$ in this norm. The right action in (3.1.2) and inner product in (3.1.3) extend to completions to give ${C}^{*}_r (M;E)$ the structure of a Hilbert ${C}^{*}$ -module over ${C}^{*}_r (G)$ . See [Reference Lance34] for background information on Hilbert ${C}^{*}$ -modules. The above constructions are adapted from [Reference Baum, Connes and Higson5, Sec. 3], but they are due to Kasparov [Reference Kasparov28].

For the rest of this section we shall specialize to the situation where $K$ is a compact subgroup of $G$ and $M = K \backslash G$ (the space of right $K$ -cosets). Each finite-dimensional unitary representation $V$ of $K$ determines an induced $G$ -equivariant Hermitian vector bundle

$$\begin{align*}K \backslash ( V \times G) \to K \backslash G. \end{align*}$$

Its fiber over $Kg\in K\backslash G$ is the set of all equivalence classes $[v,h]$ of elements in $ V\times Kg$ under the equivalence relation

$$\begin{align*}(v,h)\sim (kv,kh). \end{align*}$$

If $C_c^\infty (G)$ is equipped with the usual left-translation action of $K$ (arranged to be a left action), then there is a unique isomorphism of $G$ -vector spaces

(3.1.4) $$ \begin{align} \bigl [ V \otimes C_c^\infty (G) \bigr ]^K \stackrel \cong \longrightarrow C_c^\infty \bigl (K \backslash G; K \backslash ( V {\times} G) \bigr) \end{align} $$

such that

$$\begin{align*}\sum_j v_j \otimes f_j \longmapsto \Bigl [ Kg \mapsto \sum_j f_j(g) [ v_j,g]\Bigr]. \end{align*}$$

We shall mostly use this view of the sections of the induced bundle from now on.

Under the isomorphism (3.1.4), if $E$ is the Hermitian vector bundle over $M$ induced from $V$ , then the right action of $C_c^\infty (G)$ on $ C_c^\infty (M;E) $ given by (3.1.2) corresponds to the right action on the left-hand side in (3.1.4) given by the simple formula

$$\begin{align*}\left ( \sum_j v_j \otimes f_j \right) * f = \sum_j v_j \otimes (f_j* f) , \end{align*}$$

while the inner product (3.1.3) corresponds to

$$\begin{align*}\left \langle \sum_j v_j\otimes f_j , \sum_k w_k\otimes h_k \right \rangle_{C_c^\infty (G)} = \sum_{j,k} \langle v_j,w_k\rangle_{V} \ ( f_j{}^{*} * h_k ). \end{align*}$$

The isomorphism (3.1.4) extends to a unitary isomorphism

(3.1.5) $$ \begin{align} \bigl [ {C}^{*}_r (G) \otimes V\bigr]^K \stackrel \cong\longrightarrow {C}^{*}_r \bigl (K \backslash G, K \backslash ( V {\times} G) \bigr ) \end{align} $$

between right Hilbert ${C}^{*}$ -modules over ${C}^{*}_r (G)$ .

3.2 Properly supported operators

Let $M$ be a general smooth manifold (for a moment) and let $E_1$ and $E_2$ be smooth vector bundles over $M$ . A complex-linear operator

$$\begin{align*}T \colon C_c^\infty (M;E_1) \longrightarrow C^\infty (M;E_2) \end{align*}$$

is properly supported if for every $\varphi \in C_c^\infty (M)$ there exists some $\psi \in C_c^\infty (M)$ such that

$$\begin{align*}(1-\psi) T \varphi = 0 \quad\text{and} \quad \varphi T (1 - \psi) = 0. \end{align*}$$

In this case $T$ actually maps smooth, compactly supported sections of $E_1$ to smooth, compactly supported sections of $E_2$ . Moreover there is a unique extension of $T$ to a linear operator

$$\begin{align*}T \colon C^\infty (M;E_1) \longrightarrow C^\infty (M;E_2) \end{align*}$$

with the property that if $\varphi $ and $\psi $ are smooth compactly supported functions, and if $\varphi T (1 {-} \psi ) = 0$ , then

$$\begin{align*}\varphi T s = \varphi T \psi s \quad \forall s \in C^\infty (M;E_1). \end{align*}$$

Now assume that $E_1$ and $E_2$ are equipped with Hermitian structures, and that $M$ is equipped with a smooth measure. The operator $T$ is adjointable if there exists an operator

$$\begin{align*}T^{*} \colon C_c^\infty (M;E_2) \longrightarrow C^\infty (M;E_1) \end{align*}$$

such that

$$\begin{align*}\langle Ts_1,s_2\rangle _{L^2(M;E_2)} = \langle s_1,T^{*}s_2\rangle _{L^2(M;E_1)} \end{align*}$$

for all $s_1\in C_c^\infty (M;E_1)$ and $s_2\in C_c^\infty (M;E_2)$ . The property of being adjointable is independent of the choice of measure (but the adjoint operator depends on the choice).

Proposition 3.2.1. Let $K$ be a compact subgroup of $G$ and let $V_1$ and $V_2 $ be finite-dimensional unitary representations of $K$ . If $T\colon [C_c^\infty (G)\otimes V_1]^K \to [C_c^\infty (G)\otimes V_2]^K $ is properly supported, adjointable and equivariant, then $T$ extends to a bounded operator

$$\begin{align*}T\colon [L^2 (G)\otimes V_1]^K \to [L^2 (G)\otimes V_2]^K, \end{align*}$$

if and only if it extends to a bounded and adjointable operator between Hilbert ${C}^{*}$ -modules over ${C}^{*}_r (G)$ ,

$$\begin{align*}T\colon [{C}^{*}_r (G)\otimes V_1]^K \to [{C}^{*}_r (G)\otimes V_2]^K. \end{align*}$$

In this case, the two extensions have the same norm.

Proof. To simplify the notation let us consider only the case where $K =\{ e\}$ and the representations $V_1$ and $V_2$ are (necessarily) trivial, and of dimension one. Apart from issues of notation, the general case is no different.

If $f\in C_c^\infty (G)$ , then

$$\begin{align*}\| Tf \|_{{C}^{*}_r (G)} = \sup \, \bigl \{ \, \| (Tf)h\| _{L^2(G)} : h\in C_c^\infty (G),\,\, \| h\|_{L^2 (G)}=1\,\bigr \}. \end{align*}$$

But

$$ \begin{align*} \| (Tf)h\| _{L^2(G)} & = \| T(fh)\| _{L^2(G)}\\ & \le \|T\|_{L^2 \to L^2} \|fh\|_{L^2(G)} \le \|T\|_{L^2 \to L^2} \|f\|_{{C}^{*}_r (G)}\|h \|_{L^2(G)}, \end{align*} $$

which shows that $\|T\|_{{C}^{*}_r \to {C}^{*}_r }\le \|T\|_{L^2 \to L^2} $ .

In the opposite direction let $\{f_n\}$ be an approximate unit for ${C}^{*}_r(G)$ that consists of elements in $C^\infty _c(G)$ . If $k\in C_c^\infty (G)$ , then $f_n k \to k$ in the $L^2$ -norm, and so for any $h\in C_c^\infty (G)$ ,

$$ \begin{align*} \| Th \|_{L^2(G)}^2 & = \bigl \langle Th , Th \bigr \rangle_{L^2 (G)} = \bigl \langle h , T^{*}Th \bigr \rangle_{L^2 (G)} = \lim_{n\to \infty}\bigl \langle f_n h , T^{*}Th \bigr \rangle_{L^2 (G)} \\ & = \lim_{n\to \infty}\bigl \langle T^{*}T (f_n h) , h \bigr \rangle_{L^2 (G)} = \lim_{n\to \infty}\bigl \langle (T^{*}T f_n) h , h \bigr \rangle_{L^2 (G)}. \end{align*} $$

So the Cauchy-Schwarz inequality gives

$$ \begin{align*} \| Th \|_{L^2(G)}^2 & \le \limsup \| (T^{*}T f_n) h\| _{L^2(G)} \cdot \| h\| _{L^2(G)} \\ & \le \limsup \| (T^{*}T f_n)\| _{{C}^{*}_r(G)}\cdot \| h\|^2 _{L^2(G)} \le \|T\|^2 _{{C}^{*}_r \to {C}^{*}_r} \| h\|^2 _{L^2 (G)}. \end{align*} $$

This shows that $\|T\|_{L^2 \to L^2} \le \|T\|_{{C}^{*}_r \to {C}^{*}_r } $ .

3.3 Smoothing operators

Let $M$ be a smooth manifold. If k is a smooth section of the bundle $\operatorname { {Hom}} (E_1, E_2)$ over $M\times M$ (where $E_2$ is pulled back to the product $M\times M$ along the left coordinate projection, and $E_1$ along the right), then the formula

$$\begin{align*}(T s)(x) = \int _M k(x,y)\cdot s(y) \, dy , \end{align*}$$

where the dot represents the contraction operation from $\operatorname { {Hom}} (E_1,E_2)\otimes E_1$ to $E_2$ , defines a smoothing operator

$$\begin{align*}T \colon C_c^\infty (M;E_1) \longrightarrow C^\infty (M;E_2) \end{align*}$$

acting between sections of the bundles $E_1$ and $E_2$ . The operator $T$ is properly supported if and only if k is properly supported, in which case $T$ is also automatically adjointable. It is equivariant if and only if k is a $G$ -equivariant section for the diagonal action of $G$ on $M\times M$ .

The purpose of this section is to describe the smoothing operators in the case where $M = K \backslash G$ , and where $E_1$ and $E_2$ are the equivariant vector bundles induced from finite-dimensional unitary representations $V_1$ and $V_2$ of the compact group $K$ .

The space of all smooth integral kernels may be described in this case as follows. Let $K{\times } K$ act by left translation on $G{\times }G $ in the obvious way, and on the vector space $\operatorname { {Hom}} (V_1,V_2)$ by $(k_1,k_2) \cdot T = k_1 Tk_2^{-1}$ . There is a unique isomorphism of $G$ -vector spaces

(3.3.1) $$ \begin{align} [ C^\infty (G{\times} G)\otimes \operatorname{{Hom}} (V_1,V_2) ]^{K\times K} \stackrel \cong \longrightarrow C^\infty (M{\times} M; \operatorname{{Hom}} (E_1,E_2 )) \end{align} $$

for the diagonal action of $G$ such that

(3.3.2) $$ \begin{align} \sum_j F_j \otimes T_j \mapsto h, \qquad h (Kg_1,Kg_2)\colon [v,g_1]\mapsto \sum_j F_j(g_1,g_2) [T_jv,g_1]. \end{align} $$

The action of $G$ on (3.3.1) is through the diagonal right-translation action on $G\times G$ and on $M\times M$ . The map in (3.3.1) determines an isomorphism

(3.3.3) $$ \begin{align} [ C^\infty (G )\otimes \operatorname{{Hom}} (V_1,V_2) ]^{K\times K} \stackrel \cong \longrightarrow C^\infty (M{\times} M; \operatorname{{Hom}} (E_1,E_2 ))^G \end{align} $$

such that

(3.3.4) $$ \begin{align} \sum_j f_j \otimes T_j \mapsto h; \qquad h (Kg_1,Kg_2)\colon [v,g_1]\mapsto \sum_j f_j(g_2g_1^{-1}) [T_jv,g_1]. \end{align} $$

The image of the restriction of (3.3.3) to $C_c^\infty (G)$ is the space of properly supported, $G$ -equivariant smooth integral kernels.

