Amenability and representation theory of pro-Lie groups

Abstract

We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups. The usual $C^*$-algebraic approach to group representation theory positively breaks down for infinite direct products of non-compact locally compact groups, hence the description of their unitary duals in terms of coadjoint orbits is particularly important whenever it is available, being the only description known so far.

Appendix: Complements on representations of nilpotent Lie groups

Our basic references for representation theory of nilpotent Lie groups are [16] and [8]. In this section we record a few results that are needed in the main body of our paper. For the reader’s convenience we provide self-contained proofs for some of the results that we found more difficult to locate in the literature.

Lemma A.1

Let $p:G_1\rightarrow G_2$ be a continuous surjective morphism of connected and simply connected nilpotent Lie groups. Let $\pi _2:G_2\rightarrow {{\mathcal {B}}}({{\mathcal {H}}})$ be any unitary irreducible representation of $G_2$, associated with a coadjoint orbit ${{\mathcal {O}}}_2\subseteq \mathbf{L }(G_2)^*$. Then $\pi _1:=\pi _2\circ p :G_1\rightarrow {{\mathcal {B}}}({{\mathcal {H}}})$is a unitary irreducible representation of $G_1$, and its corresponding coadjoint orbit is ${{\mathcal {O}}}_1:=\mathbf{L }(p)^*({{\mathcal {O}}}_2)\subseteq \mathbf{L }(G_1)^*$.

Proof

It follows by Lemma 4.4 that ${{\mathcal {O}}}_1$ is a coadjoint orbit of $G_1$.

Let $\ell _2\in {{\mathcal {O}}}_2$ and $\ell _1=\mathbf{L }(p)^*(\ell _2)$. To construct $\pi _2$, we select a real polarization ${{\mathfrak {h}}}_2$ in $\ell _2$, i.e ${{\mathfrak {h}}}_2\subset \mathbf{L }(G_2)$ such that:

• 1. ${{\mathfrak {h}}}_2$ is a subalgebra of $\mathbf{L }(G_2)$,

• 2. ${{\mathfrak {h}}}_2$ is subordinated to $\ell _2$, i.e $\langle \ell _2,[{{\mathfrak {h}}}_2,{{\mathfrak {h}}}_2]\rangle =0$,

• 3. ${{\mathfrak {h}}}_2$ is a maximal isotropic subspace, i.e.,

  $$\begin{aligned} \dim {{\mathfrak {h}}}_2=\frac{1}{2}\dim \mathbf{L }(G_2)+\frac{1}{2}\dim \mathbf{L }(G_2)(\ell _2). \end{aligned}$$

We denote by $H_2=\exp {{\mathfrak {h}}}_2$ the analytic subgroup with Lie algebra ${{\mathfrak {h}}}_2$. The exponential of the character $i\ell _2$ of ${{\mathfrak {h}}}_2$ is a character $\chi _{\ell _2}$ of $H_2$, defined by:

$$\begin{aligned} \chi _{\ell _2}(\exp ~X)=e^{i\langle \ell _2,X\rangle }\qquad (X\in {{\mathfrak {h}}}_2). \end{aligned}$$

The representation $\pi _2$ is an induced unitary representation:

$$\begin{aligned} \pi _2=\text {Ind}_{H_2}^{G_2}~\chi _{\ell _2}. \end{aligned}$$

It is realized in the completion of the space ${{\mathcal {C}}}^\infty (G_2:H_2)$ of $C^\infty$ functions $\varphi$ on $G_2$, with compact support modulo $H_2$, satisfying

$$\begin{aligned} \varphi (xh)=\chi _{\ell _2}(h)^{-1}\varphi (x)\quad \text { for }x\in G_2,\ h\in H_2,\quad \text {and}\quad \Vert \varphi \Vert ^2=\int _{G_2/H_2}|\varphi ({\dot{x}})|^2~d{\dot{x}} \end{aligned}$$

The representation $\pi _2$ is defined by:

$$\begin{aligned} \left( \pi _2(x)\varphi \right) (y)=\varphi (x^{-1}y),\quad \text { for }\varphi \in {{\mathcal {C}}}^\infty (G_2:H_2). \end{aligned}$$

