Note on Trace Class Groups

TL;DR

This paper investigates conditions under which certain semidirect product Lie groups are trace class, establishing that non-compact semisimple algebraic subgroups lead to non-trace class groups, with initial exploration of Heisenberg-type groups.

Contribution

It proves that semidirect products of R^n with non-compact semisimple algebraic groups are not trace class, and begins studying similar properties for Heisenberg-type groups.

Findings

Semidirect products with non-compact G are not trace class.

The converse for compact G is known from previous work.

Initial analysis of semidirect products with Heisenberg groups.

Abstract

A Lie group G is called a trace class group if for every irreducible unitary representation R of G and every C-infinity function f with compact support the operator R(f) is of trace class. In this note we prove that the semidirect product of R^n and a real semisimple algebraic subgroup G of GL(n;R) is a trace class group only if G is compact. The converse has been shown elsewhere. We also make a descent start with the study of semidirect products with Heisenberg-type groups.

Equations

$$q(e)=1,$$

$$q(ng)={\Delta }_N(n){\Delta }_G(n^{-1})q(g)(n\in N,q\in G),$$

$${\int }_{G}f(a^{-1}g)dg={\Delta }_G(a){\int }_{G}f(g)dg$$

$${\int }_{G/N}{T}_{N}f(\dot{g})d_q(\dot{g})={\int }_{G}f(g)q(g)dg.$$

$$f(ng)=\gamma (n)f(g)\mathrm{and}{\int }_{G/N}\Vert f(\dot{g}){\Vert }^2{d}_q\dot{g}<\infty .$$

$$\pi (g)f(x)=f(xg)[q(xg)/q(x){]}^{1/2}.$$

$$\mathrm{tr}\pi (\psi )=$$

$${\int }_{G/N}{\Delta }_G(g^{-1})q(g^{-1})\mathrm{tr}[{\int }_N\psi (g^{-1}ng)\gamma (n){\Delta }_G(n{)}^{1/2}{\Delta }_N(n{)}^{-1/2}dn]d_q\dot{g}$$

$$(v,h)(v^{\prime },h^{\prime })=(v+h\cdot v^{\prime },h{h}^{\prime })\mathrm{for}v,v^{\prime }\in V,h,h^{\prime }\in H.$$

$$v\mapsto \chi (h\cdot v)(v\in V).$$

$$H_{\chi }=\{h\in H:\chi \cdot h=\chi \}$$

$$(v,h)\mapsto \chi (v)\rho (h)(v\in V,h\in H_{\chi }).$$

$${\Delta }_G(v,h)=\det (h).{\Delta }_H(h)(v\in V,h\in H)$$

$${\Delta }_N(v,h)=\det (h){\Delta }_{H_{\chi }}(h)(v\in V,h\in H_{\chi }).$$

$$Q(h_{0}h)={\Delta }_{H_{\chi }}(h_0){\Delta }_H(h_0^{-1})Q(h)$$

$$\mathrm{tr}\pi (\psi )={\int }_{H/H_{\chi }}{\Delta }_H(h{)}^{-1}Q(h{)}^{-1}.$$

$$\mathrm{tr}[{\int }_V{\int }_{H_{\chi }}\psi (v,h^{-1}{h}_{o}h)(\chi \cdot h)(v)\rho (h_0){\Delta }_H(h_0{)}^{1/2}{\Delta }_{H_{\chi }}(h_0{)}^{-1/2}dvd{h}_0]d_Q\dot{h}.$$

$$\lambda (H)=B(H,H_1)(H\in \mathfrak{a}),$$

$$dg=d(g_0{a}_{1}k)={\Delta }_{G_0}(g_0^{-1})\delta (a_1^{-1})d{g}_{0}d{a}_{1}dk.$$