Under the isomorphisms (3.1.4) and (3.3.4), the action of equivariant smoothing operators on the smooth compactly supported sections of $E_1$ is given by the natural map

(3.3.5) $$ \begin{align} [ C^\infty (G)\otimes \operatorname{{Hom}} (V_1,V_2) ]^{K\times K} \times [ C^\infty_c (G) \otimes V_1]^{K } \longrightarrow [ C^\infty (G)\otimes V_2 ]^{K } \end{align} $$

characterized by

(3.3.6) $$ \begin{align} \left ( \sum_j f_j \otimes T_j , \sum _k h_k\otimes v_k\right ) \longmapsto \sum_{j,k} (f_j {*} h_k) \otimes T_j v_k. \end{align} $$

This is under the assumption that the invariant measure on $M = K \backslash G$ is chosen so that the integral of any function on $M$ is equal to the Haar integral of its pullback to $G$ .

Of course, properly supported smoothing operators may composed, and their adjoints taken. We omit the natural formulas for these operations under the isomorphism (3.3.4). It is clear from the explicit formula (3.3.6) that:

Lemma 3.3.7. Every $G$ -equivariant, properly supported smoothing operator

$$\begin{align*}T \colon [ C_c^\infty (G)\otimes V_1]^K \longrightarrow [ C_c^\infty (G)\otimes V_2]^K \end{align*}$$

extends to bounded operator $ [ L^2 (G)\otimes V_1]^K \to [L^2 (G)\otimes V_2]^K$ , and to a bounded and adjointable operator $[C_r^{*} (G)\otimes V_1]^K\to [ C_r^{*} (G)\otimes V_2]^K$ .

The operator-norm completion of the properly supported, $G$ -equivariant smoothing operators is isomorphic to the space

(3.3.8) $$ \begin{align} \bigl [ {C}^{*}_r (G)\otimes \operatorname{{Hom}} (V_1,V_2)\bigr ]^{K\times K} \subseteq \mathfrak{B} \bigl ( [ {C}^{*}_r(G)\otimes V_1 ]^K, [ {C}^{*}_r(G) \otimes V_2 ]^K\bigr ) \end{align} $$

under the identifications (3.1.4) and (3.3.3)-(3.3.6).

3.4 Pseudodifferential operators on proper homogeneous spaces

We are interested in properly supported, $G$ -equivariant, classical pseudodifferential operators on the homogeneous space $K\backslash G$ . We shall use the following notation:

Definition 3.4.1. Let $V_1$ and $V_2$ be finite-dimensional unitary representations of $K$ and let $m\in \mathbb {Z}$ . We shall denote by $\Psi ^{m}_{G,K}(V_1,V_2)$ the space of $G$ -equivariant, properly supported, classical pseudodifferential operators

$$\begin{align*}A \colon [ C_c^\infty (G)\otimes V_1]^K \longrightarrow [ C_c^\infty (G)\otimes V_2]^K \end{align*}$$

of order $m$ .

Let us consider the case $m{=}0$ . As we have seen, the principal symbol of an operator in $\Psi ^0_{G,K}(V_1,V_2)$ can be viewed as a smooth section over $\partial T^{*} M$ , where $M=K \backslash G$ , of the $G$ -equivariant vector bundle whose fiber over each point of $\partial T^{*}_e M$ is the space $\operatorname { {Hom}} (V_1,V_2)$ . Of course it must be a $G$ -equivariant section, and therefore the restriction to the basepoint $e\in K \backslash G$ determines the symbol. This restriction is a smooth, $K$ -equivariant function

$$\begin{align*}a_0 \colon \partial (\mathfrak{g}/\mathfrak{k})^{*} \longrightarrow \operatorname{{Hom}} (V_1,V_2) \end{align*}$$

(we are using the standard identification of $T_eM$ with $\mathfrak {g} / \mathfrak {k}$ ). Conversely, it is shown in [Reference Stetkaer44] how to construct an operator in $\Psi ^0_{G,K}(V_1,V_2)$ with any given principal symbol function $a_0$ , as above, by a simple averaging procedure (and we shall see the same averaging procedure below).

Definition 3.4.2. We shall denote by

$$\begin{align*}\Sigma^0_{G,K} (V_1,V_2)= C^\infty \bigl ( \partial (\mathfrak{g}/\mathfrak{k})^{*} , \operatorname{{Hom}}(V_1,V_2)\bigr )^K \end{align*}$$

the space of principal symbols of operators in $\Psi ^0_{G,K}(V_1,V_2)$ .

The discussion above provides a short exact sequence of vector spaces

(3.4.3) $$ \begin{align} 0 \longrightarrow \Psi^{-1}_{G,K}(V_1,V_2) \longrightarrow \Psi^0_{G,K}(V_1,V_2) \longrightarrow \Sigma^0_{G,K} (V_1,V_2) \longrightarrow 0 \end{align} $$

involving the space of properly supported, order $0$ , classical pseudodifferential operators acting between the spaces of smooth sectionsFootnote ³ of $G$ -equivariant complex vector bundles on the symmetric space $K \backslash G$ .

Proposition 3.4.4. Every properly supported, $G$ -equivariant, order zero classical pseudodifferential operator

$$\begin{align*}A \colon [C_c^\infty ( G)\otimes V_1]^K \longrightarrow [C_c^\infty ( G)\otimes V_2]^K \end{align*}$$

extends to a bounded operator between $L^2$ -spaces.

Proof. Let $\varphi $ be a smooth and compactly supported function on $K \backslash G$ such that

$$\begin{align*}\int_G \varphi (xg) \, dg = 1 \quad \forall x\in K\backslash G, \end{align*}$$

and let

$$\begin{align*}A_e {=} A\varphi \colon [C_c^\infty ( G)\otimes V_1]^K \longrightarrow [C_c^\infty ( G)\otimes V_2]^K. \end{align*}$$

This is a compactly supported, order zero classical pseudodifferential operator on $M{=}K\backslash G$ , and therefore it is bounded in the $L^2$ -operator norm. If we set $A_g = g(A_e)$ , then

$$\begin{align*}(A f)(x) = \int_G (A_g f)(x) \, dg \quad \forall x \in K \backslash G,\quad \forall f\in [C_c^\infty ( G)\otimes V_1]^K. \end{align*}$$

The operators $A_g$ are unitarily equivalent to one another, and therefore they have the same $L^2$ -operator norm:

$$\begin{align*}\| A_g \| = \| A_e\| \quad \forall g\in G. \end{align*}$$

Let $C\subseteq K \backslash G$ be a compact subset such that the support of $A_e$ is included in $C{\times } C$ , and let

$$\begin{align*}S = \{ \, g\in G : Kg \in C\, \}. \end{align*}$$

This is a compact subset of $G$ , and if $f\in [C_c^\infty ( G)\otimes V_1]^K $ , $g\in G$ and $x\in K \backslash G$ , then $(A_gf)(x) = 0$ unless $x\in Cg$ . In other words, $(A_gf)(x) = 0$ unless $g\in x^{-1}S$ . So by the Cauchy-Schwarz inequality,

$$\begin{align*}\begin{aligned} \| A f\|_{L^2}^2 & = \int _M \Bigl \| \int_G (A_gf)(x)\, dg \Bigr\|_{V_2}^2 \, dx \\ & \le \int _M \left ( \int_{x^{-1} S} 1 \, dg\right ) \left ( \int_{x^{-1}S} \bigl \|(A_gf)(x)\bigr \|_{V_2} ^2 \, dg\right ) \, dx \\ & = \operatorname{vol}(S)\cdot \int _M \left (\int_{x^{-1}S} \bigl \|(A_gf)(x)\bigr \|_{V_2} ^2 \, dg \right ) \, dx. \end{aligned} \end{align*}$$

But now, using Fubini’s theorem, we can write

$$\begin{align*}&\int _M \left (\int_{x^{-1}S} \bigl \|(A_gf)(x)\bigr \|_{V_2} ^2 \, dg \right ) \, dx \\&\quad = \int _G \left (\int_{Cg} \bigl \|(A_gf)(x)\bigr \|_{V_2} ^2 \, dx \right ) \, dg. \end{align*}$$

The inner integral on the right may be estimated by

$$\begin{align*}\int_{Cg} \bigl \|(A_gf)(x)\bigr \|_{V_2} ^2 \, dx \le \| A_e\|^2\cdot \int_{Cg} \bigl \|f(x)\bigr \|_{V_1} ^2 \, dx , \end{align*}$$

and so, by Fubini once again,

$$\begin{align*}\begin{aligned} \| A f\|_{L^2}^2 & \le \operatorname{vol}(S)\cdot \| A_e\|^2 \cdot \int _G \Bigl (\int_{Cg} \|f(x) \|_{V_1} ^2 \, dx \Bigr ) \, dg \\ & = \operatorname{vol}(S)\cdot \| A_e\|^2 \cdot \int _M \Bigl (\int_{x^{-1}S} \|f(x) \|_{V_1} ^2 \, dg \Bigr ) \, dx \\ & = \operatorname{vol}(S)^2\cdot \| A_e\|^2 \cdot \int _M \|f(x) \|_{V_1} ^2 \, dx \\ & = \operatorname{vol}(S)^2 \cdot \| A_e\|^2 \cdot \| f\|_{L^2}^2, \end{aligned} \end{align*}$$

which proves that $A$ is $L^2$ -bounded, as required.

We noted in Theorem 2.4.3 that compactly supported, negative order operators are compact operators. The analogue of this result in the current context is as follows:

Proposition 3.4.5. The ${L}^2$ -operator norm-closure of the properly supported, equivariant smoothing operators

$$\begin{align*}T \colon [C_c^\infty ( G)\otimes V_1]^K \longrightarrow [C_c^\infty ( G)\otimes V_2]^K \end{align*}$$

includes all the properly supported, equivariant classical pseudodifferential operators of order $-1$ or less.