First, the Lie algebra of $G_2=p(G_1)$ is $\mathbf{L }(G_2)= \mathbf{L }(p)(\mathbf{L }(G_1))$, hence $\mathbf{L }(p)$ is surjective. Moreover

$$\begin{aligned} \begin{aligned} \mathbf{L }(G_1)(\ell _1)&:=\{X\in \mathbf{L }(G_1)\mid \mathrm{ad}^*(X)\ell _1=0\} \\&=\{X\in \mathbf{L }(G_1)\mid (\forall Y\in \mathbf{L }(G_1))\ \langle (\mathbf{L }p)^*(\ell _2),[X,Y]\rangle =0\}\\&=\{X\in \mathbf{L }(G_1)\mid (\forall Y\in \mathbf{L }(G_1))\ \langle \ell _2,[\mathbf{L }(p)(X),\mathbf{L }(p)(Y)]\rangle =0\}\\&=\{X\in \mathbf{L }(G_1)\mid \mathbf{L }(p)(X)\in \mathbf{L }(G_2)(\ell _2)\}\\&=(\mathbf{L }(p))^{-1}(\mathbf{L }(G_2)(\ell _2)). \end{aligned} \end{aligned}$$

This implies that $\mathrm{Ker}\,\mathbf{L }(p)\subset \mathbf{L }(G_1)(\ell _1)$, $\mathbf{L }(p)|_{\mathbf{L }(G_1)(\ell _1)}:\mathbf{L }(G_1)(\ell _1) \rightarrow \mathbf{L }(G_2)(\ell _2)$ is surjective, and

$$\begin{aligned} \dim \mathbf{L }(G_2)(\ell _2)+\dim (\mathrm{Ker}\,\mathbf{L }(p))=\dim (\mathbf{L }(G_1)(\ell _1)). \end{aligned}$$

Then, we put ${{\mathfrak {h}}}_1:=(\mathbf{L }(p))^{-1}({{\mathfrak {h}}}_2)$. Since ${{\mathfrak {h}}}_2$ is a maximal isotropic subspace, one has $\mathbf{L }(G_2)(\ell _2)\subset {{\mathfrak {h}}}_2$, hence

$$\begin{aligned} \mathrm{Ker}\,\mathbf{L }(p)\subset \mathbf{L }(G_1)(\ell _1)=(\mathbf{L }(p))^{-1}\mathbf{L }(G_2)(\ell _2)\subset (\mathbf{L }(p))^{-1}{{\mathfrak {h}}}_2={{\mathfrak {h}}}_1. \end{aligned}$$

The same argument gives

$$\begin{aligned} \dim {{\mathfrak {h}}}_2+\dim \mathrm{Ker}\,\mathbf{L }(p)=\dim {{\mathfrak {h}}}_1, \quad \dim \mathbf{L }(G_2)+\dim \mathrm{Ker}\,\mathbf{L }(p)=\dim \mathbf{L }(G_1). \end{aligned}$$

Now ${{\mathfrak {h}}}_1$ is a polarization in $\ell _1$. In fact:

• 1. ${{\mathfrak {h}}}_1$ is a subalgebra of $\mathbf{L }(G_1)$: For all X, $Y\in {{\mathfrak {h}}}_1$,

  $$\begin{aligned} \mathbf{L }(p)([X,Y])=[\mathbf{L }(p)(X),\mathbf{L }(p)(Y)]\in [{{\mathfrak {h}}}_2,{{\mathfrak {h}}}_2]\subset {{\mathfrak {h}}}_2, \end{aligned}$$

  then $[X,Y]\in {{\mathfrak {h}}}_1$.

• 2. ${{\mathfrak {h}}}_1$ is subordinated to $\ell _1$:

  $$\begin{aligned} \begin{aligned} \langle \ell _1,[{{\mathfrak {h}}}_1,{{\mathfrak {h}}}_1]\rangle&=\langle \mathbf{L }(p)^*\ell _2,[{{\mathfrak {h}}}_1,{{\mathfrak {h}}}_1]\rangle =\langle \ell _2,\mathbf{L }(p)([{{\mathfrak {h}}}_1,{{\mathfrak {h}}}_1])\rangle \\&=\langle \ell _2,[\mathbf{L }(p)({{\mathfrak {h}}}_1),\mathbf{L }(p)({{\mathfrak {h}}}_1)]\rangle \subset \langle \ell _2,[{{\mathfrak {h}}}_2,{{\mathfrak {h}}}_2]\rangle =0. \end{aligned} \end{aligned}$$