$${\chi }_0(v)={\mathrm{e}}^{-2\pi i⟨{\xi }_0,,v⟩}(v\in V).$$

$$dg=d(h_0{a}_{1}k)={\Delta }_{H_0}(h_0){\delta }_0(a_1)d{h}_{0}d{a}_{1}dk,$$

$$\pi =\mathrm{Ind}{}_{V⋊H_0}^{V⋊G}{\chi }_0\otimes 1$$

$$Q(h_0{a}_{1}k)={\Delta }_{H_0}(h_0){\delta }_0(a_1).$$

$$\mathrm{tr}(\psi )={\int }_{A_1}{\hat{\psi }}_1(a_1\cdot {\xi }_0)[{\int }_{H_0}{\psi }_2(h_0){\Delta }_{H_0}(h_0{)}^{-1/2}d{h}_0]d{a}_1$$

$${\int }_{A_1}{\hat{\psi }}_1(a_1\cdot {\xi }_0)d{a}_1={\int }_0^{\infty }{\hat{\psi }}_1(\mu {\xi }_0)\frac{d\mu }{\mu }$$

$$T_{N}f(\dot{x})={\int }_{G/N}f(nx)dn.$$

$$\pi (f)=\dot{\pi }(T_{N}f)(f\in C_c^{\infty }(G)).$$

$$h\cdot \dot{v}=\dot{\widetilde{h\cdot v}}(h\in H,v\in V).$$

$$V_1\supset V_2\supset \cdots \supset V_k\supset \{1\}$$

$$L=\{h\in H:h=id\mathrm{on}{V}_i/V_{i+1}\mathrm{for}\mathrm{all}i=1,\dots ,k\}$$

Full text

Mathematics Subject Classification 2010: 22D10, 43A80, 46C05.

Keywords and Phrases: Trace class group, semidirect product, induced representation, Heisenberg group.

1 Introduction

In this note we resume the study of trace class groups from [4]. An irreducible unitary representation $\pi$ of a Lie group is said to be of trace class if for every $C^{\infty}$ function $f$ with compact support the operator $\pi(f)$ is of trace class. A Lie group is said to be of trace class, or briefly, a trace class group, if every irreducible unitary representation is of trace class. Well-known examples of such groups are reductive Lie groups and unipotent Lie groups. In general each (real algebraic) Lie group is a semidirect product of a reductive and a unipotent Lie group. One of the highlights of [4] is the theorem that the semidirect product of a real algebraic semisimple Lie group and its Lie algebra is a trace class group if and only if the group is compact. In this note we prove a generalization of this theorem, provided by the case of a semisimple real algebraic group $G$ acting on a real finite-dimensional vector space $V$ by linear transformations and considering the semidirect product of $V$ and $G$. We also make a beginning with the study of real algebraic groups with unipotent radical equal to a Heisenberg group.

2 Formula for the character of an induced representation

Let $G$ be a locally compact group and $N$ a closed subgroup. Choose right Haar measures $dg$ on $G$ and $dn$ on $N$. We may find a strictly positive continuous function $q$ on $G$ satisfying

[TABLE]

[TABLE]

where $\Delta_{N},\Delta_{G}$ denote the modular functions on $N,G$. For example

[TABLE]

for all $a\in G$ and $f\in C_{c}(G)$.

The function $q$ defines a quasi-invariant measure $d_{q}\dot{g}$ on $\dot{G}=G/N$ (the space of right cosets with respect to $N$) as follows. For $f\in C_{c}(G)$ set $T_{N}f({\dot{g}})=\int_{N}f(ng)dn,\,{\dot{g}}=Ng$. Then $d_{q}({\dot{g}})$ is defined by

[TABLE]

Let $\gamma$ be a unitary representation of $N$ and set $\pi={\rm Ind}_{N}^{G}\gamma$. We write down a formula for the character of $\pi$ in terms of that for $\gamma$. Let us give the definition of $\pi$. Let $\mathcal{H}$ be the Hilbert space of $\gamma$. Then $\pi$ acts on the space ${\mathcal{H}}_{\pi}$ of function $f:G\to{\mathcal{H}}$ satisfying

[TABLE]

The action of $\pi$ is

[TABLE]

**Theorem **2.1

([2], Theorem 3.2). Let $\varphi\in C_{c}(G),\varphi^{\ast}(g)=\overline{\varphi(g^{-1})}\Delta_{G}(g^{-1})$ and set $\psi=\varphi\ast_{G}\varphi^{\ast}$. Then

[TABLE]

[TABLE]

in the sense that both sides are finite and equal or both $+\infty$.