Proof. If $A$ is any pseudodifferential operator on $\mathbb {R}^n$ of order less than $-n$ , then it follows from the integral formula (2.1.2) that the Schwartz kernel of $A$ is a continuous function. It follows that if $A$ is a properly supported and $G$ -equivariant pseudodifferential operator, as in the statement of the proposition, and if the order of $A$ is less than $-\dim (K\backslash G)$ , then the Schwartz kernel of $A$ is an equivariant, continuous and properly supported section of the bundle $\operatorname { {Hom}}( V_1,V_2)$ over $K\backslash G \times K\backslash G$ . So it corresponds to an element of

$$\begin{align*}[ C_c (G) \otimes \operatorname{{Hom}} (V_1,V_2)]^{K\times K} \end{align*}$$

under the isomorphism (3.3.3), and $A$ is therefore in the operator-norm closure of the $G$ -equivariant, properly supported smoothing operators.

Suppose now $A$ has order $-1$ or less. For $V= V_1\oplus V_2$ we can think of $A$ as an operator

$$\begin{align*}A \colon [C_c^\infty ( G)\otimes V]^K \longrightarrow [C_c^\infty ( G)\otimes V]^K \end{align*}$$

that is zero on sections with values in $V_2$ and has range in the sections valued in $V_2$ . The above shows that some power of $A^{*}A$ is in the operator-norm closure of the $G$ -equivariant, properly supported smoothing operators. But $A$ belongs to a ${C}^{*}$ -algebra of operators on $[L^2( G)\otimes V]^K $ that includes the norm closure of the $G$ -equivariant, properly supported smoothing operators as an ideal. The above shows that the self-adjoint element $A^{*}A$ is nilpotent in the quotient ${C}^{*}$ -algebra, and is therefore zero in the quotient ${C}^{*}$ -algebra.

3.5 Pseudodifferential extension in the equivariant context

Lemma 3.5.2. Let $V_1$ and $V_2$ be finite-dimensional unitary representations of $K$ and let

$$\begin{align*}A \colon [C_c^\infty ( G)\otimes V_1]^K \longrightarrow [C_c^\infty ( G)\otimes V_2]^K \end{align*}$$

be a properly supported, $G$ -equivariant, order zero, classical pseudodifferential operator. If $a_0$ is the principal symbol of $A$ , then

$$\begin{align*}\| A \|_{L^2\to L^2} \ge \sup_{\xi \ne 0} \bigl \| a_0 (e,\xi)\bigr \|. \end{align*}$$

As a result, the principal symbol map extends to a bounded linear map

$$\begin{align*}\mathsf{P}^{*}_{G,K}( V_1,V_2) \longrightarrow \mathsf{S}^{*}_{G,K}( V_1,V_2). \end{align*}$$

Proof. Let $\varphi $ be a smooth and compactly supported function on $K \backslash G$ such that $1 \ge \varphi \ge 0$ and $\varphi (e) =1$ . Then $\varphi A \varphi $ is a compactly supported classical pseudodifferential operator. Moreover $ \| A\|\ge \| \varphi A \varphi \| $ and the principal symbols of $A$ and $\varphi A \varphi $ agree at $e\in K\backslash G$ . It follows from Theorem 2.4.4 that

$$\begin{align*}\| A\| \ge \| \varphi A \varphi \| \ge \sup_{\xi \ne 0} \bigl \| a_0 (e,\xi)\bigr \| , \end{align*}$$

as required.

Proposition 3.5.3. Let $V_1$ and $V_2$ be finite-dimensional unitary representations of $K$ . By taking $L^2$ -operator norm closures, we obtain from the exact sequence

$$\begin{align*}0 \longrightarrow \Psi^{-1}_{G,K}(V_1,V_2) \longrightarrow \Psi^0_{G,K}(V_1,V_2) \longrightarrow \Sigma^0_{G,K} (V_1,V_2) \longrightarrow 0 \end{align*}$$

an exact sequence

$$\begin{align*}0 \longrightarrow \mathsf{C}^{*}_{G,K}(V_1,V_2) \longrightarrow \mathsf{P}^{*}_{G,K}(V_1,V_2) \longrightarrow \mathsf{S}^{*}_{G,K} (V_1,V_2) \longrightarrow 0. \end{align*}$$

To prove the proposition we shall use the following two lemmas. The first may be proved using the method of the proof of Proposition 3.4.4; the second is straightforward. Both proofs will be omitted.

Lemma 3.5.4. Let $\varphi $ be a continuous, compactly supported function on $K \backslash G$ . If $V_1$ and $V_2$ are any finite-dimensional unitary representations of $K$ , and if

$$\begin{align*}T \colon [L^2 ( G)\otimes V_1]^K \longrightarrow [L^2 ( G)\otimes V_2]^K \end{align*}$$

is any bounded operator, then the averaging formula

$$\begin{align*}\mathsf{Av}(\varphi T \varphi )s = \int _G g( \varphi T \varphi)s \, dg \end{align*}$$

defines a bounded operator

$$\begin{align*}\mathsf{Av}(\varphi T \varphi )\colon [L^2 ( G )\otimes V_1]^K \longrightarrow [L^2 ( G )\otimes V_2]^K. \end{align*}$$

Moreover the linear map $T \mapsto \mathsf {Av}(\varphi T \varphi )$ (with $\varphi $ fixed) is operator norm-continuous.

Lemma 3.5.5. If $V_1$ and $V_2$ are any finite-dimensional unitary representations of $K$ , and if

$$\begin{align*}A \colon [C_c^\infty (G)\otimes V_1]^K \longrightarrow [C_c^\infty (G)\otimes V_2]^K \end{align*}$$

is any equivariant, properly supported, order zero classical pseudodifferential operator, then $ \mathsf {Av}(\varphi T \varphi )$ is also an equivariant, properly supported, order zero classical pseudodifferential operator. Moreover if $\mathsf {Av}( \varphi ^2)=\operatorname {id}$ , then $ \mathsf {Av}(\varphi T \varphi )$ has the same principal symbol as $A$ .

Proof of Proposition 3.5.3

If we set $V = V_1\oplus V_2$ , then we can regard the space $\mathsf {P}^{*}_{G,K}(V_1,V_2)$ as a subspace of $\mathsf {P}^{*}(V,V)$ , as in the proof of Proposition 3.4.5. We can do the same for the $\mathsf {C}^{*}$ - and $\mathsf {S}^{*}$ -spaces, with the result that it suffices to prove exactness of the sequence

$$\begin{align*}0 \longrightarrow \mathsf{C}^{*}_{G,K}(V,V) \longrightarrow \mathsf{P}^{*}_{G,K}(V,V) \longrightarrow \mathsf{S}_{G,K}^{*} (V,V) \longrightarrow 0 , \end{align*}$$

which is a sequence of ${C}^{*}$ -algebras and $*$ -homomorphisms.

The morphism from $\mathsf {C}^{*}_{G,K}(V,V)$ to $\mathsf {P}^{*}_{G,K}(V,V)$ is simply an inclusion, so it is certainly injective, and the morphism from $\mathsf {P}^{*}_{G,K}(V,V)$ to $\mathsf {S}_{G,K}^{*}(V,V)$ is certainly surjective, since it has dense range, and the images of ${C}^{*}$ -algebra morphisms are always closed. The continuity of the symbol mapping implies that the composition

$$\begin{align*}\mathsf{C}^{*}_{G,K}(V,V) \longrightarrow \mathsf{P}^{*}_{G,K}(V,V) \longrightarrow \mathsf{S}_{G,K}^{*} (V,V) \end{align*}$$

is zero. So it only remains to show that this part of the sequence is exact.

Suppose then that $T\in \mathsf {P}^{*}_{G,K}(V,V)$ and that the principal symbol of $T$ is zero. Write $T$ as an $L^2$ -operator norm limit of pseudodifferential operators $A_n\in \Psi ^0_{G,K}(V,V)$ . Then choose a smooth compactly supported function $\varphi $ on $K \backslash G$ such that $\mathsf {Av}(\varphi ^2) = \operatorname {id}$ , and consider the operators

$$\begin{align*}\mathsf{Av}(\varphi A_n \varphi) \colon [L^2 (G)\otimes V]^K \longrightarrow [L^2 (G)\otimes V]^K. \end{align*}$$

It follows from Lemma 3.5.5 that the operators $\mathsf {Av}(\varphi A_n \varphi )$ are elements of $\Psi ^0_{G,K}(V,V)$ , and furthermore that

$$\begin{align*}\mathsf{Av}(\varphi A_n\varphi) - A_n \in \Psi^{-1}_{G,K}(V,V), \end{align*}$$

since the principal symbol of the difference is zero. It therefore follows from Proposition 3.4.5 that

(3.5.6) $$ \begin{align} \mathsf{Av}(\varphi A_n\varphi) - A_n \in \mathsf{C}^{*}_{G,K}(V,V) \end{align} $$

for all $n$ . We therefore find that

(3.5.7) $$ \begin{align} \mathsf{Av}(\varphi T\varphi) - T \in \mathsf{C}^{*}_{G,K}(V,V) , \end{align} $$

since by Lemma 3.5.4 the operator in (3.5.7) is the norm limit of the operators in (3.5.6). So to complete the proof it suffices to show that

(3.5.8) $$ \begin{align} \mathsf{Av}(\varphi T \varphi)\in \mathsf{C}^{*}_{G,K}(V,V). \end{align} $$

For this we may invoke known facts about compactly supported pseudodifferential operators, as follows.

The compactly supported operator $\varphi T\varphi $ lies in the operator norm-closure of the algebra of compactly supported, order zero classical pseudodifferential operators, and has vanishing principal symbol. It is therefore a norm-limit of compactly supported smoothing operators $B_n$ , by Theorem 2.4.3. Choosing a smooth, compactly supported function $\psi $ with $\psi \varphi = \varphi $ , we now have

$$\begin{align*}\lim_{n\to \infty}\bigl \| \mathsf{Av}( \varphi T \varphi) - \mathsf{Av}( \psi B_n \psi) \bigr \| = \lim_{n\to \infty} \bigl \| \mathsf{Av}( \psi \varphi T \varphi\psi ) - \mathsf{Av}( \psi B_n \psi) \bigr \| = 0, \end{align*}$$

where the second equality is a consequence of Lemma 3.5.4. This proves (3.5.8) and therefore the proposition.

4.1 C*-categories from pseudodifferential operators

Here are the ${C}^{*}$ -categories that we wish to study.