• 3. ${{\mathfrak {h}}}_1$ is a maximal isotropic subspace:

  $$\begin{aligned} \begin{aligned} \dim {{\mathfrak {h}}}_1&=\dim {{\mathfrak {h}}}_2+\dim \mathrm{Ker}\,\mathbf{L }(p)\\&=\frac{1}{2}\left( \dim \mathbf{L }(G_2)+\dim \mathrm{Ker}\,\mathbf{L }(p)+\dim \mathbf{L }(G_2)(\ell _2)+\dim \mathrm{Ker}\,\mathbf{L }(p)\right) \\&=\frac{1}{2}\left( \dim \mathbf{L }(G_1)+\dim \mathbf{L }(G_1)(\ell _1)\right) . \end{aligned} \end{aligned}$$

Then the representation associated to the coadjoint orbit ${{\mathcal {O}}}_1$ is (with the same notation as above):

$$\begin{aligned} \rho _1=\text {Ind}_{H_1}^{G_1}~\chi _{\ell _1}. \end{aligned}$$

Note that as $\mathrm{Ker}\,\mathbf{L }(p)\subset {{\mathfrak {h}}}_1$, one has $\mathrm{Ker}\,p=\exp (\mathrm{Ker}\,\mathbf{L }(p))\subset \exp ({{\mathfrak {h}}}_1)=H_1$. Hence, $\forall \psi \in {{\mathcal {C}}}^\infty (G_1:H_1)$, $x\in G_1$ and $k\in \mathrm{Ker}\,p$, if $k=\exp X$ then

$$\begin{aligned} \begin{aligned} \psi (xk)=\chi _{\ell _1}(k)^{-1}\psi (x)&=\mathrm{e}^{-\mathrm{i}\langle \ell _1,X\rangle }\psi (x) = \mathrm{e}^{-\mathrm{i}\langle \mathbf{L }(p)^* \ell _2,X\rangle }\psi (x) \\&= \mathrm{e}^{-\mathrm{i}\langle \ell _2,\mathbf{L }(p)(X)\rangle }\psi (x) =\psi (x). \end{aligned} \end{aligned}$$

The function $\psi$ passes to the quotient, i.e., there exists $\varphi :G_2=G_1/\mathrm{Ker}\,p\rightarrow {{\mathbb {C}}}$ such that $\varphi \circ p=\psi$. Moreover, if $h=\exp X\in H_2$, then $X\in {{\mathfrak {h}}}_2=\mathbf{L }(p)({{\mathfrak {h}}}_1)$, $X=\mathbf{L }(p)(Y)$, for each $x=p(y)$ in $G_2$, we have:

$$\begin{aligned} \varphi (xh)=\varphi (p(y)p(\exp Y))=\psi (y\exp Y)=\mathrm{e}^{-\mathrm{i}\langle \ell _1,Y\rangle }\psi (y) =\mathrm{e}^{-\mathrm{i}\langle \ell _2,X\rangle }\varphi (x). \end{aligned}$$

In other words, $\varphi \in {{\mathcal {C}}}^\infty (G_2:H_2)$. Conversely, if $\varphi \in {{\mathcal {C}}}^\infty (G_2:H_2)$, then $\varphi \circ p\in {{\mathcal {C}}}^\infty (G_1:H_1)$. Thus, there is a linear bijection ${{\mathcal {C}}}^\infty (G_2:H_2)\rightarrow {{\mathcal {C}}}^\infty (G_1:H_1)$, $\varphi \mapsto \varphi \circ p$.

On the other hand, we have:

$$\begin{aligned} G_2/H_2\simeq (G_1/\mathrm{Ker}\,p)/(H_1/\mathrm{Ker}\,p)\simeq G_1/H_1, \end{aligned}$$

and

$$\begin{aligned} \Vert \psi \Vert ^2=\int _{G_1/H_1}|\psi ({\dot{x}})|^2~d{\dot{x}}=\int _{G_2/H_2}|\varphi ({\dot{x}})|^2~d{\dot{x}}=\Vert \varphi \Vert ^2. \end{aligned}$$

The bijection above extends uniquely to a unitary operator ${{\mathcal {H}}}_{\pi _2}\rightarrow {{\mathcal {H}}}_{\rho _1}$.