A group $G$ is called unimodular if $\Delta_{G}=1$. If $G$ is a unimodualr Lie group we have ${\rm tr}\,\pi(\varphi)$ is finite for all functions $\varphi\in C_{c}^{\infty}(G)$ if and if ${\rm tr}\,\pi(\psi)$ is for all fuctions $\psi$ of the form $\psi=\varphi\ast_{G}\varphi^{\ast}$ with $\varphi\in C_{c}^{\infty}(G)$. This is because any $\varphi$ is a finite sum of functions of the form $\psi$ by [1].

3 Application to semidirect products

Let $V$ be a finite-dimensional real vector space and $H$ a closed subgroup of ${\rm GL}(V)$. Set $G=V\rtimes H$. The product in $G$ is given by

[TABLE]

If $dh$ is a right Haar measure on $H$ and $dv$ one on $V$, then $dg=dvdh$ is a right haar measure on $G$. Denote by ${\widehat{V}}$ the space of continuous unitary characters of $V$. For each $\chi\in\widehat{V}$ and each $h\in H$ consider the function

[TABLE]

This is again a continuous unitary character of $V$, which we call $\chi\cdot h$. The set of all $\chi\cdot h$ is called the orbit of $\chi$ in $\widehat{V}$ and

[TABLE]

the stability subgroup of $\chi$. Choose an irreducible unitary representation $\rho$ of $H_{\chi}$ and define $\chi\otimes\rho$ by

[TABLE]

Then $\chi\otimes\rho$ is an irreducible unitary representation of $V\rtimes H_{\chi}$ and the induced representation $\pi_{\chi,\rho}$ is an irreducible unitary representation of $V\rtimes H$, see [3], p. 43. Now apply (2) with $G=V\rtimes H$, $N=V\rtimes H_{\chi}$ and $\pi=\pi_{\chi,\rho}$. Choose as before $dvdh$ and similarly $dvdh_{\chi}$ as right Haar mesures on $G$ and $N$. Then $\Delta_{G}$ and $\Delta_{N}$ are given by

[TABLE]

and similarly

[TABLE]

Notice that $q$ is left $V$-invariant, so we may write $q(v,h)=Q(h)$. $Q$ satisfies

[TABLE]

for $h\in H,\,h_{0}\in H_{\chi}$. Let us now rewrite (2) for the above particular case.

**Theorem **3.1

Let $\psi$ be as in Theorem 2.1. Then

[TABLE]

[TABLE]

4 A special case

Let $V$ be a finite-dimensional real vector space with inner product $\langle\ ,\ \rangle$ and with complexification $\bf V$. Denote by $\bf G$ a connected , complex, semisimple, linear algebraic subgroup of ${\rm GL}(\bf V)$. Assume that $\bf G$ is defined over $\mathbb{R}$ and set $G={\bf G}(\mathbb{R})$ for its group of real points. Then $G$ is a semisimple Lie group with finite center and finitely many connected components. For any $g\in{\rm GL}(V)$ set $\langle g\cdot v,\,w\rangle=\langle v,\,^{t}g\cdot w\rangle$ for all $v,w\in V$. Assume that $G$ is invariant under the Cartan involution $\theta$ defined by $\theta(g)=\,^{t}g^{-1}\ (g\in G)$. Write $G_{1}=V\rtimes G$. Notice that $G_{1}$ is unimodular. The purpose of this note is to show

**Theorem **4.1

The group $G_{1}$ is a trace class group only if the group $G$ is compact.