The ${C}^{*}$ -category $\mathsf {C}^{*}_{G,K}$ is not only a subcategory of $\mathsf {P}^{*}_{G,K}$ but also an ideal, in the sense that the composition of any composable pair of morphisms in $\mathsf {P}^{*}_{G,K}$ that includes at least one morphism from $\mathsf {C}^{*}_{G,K}$ is a morphism in $\mathsf {C}^{*}_{G,K}$ ; see [Reference Mitchener36, Def. 4.2]. In fact according to Proposition 3.5.3, the category $\mathsf {C}^{*}_{G,K}$ is the kernel of the principal symbol functor from $\mathsf {P}^{*}_{G,K}$ to $\mathsf {S}^{*}_{G,K}$ (by kernel we mean the subcategory of $\mathsf {P}^{*}_{G,K}$ comprised of all objects of $\mathsf {P}^{*}_{G,K}$ and all morphisms that are mapped to $0$ by the symbol functor). Moreover $\mathsf {S}^{*}_{G,K}$ is the quotient ${C}^{*}$ -category $\mathsf {P}^{*}_{G,K} / \mathsf {C}^{*}_{G,K}$ in the sense of [Reference Mitchener36, Def. 4.4]. We shall express this by saying that there is an extension of ${C}^{*}$ -categories

$$\begin{align*}0 \longrightarrow \mathsf{C}^{*}_{G,K} \longrightarrow \mathsf{P}^{*}_{G,K} \longrightarrow \mathsf{S}^{*}_{G,K} \longrightarrow 0. \end{align*}$$

4.2 $K$ -Theory for the C*-category generated by smoothing operators

We turn now to $K$ -theory. The ${C}^{*}$ -categories that we defined in the previous section are all additive, and for an additive ${C}^{*}$ -category $\mathsf A$ with unit morphisms, probably the simplest way to define the $K_0$ -group is as the Grothendieck group of equivalence classes of idempotent morphisms in $\mathsf A$ .Footnote ⁵ If $\mathsf A$ does not have unit morphisms, one may embed $\mathsf A$ as an ideal in a larger ${C}^{*}$ -category $\mathsf B$ with unit morphisms, and then define $K_0(\mathsf {A})$ as the relative $K_0$ group for the function $\mathsf {B}\to \mathsf {B}/\mathsf {A}$ ; compare [Reference Karoubi24, Sec. II.2]. Then one may define $K_1$ using a suspension construction, in which a morphism in the suspended category is a path of morphisms in the original category that begins and ends at the zero morphism. See [Reference Karoubi24, Sec. II.3].

All this is exactly analogous to how one may define $K$ -theory for ${C}^{*}$ -algebras. For other approaches, which are analogous to various other approaches to ${C}^{*}$ -algebra $K$ -theory, see [Reference Mitchener35, Reference Joachim23].

In this section, we shall prove that the $K$ -theory groups of the ${C}^{*}$ -category $\mathsf {C}^{*}_{G,K}$ coincide with those of ${C}^{*}_r(G)$ .

Definition 4.2.1. If $S \subseteq \widehat {K}$ is a finite subset, and if $H$ is any Hilbert space that is equipped with a unitary representation of $K$ , then we shall denote by $P_S\colon H\to H$ the operator of convolution over $K$ with the function

$$\begin{align*}k\longmapsto \sum_{\pi \in S} \frac{\dim(\pi)}{\mu(K)} \chi_{\pi}(k^{-1}), \end{align*}$$

where $\chi _\pi $ is the character of $\pi $ , so that $P_S$ is the orthogonal projection onto the direct sum of the $\pi $ -isotypical components of $H$ , for $\pi \in S$ . In addition, we shall denote by

$$\begin{align*}{C}^{*}_r(G;S)\subseteq \mathfrak{B}(L^2 (G)) \end{align*}$$

the compression $P_S{C}^{*}_r(G)P_S$ , where we let $K$ act on $L^2(G)$ by restriction of the left-regular representation of $G$ . This is a ${C}^{*}$ -subalgebra of ${C}^{*}_r(G)$ .

Lemma 4.2.2. The inclusions of all ${C}^{*}_r (G;S)$ into ${C}^{*}_r (G)$ induce an isomorphism

$$\begin{align*}\varinjlim_{S} K_{*}({C}^{*}_r(G;S)) \stackrel \cong \longrightarrow K_{*}({C}^{*}_r(G)) \end{align*}$$

in $K$ -theory.

A similar result holds for the ${C}^{*}$ -category $\mathsf {C}^{*}_{G,K}$ .

Lemma 4.2.4. The inclusions of the subcategories $\mathsf {C}^{*}_{G,K}(S)$ into $\mathsf {C}^{*}_{G,K}$ induce an isomorphism

$$\begin{align*}\varinjlim K_{*}(\mathsf{C}^{*}_{G,K}(S)) \stackrel \cong \longrightarrow K_{*}(\mathsf{C}^{*}_{G,K}) \end{align*}$$

in $K$ -theory.

Now, the projection $P_S$ in Definition 4.2.1 is (the operator of convolution with) a smooth function on $K$ , and a central projection in ${C}^{*}_r(K)$ , and the inclusion morphisms induce vector space isomorphisms

(4.2.5) $$ \begin{align} P_S {C}^{*}_r (K) \stackrel\cong\longleftarrow P_S C^\infty (K) \stackrel \cong \longrightarrow P_SL^2(K). \end{align} $$

Let us write

$$\begin{align*}V_S = P_S L^2(K); \end{align*}$$

this is a finite-dimensional unitary representation of $K$ . The inner product on $V_S$ is transported, via the isomorphisms in (4.2.5), to the inner product

$$\begin{align*}\langle f,h\rangle = (f^{*}h)(e) \end{align*}$$

on $P_S {C}^{*}_r (K)$ , and the correspondence

$$\begin{align*}{C}^{*}_r(K)P_S \ni f \longmapsto \bigl [ h\mapsto \langle f^{*}, h\rangle \bigr] \in V_S^{*} \end{align*}$$

identifies ${C}^{*}_r(K)P_S$ with the dual of $V_S$ , as unitary representations of $K$ . From this we obtain a $K$ -bi-equivariant isomorphism

$$\begin{align*}P_S{C}^{*}_r (K)\otimes {C}^{*}_r(K)P_S \stackrel \cong \longrightarrow V_S \otimes V_S^{*} \stackrel \cong \longrightarrow \operatorname{{End}} (V_S). \end{align*}$$

From this we obtain, by rearranging the tensor factors, a linear isomorphism

(4.2.6) $$ \begin{align} \left[P_S {C}^{*}_r(K) \otimes {C}^{*}_r(G) \otimes {C}^{*}_r(K)P_S\right]^{K \times K} \stackrel \cong\longrightarrow \left[ {C}^{*}_r(G) \otimes {End}(V_S) \right]^{K \times K}, \end{align} $$

where, on the left-hand side, the first factor of $K$ acts on the left of $P_S {C}^{*}_r (K)$ and on the left of ${C}^{*}_r(G)$ , while the second factor of $K$ acts on the right of ${C}^{*}_r(G)$ and on the right of ${C}^{*}_r (K)P_S$ .

The left-hand side in (4.2.6) has a natural $*$ -algebra structure, for which the product is

$$\begin{align*}(f_1 \otimes f_2 \otimes f_3) \cdot (h_1 \otimes h_2 \otimes h_3) = f_1 \otimes (f_2 f_3 h_1 h_2) \otimes h_3 \end{align*}$$

and the $*$ -operation is

$$\begin{align*} (f_1 \otimes f_2 \otimes f_3)^{*} = (f_3^{*} \otimes f_2^{*} \otimes f_1^{*}). \end{align*}$$

Lemma 4.2.7. The formula

$$\begin{align*}(f_1 \otimes f_2 \otimes f_3) \mapsto f_1 f_2 f_3 \end{align*}$$

defines an isomorphism of $*$ -algebras

$$\begin{align*} \left[P_S {C}^{*}_r(K) \otimes {C}^{*}_r(G) \otimes {C}^{*}_r(K)P_S\right]^{K \times K} \stackrel \cong \longrightarrow P_S{C}^{*}_r(G)P_S. \end{align*}$$

We obtain from (4.2.6) and Lemma 4.2.7 an isomorphism of $*$ -algebras

(4.2.8) $$ \begin{align} {C}^{*}_r(G;S) \stackrel \cong \longrightarrow \left[ {C}^{*}_r(G) \otimes {End}(V_S) \right]^{K \times K} , \end{align} $$

which is necessarily also an isomorphism of ${C}^{*}$ -algebras. Now, we noted in (3.3.8) that the right-hand side of (4.2.8) is the endomorphism ${C}^{*}$ -algebra $\mathsf {C}^{*}_{G,K}(V_S,V_S)$ in the ${C}^{*}$ -category $\mathsf {C}^{*}_{G,K}$ . So since each ${C}^{*}$ -algebra is a ${C}^{*}$ -category on one object, the isomorphism (4.2.8) can be viewed as a functor between ${C}^{*}$ -categories, and it therefore induces a morphism of $K$ -theory groups

(4.2.9) $$ \begin{align} K_*({C}^{*}_r(G;S)) \longrightarrow K_*(\mathsf{C}^{*}_{G,K}(S)). \end{align} $$

This is indeed an isomorphism, because $V_S$ is a finite projective generator for the category $\mathsf {C}^{*}_{G,K}(S)$ .

Theorem 4.2.10. Let $G$ be a Lie group and let $K$ be a compact subgroup of $G$ . There is a unique isomorphism of $K$ -theory groups

$$\begin{align*}K_*(C_r^{*}(G)) \stackrel \cong \longrightarrow K_*(\mathsf{C}^{*}_{G,K}) \end{align*}$$

such that, for every $S$ , the diagram

is commutative (the vertical arrows are induced from inclusions).

5.1 The deformation to the normal cone and its C*-algebra

The motion group associated to $G$ and $K$ is the semidirect product group

$$\begin{align*}G_0 = K \ltimes ( \mathfrak{g}/\mathfrak{k} ) \end{align*}$$

constructed from the adjoint action of $K$ on the quotient vector space $\mathfrak {g}/\mathfrak {k}$ . The motion group fits into a smooth, one-parameter family of groups $\{ G_t \} _{t\in \mathbb {R}}$ with

$$\begin{align*}G_t = \begin{cases} G & t\ne 0 \\ K \ltimes ( \mathfrak{g}/\mathfrak{k} ) & t=0. \end{cases} \end{align*}$$

That is, there is a smooth manifold $\pmb {G} $ and a submersion $\pmb {G}\to \mathbb {R}$ whose fibers are these groups, for which the fiberwise-defined group operations of multiplication, inverse, and inclusion of the group identity element,

$$\begin{align*}\pmb{G} \times _{\mathbb{R}} \pmb{G} \longrightarrow \pmb{G} , \quad \pmb{G} \longrightarrow \pmb{G} \quad \text{and}\quad \mathbb{\mathbb{R}} \longrightarrow \pmb{G} \end{align*}$$

are smooth maps. This is an instance of the deformation to the normal cone construction; see [Reference Debord and Skandalis13, Sec. 2.4] for a concise introduction to the topic.