Note that, if $x,~y\in G_1$

$$\begin{aligned} \begin{aligned} \big (\rho _1(x)U\varphi \big )(y)&=(U\varphi )(x^{-1}y)=(\varphi \circ p)(x^{-1}y) =\varphi (p(x)^{-1}p(y)) \\&=\Big ((\pi _2\circ p)(x)(\varphi \circ p)\Big )(y) =\Big ((\pi _2\circ p)(x)(U\varphi )\Big )(y), \end{aligned} \end{aligned}$$

that is, $\rho _1(x)=(\pi _2\circ p)(x)=\pi _1(x)$, $\rho _1=\pi _1$, and this completes the proof. $\square$

Lemma A.2

Let G be any connected and simply connected nilpotent Lie group with a discrete normal subgroup $\Gamma \subseteq G$, and denote by $p:G\rightarrow G/\Gamma$ the quotient map. Denote by $\log _G:{{\mathfrak {g}}}\rightarrow G$ the inverse of the exponential map of G. Then for every unitary irreducible representation $\pi :G/\Gamma \rightarrow {{\mathcal {B}}}({{\mathcal {H}}})$ the representation $\pi \circ p :G\rightarrow {{\mathcal {B}}}({{\mathcal {H}}})$ is irreducible, and its corresponding coadjoint orbit ${{\mathcal {O}}}\subseteq {{\mathfrak {g}}}^*$ has the property that for all $\xi \in {{\mathcal {O}}}$ one has $\xi (\log _G(\Gamma ))\subseteq {{\mathbb {Z}}}$. One thus obtains a bijective correspondence from $\widehat{G/\Gamma }$ onto the set of all coadjoint G-orbits contained in the G-invariant set ${{\mathfrak {g}}}^*_{{{\mathbb {Z}}}}:=\{\xi \in {{\mathfrak {g}}}^*\mid \xi (\log _G(\Gamma ))\subseteq {{\mathbb {Z}}}\}$.

Proof

See [16, Th. 8.1] and [8, Th. 4.4.2]. $\square$

Now we can prove the following generalization of Lemma A.1 to connected nilpotent Lie groups that may not be simply connected.

Proposition A.3

Let $p:G_1\rightarrow G_2$ be a continuous surjective morphism of connected nilpotent Lie groups. Let $\pi _2:G_2\rightarrow {{\mathcal {B}}}({{\mathcal {H}}})$ be any unitary irreducible representation of $G_2$, associated with a coadjoint orbit ${{\mathcal {O}}}_2\subseteq \mathbf{L }(G_2)^*$. Then $\pi _1:=\pi _2\circ p :G_1\rightarrow {{\mathcal {B}}}({{\mathcal {H}}})$is a unitary irreducible representation of $G_1$, and its corresponding coadjoint orbit is ${{\mathcal {O}}}_1:=\mathbf{L }(p)^*({{\mathcal {O}}}_2)\subseteq \mathbf{L }(G_1)^*$.

Proof

It follows by Lemma 4.4 that ${{\mathcal {O}}}_1$ is a coadjoint orbit of $G_1$. Let ${\widetilde{G}}_j$ be the universal covering group of $G_j$, with a suitable discrete central subgroup $\Gamma _j$ and the quotient map $p_j:{\widetilde{G}}_j\rightarrow {\widetilde{G}}_j/\Gamma _j=G_j$. Since ${\widetilde{G}}_1$ is simply connected, it follows that the Lie group morphism $p\circ p_1:{\widetilde{G}}_1\rightarrow G_2$ lifts to a unique Lie group morphism ${\widetilde{p}}:{\widetilde{G}}_1\rightarrow {\widetilde{G}}_2$ for which the diagram

(A.1)

is commutative, hence $p_2\circ {\widetilde{p}}=p\circ p_1$. Since $\Gamma _j=\mathrm{Ker}\,p_j$, it then follows that ${\widetilde{p}}(\Gamma _1)=\Gamma _2$. Now the conclusion follows, using of Lemmas A.1 and A.2.