The converse of this theorem has been proved in [4], Lemma 14.2. Proof. Assume $G$ to be non-compact. The Cartan involution of $G$ gives rise to a Cartan involution of the Lie algebra $\mathfrak{g}$ of $G$, that we again denote by $\theta$. Write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the decomposition of $\mathfrak{g}$ into $\pm 1$-eigenspaces of $\theta$. Then $\mathfrak{k}$ consists of anti-symmetric and $\mathfrak{p}$ of symmetric elements. Set $K=G\cap{\rm O}(n,\mathbb{R})$. Then $\mathfrak{k}$ is the Lie algebra of $K$. Select a non-trivial maximal Abelian subspace $\mathfrak{a}$ of $\mathfrak{p}$, which exists because $G$ is non-compact, and let $\Sigma$ denote the set of roots of $(\mathfrak{g},\,\mathfrak{a})$. Then $\Sigma$ is root system (with multiplicities). Let $\Delta$ be a set of simple roots and $\Sigma^{+}$ the set of positive roots with respect to $\Delta$. Denote by $\mathfrak{n}$ the Lie subalgebra spanned by the positive root vectors and by $N$ the corresponding algebraic subgroup. Then one has $\mathfrak{g}=\mathfrak{k}+\mathfrak{a}+\mathfrak{n}$ and similarly $G=KAN$, the Iwasawa decomposition of $G$. Let $\xi_{0}$ be a highest weight vector in $V$ with highest weight $\lambda\not=1$ (with respect to $N$ and $A$). Such a vector exists. Indeed, if all highest weight vectors have weight is equal to one, then $G=\{1\}$, which is not the case. Set $G_{0}$ for the stabilizer of $\xi_{0}$ in $G$. Denote by $B$ the Killing form of $\mathfrak{g}$ and define $H_{1}$ by

[TABLE]

and set $A_{1}={\rm exp}\,\mathbb{R}H_{1}$. Let us denote by $P$ the stabilizer of the half-line ${\mathbb{R}}^{\ast}_{+}\,\xi_{0}$. Then $P=A_{1}G_{0}$, where $A_{1}\cap G_{0}=\{1\},\,a_{1}G_{0}a_{1}^{-1}=G_{0}$ for all $a_{1}\in A_{1}$. Let $da_{1}$ and $dg_{0}$ denote right Haar measures on $A_{1}$ and $G_{0}$ respectively. Then $dp=\delta(a_{1})da_{1}dg_{0}$, where $\delta(a_{1})={\rm det}\,{\rm Ad}(a_{1}^{-1})|_{\mathfrak{g}_{0}}$ is a right Haar measure on $P$. Since $A\subset P$ and $N\subset P$ we have $G=PK$ and

[TABLE]

Define the character $\chi_{0}$ of $V$ by

[TABLE]

Then ${\rm Stab}\,\chi_{0}=\theta(G_{0})$. Let us write $H_{0}=\theta(G_{0})$, and let $dh_{0}$ be a right Haar measure on $H_{0}$. In a similar way as above we have

[TABLE]

where $\delta_{0}(a_{1})={\rm det}\,{\rm Ad}(a_{1})|_{\mathfrak{h}_{0}}.$ We will consider the representation $\pi$ given by

[TABLE]

and determine whether its trace exists. Let

[TABLE]

Equation (3) then becomes:

[TABLE]

where we toke $\psi\in C_{c}^{\infty}(G)$ of the form $\psi(v,g)=\psi_{1}(v)\,\psi_{2}(g)\ (v\in V,g\in G)$, both $\psi_{1}$ and $\psi_{2}$ $K$-invariant and $\widehat{\psi}_{1}(w)=\int_{V}\psi_{1}(v)\,{\rm e}^{-2\pi i\langle v,\,w\rangle}dv$ for $w\in V$. Clearly the integral

[TABLE]

diverges for suitable $\psi_{1}$. So we may conclude that $G$ cannot be non-compact. This concludes the proof of the theorem.

5 Semidirect products with Heisenberg groups

In this section we extend our scope to semidirect products $G=V\rtimes H$ with $V$ a non-necessarily Abelian normal subgroup of $G$. Notice that any real algebraic group is of this form according to the Levi decomposition, see [4], Proposition 2.1. Let us begin with some preparations.

**Lemma **5.1

Let $G$ be a Lie group and $N$ a closed normal subgroup of $G$. If $G$ is a trace class group, then the quotient group $\dot{G}=G/H$ is

Proof. Let $\dot{\pi}$ be an irreducible unitary represenatation of $\dot{G}$ and $\pi$ the corresponding representation of $G$. Choose a right Haarmeasure $dn$ on $N$ and define for $f\in C_{c}^{\infty}(G)$

[TABLE]

Then $f\mapsto T_{N}f$ is a continuous surjective linear map $C_{c}^{\infty}(G)\to C_{c}^{\infty}(\dot{G})$ and one has

[TABLE]

So the result follows.  