The smooth manifold structure on $\pmb {G}$ may be characterized as follows. First, the complement of $G_0$ is an open subset, and carries the standard smooth structure of $G\times \mathbb {R}^{\times }$ . Second, if $V$ is any finite-dimensional real vector space, and if $E\colon V \to G$ is any smooth map such that

1. (i) $E(0) = e$ , and

2. (ii) the map

   $$\begin{align*}\begin{aligned} K\times V \longrightarrow G \\ (k,X) \longmapsto k \cdot E(X) \end{aligned} \end{align*}$$
   is a diffeomorphism onto an open subset of G,

then the map

$$\begin{align*}K \times V \times \mathbb{R} \longrightarrow \pmb{G} \end{align*}$$

defined by

$$\begin{align*}(k,v,t) \longmapsto \begin{cases} k \cdot E(tv) \in G_t & t \ne 0 \\ (k,dE(v))\in G_0 & t=0 \end{cases} \end{align*}$$

is a diffeomorphism onto an open subset of $\pmb {G}$ . Here $dE \colon V \to \mathfrak {g} / \mathfrak {k}$ is the composition of the derivative of the smooth map $E$ at $0\in V$ with the projection map from $\mathfrak {g} $ to the quotient vector space $\mathfrak {g}/\mathfrak {k}$ .

Associated to $\pmb {G} $ is a continuous field of ${C}^{*}$ -algebras $\{ {C}^{*}_r (G_t)\} _{t\in \mathbb {R}}$ (see [Reference Dixmier14, Ch. 10] for background information on continuous fields). It is constructed by choosing a smoothly varying family of Haar measures on the fiber groups $G_t$ , and then decreeing that the continuous sections of the continuous field be generated by the smooth, compactly supported functions on $\pmb {G} $ : from each such function $f$ one obtains by restriction a family of functions $f_t$ on the groups $G_t$ , and it is proved in [Reference Higson17, Lemma 6.13] that the norms $\| f_t\|_{{C}^{*}_r (G_t)}$ vary continuously with $t$ .

The following difficult and important result will be essential to everything that follows:

The case $t{=}0$ is elementary, but for all other $t{\in } [0,1]$ the theorem is by no means trivial. In fact Connes has pointed out in [Reference Connes11, Prop. 9, p.141] that the assertion in the theorem is equivalent to the Connes-Kasparov isomorphism.

5.2 $K$ -theory for the C*-category of pseudodifferential operators

In this section we shall reformulate Theorem 5.1.2, or equivalently the Connes-Kasparov isomorphism, as an assertion about the $K$ -theory of the ${C}^{*}$ -category of pseudodifferential operators that we have constructed in this paper.

Definition 5.2.2. Let $G$ be an almost-connected Lie group and let $K$ be a maximal compact subgroup of $G$ . We shall denote by

the functor of ${C}^{*}$ -categories that is the identity on objects, and maps a morphism of representations $T\colon V_1\to V_2$ to the induced morphism

$$\begin{align*}T \colon [L^2( G)\otimes V_1]^K \longrightarrow [L^2( G)\otimes V_2]^K \end{align*}$$

(which is a particularly simple example of an equivariant, properly supported, classical order $0$ pseudodifferential operator).

The name is a reference to Connes and Kasparov, for we shall prove that Theorem 5.1.2 implies, and indeed is equivalent to, the following assertion:

The main idea of the proof is to reduce to the case in which $G$ is replaced by its motion group $G_0$ . To make this reduction we shall avail ourselves of the smooth family of groups $\pmb {G}=\{ G_t\}$ and the associated continuous field of ${C}^{*}$ -algebras $\{ {C}^{*}_r (G_t)\}$ . We shall begin preparations for the proof by considering not order zero pseudodifferential operators, but smoothing operators.

Definition 5.2.4. Let $G$ be an almost-connected Lie group with maximal compact subgroup $K$ . We shall denote by $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}$ the ${C}^{*}$ -category whose objects are the finite-dimensional unitary representations of $K$ , and whose morphism spaces $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}(V_1,V_2)$ are norm closures of the spaces of properly supported smooth families of equivariant smoothing operators

$$\begin{align*}T_t \colon L^2 (K \backslash G_t, V_1) \longrightarrow L^2(K \backslash G_t, V_1)\qquad (t\in [0,1]) \end{align*}$$

(see below) in the norm $\| T\| = \sup \{\, \|T_t\|: t\in [0,1]\,\}$ .

In order to be more specific about the term properly supported smooth family, we use the description of individual properly supported and equivariant smoothing operators as elements of the spaces

(5.2.5) $$ \begin{align} \bigl [C_c^\infty (G_t)\otimes \operatorname{{Hom}} (V_1,V_2) \bigr ]^{K\times K} \end{align} $$

in Section 3.3. Our requirement on properly supported, smooth families $\{ T_t\}$ of equivariant smoothing operators is that there should be a single element in the space

$$\begin{align*}\bigl [ C_c^\infty (\pmb{G}) \otimes \operatorname{{Hom}} (V_1,V_2)\bigr ]^{K\times K} \end{align*}$$

that restricts at each $t\in [0,1]$ to the element in (5.2.5) corresponding to $T_t$ .

With this definition, it is evident that there is an isomorphism of ${C}^{*}$ -categories that is the identity on objects, and on morphism spaces takes the form of isomorphisms

$$\begin{align*}\mathsf{C}^{*}_{\pmb{G},\pmb{K}}(V_1,V_2) \stackrel \cong \longrightarrow \bigl [ {C}^{*}_r (\pmb{G}) \otimes \operatorname{{Hom}} (V_1,V_2)\bigr ]^{K\times K}. \end{align*}$$

Theorem 5.2.6. The Connes-Kasparov isomorphism in Theorem 5.1.2 is equivalent to the assertion that for all $t\in [0,1]$ the functor of evaluation at $t$ ,

$$\begin{align*}\mathsf{C}^{*}_{\pmb{G},\pmb{K}} \longrightarrow \mathsf{C}^{*}_{G_t, K}, \end{align*}$$

induces an isomorphism in $K$ -theory.

Proof. We may repeat the arguments of Section 4.2 using the smooth family $\pmb {G}$ over $[0,1]$ in place of the single group $G$ . So for a finite subset $S\subseteq \widehat K$ we may define the ${C}^{*}$ -algebra ${C}^{*}_r(\pmb {G};S)$ , the ${C}^{*}$ -category $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}(S)$ , and the $K$ -theory isomorphism

$$ \begin{align*} K_*({C}^{*}_r(\pmb{G};S)) \stackrel\cong \longrightarrow K_*(\mathsf{C}^{*}_{\pmb{G},\pmb{K}}(S)). \end{align*} $$

analogous to (4.2.9). These determine an isomorphism

(5.2.7) $$ \begin{align} K_*({C}^{*}_r(\pmb{G})) \stackrel\cong \longrightarrow K_*(\mathsf{C}^{*}_{\pmb{G},\pmb{K}}) \end{align} $$

analogous to the isomorphism in Theorem 4.2.10. Now, (5.2.7) is compatible with evaluation at $t\in [0,1]$ , in the sense that the diagram

commutes. The theorem follows from this.

The proof of Theorem 5.2.3 will require a similar treatment of the order zero pseudodifferential operators: we shall need to organize families of order zero operators on the family of homogeneous spaces $K \backslash G_t$ into a ${C}^{*}$ -category, $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ . This is a bit more complicated, and to begin we shall simply assert the existence of a suitable category, and explain how it may be used to prove Theorem 5.2.3. We shall actually construct $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ in Section 5.3.

As usual, the objects of $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ will be the finite-dimensional unitary representations of $K$ . The morphisms will be suitable continuous families of equivariant, properly supported, order zero classical pseudodifferential operators

(5.2.8) $$ \begin{align} A_t \colon [ L^2 (G_t)\otimes V_1]^K \longrightarrow [ L^2 (G_t)\otimes V_2]^K\qquad (t\in [0,1]), \end{align} $$

or norm-completions of such families. Here are the properties of the category $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ that we shall require for the argument:

1. (i) The category $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ includes the category $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}$ as an ideal.

2. (ii) The individual operators in a morphism $\{ A_t : t\in [0,1]\}$ in each family all have the same principal symbol (observe that for all $t$ , the tangent space of $K \backslash G_t$ identifies naturally with $\mathfrak {g} / \mathfrak {k}$ , so that the principal symbol is, for all $t$ , a continuous function on $\partial ( \mathfrak {g} / \mathfrak {k})^{*}$ ).

3. (iii) The principal symbol functor (that associates to a family of operators the common principal symbol) fits into an extension of ${C}^{*}$ -categories

   $$\begin{align*}0 \longrightarrow \mathsf{C}^{*}_{\pmb{G},\pmb{K}} \longrightarrow \mathsf{P}^{*}_{\pmb{G},\pmb{K}}\longrightarrow \mathsf{S}^{*}_{G,K} \longrightarrow 0 \end{align*}$$

4. (iv) If $T\colon V_1 \to V_2$ is a morphism of finite-dimensional unitary representations of $K$ , then the family of induced morphisms

   $$\begin{align*}T_t \colon [ L^2 (G_t)\otimes V_1]^K \longrightarrow [ L^2 (G_t)\otimes V_2]^K \qquad (t \in [0,1]) \end{align*}$$
   is a morphism in $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ .

Taking this for granted, for now, we shall proceed to study the $K$ -theory of the category $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ .

Theorem 5.2.10. The Connes-Kasparov isomorphism in Theorem 5.1.2 is equivalent to the assertion that for all $t\in [0,1]$ the functor of evaluation at $t$ ,

$$\begin{align*}\mathsf{P}^{*}_{\pmb{G},\pmb{K}} \longrightarrow \mathsf{P}^{*}_{G_t, K}, \end{align*}$$

induces an isomorphism in $K$ -theory.

Proof. This follows from Theorem 5.2.6 and the five lemma, applied to the diagram of $K$ -theory six-term exact sequences associated to the diagram

where the vertical maps are given by evaluation at $t\in [0,1]$ .

Let us now examine the category $\mathsf {P}^{*}_{G_0,K}$ .