Specifically, since $p:G_1\rightarrow G_2$ is surjective and $\pi _2$ is irreducible, it is easily checked that $\pi _2\circ p$ is an irreducible representation of $G_1$. In order to identify the coadjoint orbit of $G_1$ associated with $\pi _2\circ p$ via Lemma A.2, we first note that

$$\begin{aligned} \pi _2\circ p\circ p_1=(\pi _2\circ p_2)\circ {\widetilde{p}} \end{aligned}$$

by (A.1). The above equality implies by Lemma A.1 that if we denote by ${\widetilde{{{\mathcal {O}}}}}_2\subseteq \mathbf{L }({\widetilde{G}}_2)^*=\mathbf{L }(G_2)^*$ the coadjoint orbit of ${\widetilde{G}}_2$ that is associated with $\pi _2\circ p_2$, then

$$\begin{aligned} {\widetilde{{{\mathcal {O}}}}}_1:=\mathbf{L }({\widetilde{p}})^*({{\mathcal {O}}}_2)\subseteq \mathbf{L }({\widetilde{G}}_1)^* \end{aligned}$$

(A.2)

is the coadjoint orbit of ${\widetilde{G}}_1$ that is associated with $\pi _2\circ p\circ p_1$, where we have used the notation ${{\mathcal {O}}}_2$ introduced in the statement. On the other hand, for $j=1,2$, the group morphism $p_j:{\widetilde{G}}_j\rightarrow G_j$ is a covering map, hence $\mathbf{L }(p_j):\mathbf{L }({\widetilde{G}}_j)\rightarrow \mathbf{L }(G_j)$ is an isomorphism of Lie algebras. By Lemma A.2, the coadjoint orbit of $G_2$ associated with $\pi _2$ is just the coadjoint orbit of ${\widetilde{G}}_2$ associated with $\pi _2\circ p_2$. More precisely, using the vector space isomorphism $\mathbf{L }(p_2)^*:\mathbf{L }(G_2)^*\rightarrow \mathbf{L }({\widetilde{G}}_2)^*$, one has

$$\begin{aligned} \mathbf{L }(p_2)^*({{\mathcal {O}}}_2)={\widetilde{{{\mathcal {O}}}}}_2. \end{aligned}$$

(A.3)

Similarly, by Lemma A.2 again, the coadjoint orbit of $G_1$ associated with $\pi _2\circ p$ is just the coadjoint orbit of ${\widetilde{G}}_1$ associated with $(\pi _2\circ p)\circ p_1$. More precisely, using the vector space isomorphism $\mathbf{L }(p_1)^*:\mathbf{L }(G_1)^*\rightarrow \mathbf{L }({\widetilde{G}}_1)^*$, one has

$$\begin{aligned} \mathbf{L }(p_1)^*({{\mathcal {O}}}_1)={\widetilde{{{\mathcal {O}}}}}_1. \end{aligned}$$

(A.4)

Also, by (A.1), one has $p\circ p_1=p_2\circ {\widetilde{p}}$, hence $\mathbf{L }(p)\circ \mathbf{L }(p_1)=\mathbf{L }(p_2)\circ \mathbf{L }({\widetilde{p}})$, and then $\mathbf{L }(p_1)^*\circ \mathbf{L }(p)^*=\mathbf{L }({\widetilde{p}})^*\circ \mathbf{L }(p_2)^*$, which further implies by (A.3), (A.2), and (A.4),

$$\begin{aligned} \mathbf{L }(p_1)^*(\mathbf{L }(p)^*({{\mathcal {O}}}_2))=\mathbf{L }({\widetilde{p}})^*(\mathbf{L }(p_2)^*({{\mathcal {O}}}_2)) =\mathbf{L }({\widetilde{p}})^*({\widetilde{{{\mathcal {O}}}}}_2)={\widetilde{{{\mathcal {O}}}}}_1=\mathbf{L }(p_1)^*({{\mathcal {O}}}_1). \end{aligned}$$

Now, as $\mathbf{L }(p_1)^*$ is a vector space isomorphism, we obtain $\mathbf{L }(p)^*({{\mathcal {O}}}_2)={{\mathcal {O}}}_1$, as claimed, and this completes the proof. $\square$