 Let us consider a special case. Denote by $G$ a Lie group, being the semidirect product $G=V\rtimes H$ where $H$ is a closed subgroup of $G$ and $V$ a closed normal subgroup of $G$. Let $W$ be a closed normal subgroup of $G$ contained in $V$. Denote by $v\mapsto\dot{v}$ the canonical map $V\to V/W=\dot{V}$. The group $H$ acts on $\dot{V}$ by

[TABLE]

Set $\dot{G}=\dot{V}\rtimes H$. Then we have

**Lemma **5.2

The map $(v,h)\mapsto(\dot{v},h)$ is a surjective homomorphism from $G$ to $\dot{G}$ with kernel $W$.

We will now specialize to real algebraic groups $G$ of the form $G=V\rtimes H$ with $V$ a unipotent and $H$ a semisimple real algebraic group. Set

[TABLE]

for the descending series of $V$, where $V_{1}=V,\,V_{i}=[V,V_{i-1}],\,V_{k}=Z$, the (non-trivial center of $V$. Notice that each $V_{i}$ is normal in $G$, so in particular $H$-invariant. Define

[TABLE]

Clearly $L$ is a closed real algebraic normal subgroup of $H$. We can now formulate a conjecture.

**Conjecture **5.3

Let $G=V\rtimes H$ with $V$ a unipotent and $H$ a semisimple real algebraic group. Then $G$ is a trace class group if and only if $H/L$ is compact.

Let us consider an example with $V$ of a special nature, namely a Heisenberg group. Such a group can be seen as the most simple choice for a non-Abelian group $V$. We shall show that the conjecture holds in this case.

**Example **5.4

Denote by $\mathcal{V}$ the $(2n+1)$-dimensional Heisenberg group with Lie algebra basis $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n},z$ with $[x_{i},\,y_{i}]=z$ and all other brackets equal to zero. Then $Z$ is one-dimensional and spanned by $z$. Let $H$ be any semisimple real algebraic group acting on $\mathcal{V}$ algebraically and set $G={\mathcal{V}}\rtimes H$. There are two kinds of irreducible unitary representations $\pi$ of $\mathcal{V}$, depending on their behaviour on $Z$. By Schur’s Lemma we have $\pi(z)=\lambda(z)\,I\ (z\in Z)$ for some character $\lambda$ of $Z$.

If $\lambda\not=1$, then $\pi$ is equivalent with an infinite-dimensional representation $\pi_{\lambda}$ satisfying $\pi_{\lambda}(z)=\lambda(z)\,I$ and $\pi_{\lambda}$ is square-integrable modulo $Z$.

If $\lambda=!$, then $\pi$ is actually a one-dimensional representation of ${\mathcal{V}}/Z\simeq{\mathbb{R}}^{2n}$, so a character $\chi$ of $\mathcal{V}$.

Let us now perform the usual construction for the determination of the irreducible unitary representations of $G$, see [2], p. 470. Let us begin with $\pi_{\lambda}$. Since $H$ acts trivially on $Z$, one has ${\rm Stab}\,\pi_{\lambda}=H$. If $\rho$ is an irreducible unitary representation of $H$ then $\pi_{\lambda}\otimes\rho$ in one of $G$, of trace class. Let now $\chi$ be a character of ${\mathcal{V}}/Z$ (hence of $\mathcal{V}$). By the ususal construction (see [4]), we always obtain a trace class representation of ${\mathcal{V}}/Z\rtimes H=G/Z$, so of $G$, if and only if $H/L_{0}$ is compact, where $L_{0}=\{h\in H:\,h=id\ {\rm on}\ {\mathcal{V}}/Z\}$. Clearly $L_{0}=L$, so if and only if $H/L$ is compact. Resuming, $G={\mathcal{V}}\rtimes H$ is trace class if and only if $H/L$ is compact.