Proof. In the case of the motion group, the homogeneous space $K\backslash G_0$ identifies with the vector space $\mathfrak {g} / \mathfrak {k}$ , and the $G_0$ -equivariant operators

$$\begin{align*}T \colon [C_c^\infty ( G_0 )\otimes V_1]^K \longrightarrow [C_c^\infty ( G_0 )\otimes V_2]^K \end{align*}$$

are precisely the operators

$$\begin{align*}T \colon C_c^\infty (\mathfrak{g}/\mathfrak{k})\otimes V_1 \longrightarrow C_c^\infty (\mathfrak{g}/\mathfrak{k}) \otimes V_2 \end{align*}$$

that are both $K$ -equivariant and translation-invariant.

Being translation-invariant, the operators $T$ above may be analyzed using the Fourier transform. We find that the properly supported, $G_0$ -equivariant, order $0$ classical pseudodifferential operators $T$ as above are precisely the operators of convolution by $\operatorname { {Hom}} (V_1,V_2)$ -valued, compactly supported distributions on $\mathfrak {g}/\mathfrak {k}$ whose Fourier transforms are $K$ -equivariant, order zero classical symbol functions on $(\mathfrak {g}/\mathfrak {k})^{*}$ with values in $\operatorname { {Hom}} (V_1,V_2)$ .

Now, every order zero, classical symbol function extends by continuity to the compactification of $(\mathfrak {g}/\mathfrak {k})^{*}$ in Definition 2.3.2, and these extended functions separate the points of the compactification. This yields an isomorphism of ${C}^{*}$ -categories from $\mathsf {P}^{*}_{G_0,K}$ to the ${C}^{*}$ -category whose objects are the finite-dimensional unitary representations of $K$ , and whose morphisms are elements of the spaces

$$\begin{align*}C \bigl (\overline {(\mathfrak{g} / \mathfrak{k})^{*}} , \operatorname{{Hom}} (V_1,V_2)\bigr )^K. \end{align*}$$

The functor corresponds to the inclusion

$$\begin{align*}\operatorname{{Hom}} (V_1,V_2)^K \longrightarrow C \bigl (\overline {(\mathfrak{g} / \mathfrak{k})^{*}} , \operatorname{{Hom}} (V_1,V_2)\bigr )^K \end{align*}$$

as constant functions. It has a left-inverse

$$\begin{align*}C \bigl (\overline {(\mathfrak{g} / \mathfrak{k})^{*}} , \operatorname{{Hom}} (V_1,V_2)\bigr )^K\longrightarrow \operatorname{{Hom}} (V_1,V_2)^K \end{align*}$$

given by evaluation of functions at $0$ . The composition of the two functors in the opposite order is the functor

$$\begin{align*}F\colon C \bigl (\overline {(\mathfrak{g} / \mathfrak{k})^{*}} , \operatorname{{Hom}} (V_1,V_2)\bigr )^K \longrightarrow C \bigl (\overline {(\mathfrak{g} / \mathfrak{k})^{*}} , \operatorname{{Hom}} (V_1,V_2)\bigr )^K \end{align*}$$

that replaces each function by the constant function with the same value at $0$ . Because the compactification $\overline {(\mathfrak {g} / \mathfrak {k})^{*}}$ is $K$ -equivariantly contractible, the composition is homotopic through a family of functors $F_t$ to the identity functor (the continuity condition on the family is that for every morphism $a$ , $F_t(a)$ is a norm-continuous function of $t$ ). It therefore follows from the homotopy-invariance of $K$ -theory that $F$ induces the identity map on $K$ -theory. Thus induces an isomorphism on $K$ -theory, as required.

With this, we can complete the proof of Theorem 5.2.3. By the assumptions (i) and (iii) above, the functor

(5.2.12) $$ \begin{align} \mathsf{P}^{*}_{\pmb{G},\pmb{K}} \longrightarrow \mathsf{P}^{*}_{G_0,K} \end{align} $$

of evaluation at $t{=}0$ is surjective on Hom-spaces. Its kernel—the ideal in $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ comprised of all morphisms that map to a zero morphism—consists of those operator families that define morphisms in $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ and that vanish at $t{=}0$ . In particular, the principal symbols of such families vanish, and so by (iii) the families are morphisms in $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}$ . Therefore the kernel is precisely the ideal in $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}$ whose morphisms are the families that vanish at $t{=}0$ . This ideal is contractible in the sense of [Reference Mitchener35, Sec. 3] and so has vanishing $K$ -theory [Reference Mitchener35, Prop. 3.23]. Therefore, by the $K$ -theory long-exact sequence associated to an extension of ${C}^{*}$ -categories [Reference Mitchener35, Cor. 3.20], the functor (5.2.12) induces an isomorphism in $K$ -theory.

Next, we may define a functor

(5.2.13)

exactly as we defined the original version in Definition 5.2.2, but using families, and using property (iv) of the category $\mathsf {P}^{*} _{\pmb {G},\pmb {K}}$ listed above. The composition

(5.2.14)

is precisely the morphism that was proved in Theorem 5.2.11 to be an isomorphism at the $K$ -theory level. So the functor (5.2.13) induces an isomorphism on $K$ -theory, too.

So far we have not used the Connes-Kasparov isomorphism in Theorem 5.1.2. But if we now use it in the equivalent form presented in Theorem 5.2.10, then we may conclude that both morphisms in the composition

(5.2.15)

are isomorphisms for all $t \in [0,1]$ . But when $t \neq 0$ , the composition (5.2.15) is precisely the morphism in the statement of Theorem 5.2.3, and so the proof of Theorem 5.2.3 is complete.

In the reverse direction, if we assume Theorem 5.2.3, which is to say if we assume that the composition in (5.2.15) is an isomorphism when $t \neq 0$ , then since we already proved, without recourse to the Connes-Kasparov isomorphism, that (5.2.13) induces an isomorphism on $K$ -theory, it follows that evaluation at $t$ induces an isomorphism

$$\begin{align*}K_*(\mathsf{P}^{*}_{\pmb{G},\pmb{K}} ) \stackrel \cong \longrightarrow K_* (\mathsf{P}^{*}_{G_t,K} ) \end{align*}$$

for all $t\in [0,1]$ (the case $t = 0$ is covered by Theorem 5.2.11), and according to Theorem 5.2.10 this implies the Connes-Kasparov isomorphism.

5.3 Construction of a C*-category from families of pseudodifferential operators

In this section we shall construct the category $\mathsf {P}_{\pmb {G},\pmb {K}}$ that is used in the proof of Theorem 5.2.3.

The main issue is to decide when a family of operators (5.2.8) acting on the fibers of the smooth family of homogeneous spaces $\pmb K \backslash \pmb G$ should be deemed to be continuous. There are a number of equivalent ways of expressing the continuity condition, and we shall choose the one that seems to us to be the quickest (what seems to us to be the most conceptual approach will be outlined in the next section). The approach that we shall follow is based on the fact, first documented in [Reference Atiyah and Singer4], that if one conjugates a compactly supported, order zero pseudodifferential operator by a smooth family of diffeomorphisms, then one obtains an $L^2$ -operator norm-continuous family of bounded Hilbert space operators:

Theorem 5.3.1 (See for instance [Reference Atiyah and Singer4, Prop. 1.3])

Let $E_1$ and $E_2$ be smooth vector bundles over smooth manifolds $M_1$ and $M_2$ , respectively. Let $V$ be an open subset of some Euclidean space and let

$$\begin{align*}\Psi_v \colon E_1 \stackrel \cong \longrightarrow E_2\qquad (v\in V) \end{align*}$$

be a smooth family of vector bundle isomorphisms that covers a smooth family of diffeomorphisms from $M_1$ to $M_2$ . If $A$ is a compactly supported, classical pseudodifferential operator of order zero on $M_2$ , acting on the sections of $E_2$ , then the operators

$$\begin{align*}\Psi_v ^{*} A \Psi_v^{*,-1} \colon L^2 (M_1; E_1) \longrightarrow L^2 (M_1; E_1) \qquad (v \in V) \end{align*}$$

vary norm-continuously with $V$ .

Definition 5.3.2. Let $\pi \colon \pmb M \to W$ be a submersion of smooth manifolds with fibers $M_w$ ( $w\in W)$ and let $\pmb E $ be a smooth vector bundle over $\pmb M$ , restricting to bundles $E_w$ over the fibers $M_w$ . We shall say that a family of classical, pseudodifferential operators

$$\begin{align*}A_w \colon C_c^\infty (M_w;E_w) \longrightarrow C^\infty (M_w; E_w)\qquad (w\in W) \end{align*}$$

of order zero is continuous if

1. (a) for every pair of open sets $\pmb U\subseteq \pmb M$ and $V\subseteq W$ ,

2. (b) for every smooth, compactly supported function $\varphi $ on $\pmb {U}$ ,

3. (c) for every commuting diagram

   
   in which $L $ is a smooth manifold, $\Phi $ is a diffeomorphism and $\pi _V$ is the projection onto the factor $V$ , and

4. (d) for every vector bundle isomorphism

   $$\begin{align*}\Psi \colon L{\times} V {\times} \mathbb{C}^k \stackrel \cong \longrightarrow \pmb E \vert _{\pmb U} \end{align*}$$
   that covers $\Phi $ ,

the family of compactly supported, order zero, classical pseudodifferential operators

$$\begin{align*}\Psi^{*}_v \varphi_v A_v \varphi_v \Psi_v^{*,-1} \colon C_c^\infty (L;\mathbb{C}^k ) \longrightarrow C_c^\infty (L;\mathbb{C}^k )\qquad (v\in V) \end{align*}$$

extends to a norm-continuous family of operators between $L^2$ -spaces. Here

$$\begin{align*}\Psi_v^{*}\colon C_c^\infty ( M_v\cap \pmb U; E) \stackrel \cong \longrightarrow C_c^\infty (L; \mathbb{C}^n) \end{align*}$$

is the isomorphism of linear spaces induced from $\Psi $ , and $\varphi _v$ is the restriction of $\varphi $ to $M_v\cap \pmb U$ .

It follows from Theorem 5.3.1 that the continuity condition in the definition above can be checked using a single covering family of local trivializations in the sense of (a)-(d) above, and that continuous families may be constructed using such covering families and partitions of unity.

We are now almost ready to define the category $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ .

Definition 5.3.3. We shall say that a family of equivariant operators

$$\begin{align*}A _t \colon [C_c^\infty ( G_t )\otimes V_1]^K \longrightarrow [C_c^\infty ( G_t )\otimes V_2]^K \qquad (t \in \mathbb{R}) \end{align*}$$

is a properly supported family if all of the individual supports of the operators $A_t$ , in $K \backslash G_t \times K\backslash G_t$ , are included in a closed subset of $\pmb {K}\backslash \pmb {G} \times _{\mathbb {R}}\pmb {K}\backslash \pmb {G}$ for which the two coordinate projections to $\pmb {K}\backslash \pmb {G}$ are proper maps.

Definition 5.3.4. We shall denote by $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ the ${C}^{*}$ -category whose objects are the finite-dimensional unitary representations of $K$ , and whose morphism spaces are the norm-closures of the collections of all restrictions to $[0,1]$ of properly supported families

$$\begin{align*}A_t \colon [C_c^\infty ( G_t )\otimes V_1]^K \longrightarrow [C_c^\infty ( G_t )\otimes V_2]^K \qquad (t \in \mathbb{R}) \end{align*}$$

of equivariant, order zero, classical pseudodifferential operators that share the same principal symbol; that are smooth in the sense their compositions on the left or on the right with any smooth family of equivariant smoothing operators are again smooth families of smoothing operators; and that are norm-continuous in the sense of Definition 5.3.2. The norm here is

$$\begin{align*}\| \{ A_t\} \| = \sup \, \{ \|A_t \|_{L^2\to L^2} : t \in [0,1]\,\}. \end{align*}$$

The category $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}$ is included in $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ as a subcategory because if $\{ A_t\}$ is a smooth family of properly supported, equivariant smoothing operators on the fibers $G_t$ of $\pmb G$ , with integral kernels $k_t(g_t,h_t)$ that vary smoothly with $t$ , $g_t$ and $h_t$ , then any of the local trivializations constructed using the data (a)-(d) of Definition 5.3.2 is a smooth family of compactly supported smoothing operators, and so certainly varies norm-continuously with $t$ . Moreover it is included as an ideal by our smoothness assumption in Definition 5.3.4, and so $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}$ has property (i) in Section 5.2.

Property (ii) is built into Definition 5.3.4. Surjectivity of the principal symbol functor (on morphism spaces), may be proved using the averaging construction in [Reference Stetkaer44] and in Section 3.4. The fact that $\mathsf {C}^{*}_{\pmb {G},\pmb {K}}$ is the kernel of the principal symbol functor is proved similarly, by a variation of the argument in Section 3.5. Finally, property (iv) is clear.

5.4 Comments on the van Erp-Yuncken theory and the Mackey bijection

Erik van Erp and Robert Yuncken have presented in [Reference van Erp and Yuncken45] a new account of the theory of classical pseudodifferential operators that is particularly well adapted to the general perspectives of this paper. We have not used their approach here because some aspects of their theory, related to $L^2$ -operator norm estimates, are not yet well-documented in the literature. But their work is so relevant to our purposes that we want to at least mention it here.

Central to the van Erp-Yuncken approach is the zoom action

$$\begin{align*}\alpha_\lambda \colon \pmb{G} \longrightarrow \pmb{G}\qquad (\lambda> 0) \end{align*}$$

of the group $\mathbb {R}^+=\{ \lambda \in \mathbb {R} : \lambda> 0\}$ of positive real numbers on the deformation space $\pmb {G}$ by diffeomorphisms. This is defined by the formula

$$\begin{align*}\begin{cases} G_t\ni g \stackrel {\alpha _\lambda} \longmapsto g\in G_{\lambda t} & t {\ne } 0 \\ G_0\ni (k,X) \stackrel {\alpha _\lambda} \longmapsto (k,\lambda ^{-1} X)\in G_{ 0} & t{=}0. \end{cases} \end{align*}$$

Van Erp and Yuncken do not consider the equivariant context that we have studied in this paper, but if they were to have done so, then they would have defined equivariant order zero pseudodifferential operators as those operators on the $t{=}1$ fiber of the family of homogeneous spaces $\pmb {K}\backslash \pmb {G}$ that extend to smooth families of operators that remain fixed, modulo smooth families of smoothing operators, under the zoom action. With this, the category $\mathsf {P}^{*}_{\pmb {G},\pmb {K}}$ that we defined above would have been built into their theory from the start—as the category whose morphism sets are generated by families that are invariant under the zoom action, modulo smooth families of smoothing operators.

Another topic that we must at least mention is the Mackey bijection for real reductive groups [Reference Higson17, Reference Afgoustidis1, Reference Afgoustidis2]. This involves the same deformation space $\pmb {G}$ that appears here, and it is also intimately connected to the Connes-Kasparov isomorphism. Not only that, but the Mackey bijection is also closely related to the concept of tempiric representation that will figure prominently in the coming sections. So it is natural to wonder if there might be a direct connection between pseudodifferential operators and the Mackey bijection. Unfortunately, for the time being at least, we can shed no light on this question.

6.1 Multiplicity spaces

Harish-Chandra showed in [Reference Harish-Chandra15, Thm. 6] that if a unitary representation $\pi $ is irreducible, then it is admissible in the sense that it includes each irreducible representation of $K$ with at most finite multiplicity. Let $\pi $ be an admissible unitary representation of a real reductive group $G$ on a Hilbert space $H_\pi $ , and denote by $H^\infty _\pi $ the space of smooth vectors in $H_\pi $ (see for instance [Reference Knapp29, Ch. III]). If $\Lambda $ is a compactly supported distribution on $G$ , then there is unique linear operator

$$\begin{align*}\pi(\Lambda) \colon H^\infty _\pi \longrightarrow H^\infty _\pi \end{align*}$$

such that

$$\begin{align*}\langle w_1 ,\pi(\Lambda) w_2\rangle = \int_G \Lambda (g) \langle w_1, \pi(g) w_2\rangle\, dg \qquad \forall w_1\in H_\pi ,\,\, \forall w_2 \in H^\infty _\pi. \end{align*}$$

Let $\mathcal {E}'(G)$ denote the space of compactly supported distributions on $G$ . From the above, if $A \in [ \mathcal {E}'(G) \otimes \operatorname { {Hom}}(V_1,V_2)]^{K\times K}$ , say $A = \sum _k \Lambda _k \otimes T_k$ , then we obtain an operator

(6.1.1) $$ \begin{align} \pi(A) \colon [ H^\infty _\pi\otimes V_1]^K \longrightarrow [ H^\infty _\pi \otimes V_2] ^K \end{align} $$

by means of the formula

$$\begin{align*}\pi(A) \colon w\otimes v \longmapsto \sum_k \pi(\Lambda_k)w\otimes T_k v. \end{align*}$$

The operator (6.1.1) may also be characterized by the formula

(6.1.2) $$ \begin{align} \pi(A) (\pi(f) F) & = \pi(Af)F \\ & \forall f \in [C_c^\infty(G)\otimes \operatorname{{End}} (V_1)]^{K\times K},\,\, \forall F \in [C_c^\infty (G) \otimes V_1]^K .\nonumber \end{align} $$

This uses the action in (3.3.5), plus the evident convolution action

$$ \begin{align*} [ \mathcal{E}'(G) \otimes \operatorname{{Hom}}(V_1,V_2)]^{K\times K} & \times [C_c^\infty(G)\otimes \operatorname{{End}} (V_1)]^{K\times K}\\ & \longrightarrow [C_c^\infty(G)\otimes \operatorname{{Hom}} (V_1,V_2)]^{K\times K}. \end{align*} $$

Of course, all of the above applies when $A$ is a properly supported, $G$ -equivariant, order zero classical pseudodifferential operator,

(6.1.4) $$ \begin{align} A \colon [C_c^\infty (G) \otimes V_1]^K \longrightarrow [C_c^\infty (G) \otimes V_2]^K. \end{align} $$

But we shall also need to extend the construction in (6.1.1) to $L^2$ -operator-norm limits order zero pseudodifferential operators. To prepare for this, let us make the following observation:

Lemma 6.1.5. If $\pi $ is an admissible representation, then the inclusion $H^\infty _\pi \hookrightarrow H_\pi $ induces an isomorphism of finite-dimensional vector spaces

$$\begin{align*}\bigl [ H_\pi^\infty \otimes V\bigr ] ^K\stackrel{\cong}\longrightarrow \bigl [ H_\pi \otimes V\bigr ] ^K. \end{align*}$$

In the following lemma we shall use the (completed) balanced tensor product from Hilbert ${C}^{*}$ -module theory. This was introduced by Kasparov [Reference Kasparov25, Sec. 2.8], but see [Reference Lance34] for an exposition of the topic.

Lemma 6.1.6. Let $\pi $ be an admissible tempered unitary representation of $G$ and let $V$ be a finite-dimensional unitary representation of $K$ . There is a unique isomorphism of Hilbert spaces

$$\begin{align*}\bigl [ C_r^{*}(G) \otimes V\bigr ] ^K \otimes_{{C}^{*}_r(G)} H_\pi \stackrel \cong \longrightarrow \bigl [ H_\pi \otimes V\bigr ] ^K \end{align*}$$

such that

$$\begin{align*}\left ( \sum _j f_j \otimes v_j \right ) \otimes \xi \longmapsto \sum _j \pi(f_j)\xi \otimes v_j. \end{align*}$$

The following definition extends the construction of $\pi (A)$ in (6.1.1).

Definition 6.1.7. Let $\pi $ be an admissible tempered unitary representation of $G$ , viewed as a representation of the ${C}^{*}$ -algebra ${C}^{*}_r (G)$ , and let

$$\begin{align*}T \colon [C_c^\infty (G)\otimes V_1]^K \longrightarrow [C_c^\infty (G)\otimes V_2]^K \end{align*}$$

be a $G$ -equivariant, properly supported, adjointable and $L^2$ -bounded operator. We shall denote by

$$\begin{align*}\pi(T) \colon \bigl [ H_\pi \otimes V_1\bigr ] ^K \longrightarrow \bigl [ H_\pi \otimes V_2\bigr ] ^K \end{align*}$$

the tensor product operator

$$\begin{align*}\pi(T)= T\otimes \mathrm{id}_{H_\pi} \colon [C_r^{*} (G)\otimes V_1]^K\otimes _{{C}^{*}_r(G)} H_\pi \longrightarrow [C_r^{*} (G)\otimes V_2]^K \otimes _{{C}^{*}_r(G)} H_\pi , \end{align*}$$

which we view an operator between multiplicity spaces using the isomorphisms in Lemma 6.1.6.

The construction just described can be dressed up in ${C}^{*}$ -category clothes, as follows.

If $\pi $ is any admissible tempered unitary representation of $G$ , then the multiplicity space construction determines a ${C}^{*}$ -functor

(6.1.9)

by means of the formula

We shall use this ${C}^{*}$ -categorical perspective on the multiplicity space construction in Section 7. But for now, roughly speaking, we may viewing the family of all multiplicity functors as a Fourier transform for the category $\mathsf {P}^{*}_{G,K}$ . We shall refine this idea in the remainder of Section 6.

6.2 Multiplicity spaces for principal series representations

When $\pi $ is a principal series representation, the constructions in the previous section may be made more explicit by using an explicit model for the representation $\pi $ . The purpose of this section is to explain this for the unitary minimal principal series (essentially the same picture emerges for all the unitary principal series representations, but in this paper we are mostly concerned with groups that are of real rank one, for which there are no higher principal series representations beyond the minimal principal series).

Let $G$ be a real reductive group and let $G=KAN$ be an Iwasawa decomposition; in addition let $M$ be the centralizer of $A$ in $K$ , so that $P=MAN$ is a minimal parabolic subgroup of $G$ . See [Reference Knapp30, Chs. VI,VII] or [Reference Vogan46]. Denote by $\rho \in \mathfrak {a}^{*}$ the half-sum of the positive restricted roots associated to $N$ , as in [Reference Knapp30] or [Reference Vogan46]. If $\sigma $ is an irreducible unitary representation of the compact group $M$ on a finite-dimensional Hilbert space $L_\sigma $ , and if $\nu \in \mathfrak {a}^{*}$ , then the completion of the space

(6.2.1) $$ \begin{align} H^\infty _{\sigma,i\nu} & = \bigl \{ \, f\colon G \stackrel{C^\infty}\to L_\sigma \, : \, f(gman) = e^{-(\rho + i \nu) (\log (a)) } \sigma(m)^{-1} f(g) \nonumber \\& \forall g\in G, m\in M , a\in A, n\in N \,\} \end{align} $$

in the norm

(6.2.2) $$ \begin{align} \| f\| ^2 = \int_K \| f(k) \|\, dk \end{align} $$

is the Hilbert space $H_{\sigma , i\nu }$ of the unitary minimal principal series representation

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

(the action of $G$ is by left translation). As the notation suggests, $H^\infty _{\sigma , i \nu }$ is in fact the space of smooth vectors for the representation in the Hilbert space $H_{\sigma , i \nu }$ .

Now let $V$ be any finite-dimensional unitary representation of $K$ and consider the linear map

(6.2.3) $$ \begin{align} H_{\sigma,\nu} ^\infty\otimes V \ni f\otimes v \longmapsto \bigl [ g\mapsto v \otimes f(g)\bigr ] \in C^\infty (G)\otimes V\otimes L_\sigma. \end{align} $$

It is a morphism of $C^\infty _c(G)$ -modules for the left convolution action of $G$ on $C^\infty (G)$ . Keeping in mind that the multiplicity space may be computed using the smooth vectors $H^\infty _{\sigma , i \nu }\subseteq H_{\sigma , i \nu }$ , we find that (6.2.3) restricts to an isomorphism

(6.2.4) $$ \begin{align} \begin{aligned} & [ H _{\sigma,i\nu}\otimes V]^K \stackrel\cong\longrightarrow \bigl \{ \, f\colon G \stackrel{C^\infty}\to V\otimes L_\sigma \, : \,f(kgman) = e^{-(\rho + i \nu) (\log (a) }\cdot \\& \quad (k\otimes \sigma(m)^{-1}) f(g)\quad \forall g\in G, m\in M , a\in A, n\in N \,\bigl \}. \end{aligned} \end{align} $$

We shall express this more succinctly as an isomorphism

(6.2.5) $$ \begin{align} [ H _{\sigma,i\nu}\otimes V]^K \stackrel{\cong}\longrightarrow [C^\infty( G)\otimes V\otimes L_\sigma ]^{K\times P} , \end{align} $$

where the $P$ -action is given by the prescription in (6.2.4):

$$\begin{align*}(p\cdot f)(g)= e^{(\rho + i \nu) (\log (a)) }\cdot (\mathrm{id}_V\otimes \sigma(m) ) f(gp)\quad \forall p=man. \end{align*}$$

Now let us return to our pseudodifferential operator $A$ in (6.1.4). Because $A$ is properly supported, it extends to an operator

$$\begin{align*}A \colon [C^\infty (G) \otimes V_1]^K \longrightarrow [C^\infty (G)\otimes V_2]^K, \end{align*}$$

which we may tensor with the identity operator on $L_\sigma $ to obtain

$$\begin{align*}A{\otimes}\operatorname{id}_{L_\sigma} \colon [C^\infty (G) \otimes V_1\otimes L_\sigma ]^K\longrightarrow [C^\infty (G) \otimes V_2\otimes L_\sigma ]^K \end{align*}$$

(we give $L_\sigma $ the trivial action of $K$ ).

Lemma 6.2.6. The diagram

is commutative.

6.3 Limit formula for order zero pseudodifferential operators

Let us continue with the same notation as the previous section. Since $G=KP$ , there are isomorphisms

(6.3.1)

in which the second arrow is evaluation at $e\in G$ . Using them, if $A$ is a pseudodifferential operator as in (6.1.4), then we may regard $\pi _{\sigma , i \nu }(A)$ as an operator

$$\begin{align*}\pi_{\sigma, i \nu}(A) \colon [V_1\otimes L_\sigma ]^M \longrightarrow [V_2\otimes L_\sigma ]^M, \end{align*}$$

and we may think of the family of operators $\pi _{\sigma , i \nu }(A)$ , as $\nu $ varies, as a function

(6.3.2) $$ \begin{align} \pi_\sigma (A) \colon \mathfrak{a} \longrightarrow \operatorname{{Hom}} \bigl ( [V_1\otimes L_\sigma ]^M,[V_2\otimes L_\sigma ]^M \bigr ). \end{align} $$

We wish to determine the behavior of this function at infinity in $\mathfrak {a}$ .

We shall use the following identification, coming from the Iwasawa decomposition ${G=KAN}$ :

(6.3.3) $$ \begin{align} \mathfrak{a}\oplus \mathfrak{n} \stackrel \cong \longrightarrow \mathfrak{g} / \mathfrak{k}. \end{align} $$

This has the following geometric consequence: if we denote by $e\in K\backslash G$ the basepoint of the symmetric space associated to $G$ , then the tangent space of the symmetric space at $e$ identifies naturally with $\mathfrak {g} / \mathfrak {k}$ , and hence, by (6.3.3), with $\mathfrak {a}\oplus \mathfrak {n}$ . By duality we obtain an isomorphism

(6.3.4) $$ \begin{align} (\mathfrak{g} / \mathfrak{k})^{*} \stackrel{\cong}\longrightarrow \mathfrak{a}^{*} \oplus \mathfrak{n}^{*} , \end{align} $$

which we shall use to identify each element $\nu \in \mathfrak {a}^{*}$ with a cotangent vector at $e\in K\backslash G$ .

Returning to the operator $A$ in (6.1.4), its principal symbol, restricted to the cotangent fiber over the basepoint $e$ is a $K$ -equivariant, $0$ -homogeneous smooth function

If we further restrict to $\mathfrak {a}^{*}$ , as above, then we obtain a $0$ -homogeneous smooth function

since $\mathfrak {a}\subseteq \mathfrak {g}/\mathfrak {k}$ is centralized by $M$ . Tensoring with the identity operator on $L_\sigma $ , we obtain from this a map

(6.3.5)

(which in general characterizes part but not all of the principal symbol).

We shall now apply the compactification operation introduced in Definition 2.3.2 to the real vector space $\mathfrak {a}^{*}$ to obtain from the above a continuous function

(6.3.6)

where $\partial \mathfrak {a}^{*}$ denotes the boundary of $\mathfrak {a}^{*}$ in $\overline {\mathfrak {a}^{*}}$ .

Theorem 6.3.7. Let $G= KAN$ be an Iwasawa decomposition of the real reductive group $G$ , let $P=MAN$ be the corresponding minimal parabolic subgroup, and let $\sigma $ be an irreducible unitary representation of the compact group $M$ on a finite-dimensional space $L_\sigma $ . Let $V_1$ and $V_2$ be finite-dimensional unitary representations of $K$ , and let $A$ be a properly supported, $G$ -equivariant, order zero classical pseudodifferential operator, as in (6.1.4). The function in (6.3.2) extends to a continuous function

$$\begin{align*}\pi_\sigma (A) \colon \overline{\mathfrak{a}^{*}}\longrightarrow \operatorname{{Hom}} ([V_1\otimes L_\sigma ]^M,[V_2\otimes L_\sigma ]^M) \end{align*}$$

whose restriction to the boundary $\partial \mathfrak {a}^{*} $ is the principal symbol function (6.3.6).

Proof. We shall use Lemma 6.2.6. The functions in the version of the multiplicity space given there, namely the space

$$\begin{align*}C^\infty (G, (V_1\otimes L_\sigma) )^{K\times P} \cong [C^\infty (G)\otimes V_1\otimes L_\sigma ]^{K\times P}, \end{align*}$$

have the form

$$\begin{align*}f_{\nu,u}\colon {kan}\longmapsto e^{-(i\nu+\rho) (\log (a))} (k\otimes \mathrm{id}) u , \end{align*}$$

where $u=f(e) \in [V_1\otimes L_\sigma ]^M $ . The single value

$$\begin{align*}(Af_{\nu,u})(e) \in [C^\infty (G) \otimes V_2\otimes L_\sigma ]^{K\times P} \end{align*}$$

determines $Af_{\nu ,u}$ , and it is a continuous function of $\nu $ , for u fixed, by virtue of the continuity of pseudodifferential operators. So the function in (6.3.2) is continuous.

Since $A$ is properly supported, there is a smooth, compactly supported function $\varphi $ on $K \backslash G$ such that $\varphi (e)=1$ and

$$\begin{align*}(Af_{\nu,u})(e) = (Af_{\nu,e}\varphi) (e)\qquad \forall \nu\in \mathfrak{a}^{*},\,\, \forall u\in [V_1\otimes L_\sigma ]^M. \end{align*}$$

If we write

$$\begin{align*}(\varphi f_{\nu,u})(kan) = e^{-i\nu (\log (a))} \cdot s(kan) , \end{align*}$$

where

$$\begin{align*}s(kan) = e^{-\rho (\log(a))} \varphi (an) \cdot (k\otimes \mathrm{id}) u , \end{align*}$$

then we may write

$$\begin{align*}(Af_{\nu,u})(e) = e^{i \nu (\log(a))} (Ae^{-i \nu (\log(a))} s)(e). \end{align*}$$

With this, the theorem becomes an immediate consequence of the formula (2.3.1) for the symbol of a classical pseudodifferential operator (applied to operators of order $0$ ).