On Square-Integrable Representations of A Lie Group of 4 Dimensional Standard Filiform Lie Algebra


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(2) (2020), Pages 84-90 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: April 15, 2020 Reviewed: April 17, 2020 Accepted: May 31, 2020 
DOI: http://dx.doi.org/10.18860/ca.v6i2.9094     

On Square-Integrable Representations of A Lie Group of 4 
Dimensional Standard Filiform Lie Algebra 

Edi Kurniadi   

Department of Mathematics of FMIPA of Universitas Padjadjaran  
Email: edi.kurniadi@unpad.ac.id 

ABSTRACT  

In this paper, we study irreducible unitary representations of a real standard filiform Lie group 
with dimension equals 4 with respect to its basis. To find this representation we apply the orbit 
method introduced by Kirillov. The corresponding orbit of this representation is generic orbits of 
dimension 2. Furthermore, we show that obtained representation of this group is square-
integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar  
identity operator. In our case  that scalar is equal to one.  

Keywords: Duflo-Moore operator; irreducible unitary representation; square-integrable 
representation, standard filiform Lie algebra.   

INTRODUCTION  

In this paper, all Lie algebras are determined over ℝ. The notion of filiform Lie 
algebras can be found in [1]. In this work, we can find the classification of filiform Lie 
algebras of dimension ≤ 8 and the computation of their index.  Roughly speaking, a Lie 
algebra 𝔤 of dimension 𝑛 is said to be filiform if there exists a smallest positive integer 𝜍 
that equals 𝑛 − 1 such that 𝔤𝜍 = {0}. We recall that 𝔤𝑛 is defined as follows :   𝔤1 = 𝔤, 𝔤2 =
[𝔤, 𝔤], … , 𝔤𝑛+1 = [𝔤, 𝔤𝑛].  Particularly, we shall work on a Lie group 𝐺 of a 4-dimensional 
standard filiform Lie algebra 𝔤. We determine the irreducible unitary representations of 
𝐺 by applying the orbit method. 

The orbit method that shall be applied in this paper comes from [2] and [3]. To 
bring it down to earth of this method, we can enjoy some examples in [4]. We review this 
method from [2] and [4] as follows.  

 
Definition 1 [5].   Let  𝐺 be a Lie group whose Lie algebra is 𝔤. Let 𝑔 be element of 𝐺. We 
define the conjugation map given by 𝐶𝑔: 𝐺 ∋ 𝑥 ↦ 𝑔𝑥𝑔

−1 ∈ 𝐺 which is  a Lie group 

homomorphism whose differential of 𝐶𝑔 is denoted by Ad(𝑔) ≔ (𝐶𝑔)∗
. The adjoint 

representations of 𝐺 is given as group homomorphism Ad ∶ 𝐺 → GL(𝔤).  
 
The infinitesimal adjoint representation of Ad is denoted by ad. This map is given 

by ad: 𝔤 → 𝔤𝑙(𝔤) which is defined by ad(𝑥)𝑦 = [𝑥, 𝑦] for each 𝑥, 𝑦 ∈ 𝔤. Moreover,  the 
representations Ad can be defined as a dual representation on a dual vector space 𝔤∗. 
Namely, we have  
 

http://dx.doi.org/10.18860/ca.v6i2.9094
mailto:edi.kurniadi@unpad.ac.id


On Square-Integrable Representations of A Lie Group of 4 Dimensional Standard Filiform Lie 
Algebra 

Edi Kurniadi 85 

〈Ad∗(𝑔)𝑓, 𝑋〉 = 〈𝑓, Ad(g−1)X〉,   (g ∈ 𝐺, 𝑓 ∈ 𝔤∗, 𝑋 ∈ 𝔤)             (1) 
 
The coadjoint orbit  of 𝑓 ∈ 𝔤∗ is defined as a set {Ad∗(𝑔)𝑓      ; 𝑔 ∈ 𝐺} ⊂ 𝔤∗.  

Let 𝐺 be a Lie group whose Lie algebra is 𝔤. Let 𝔤∗ be a dual vector space of 𝔤.  To 
obtain the formula of irreducible unitary representation of 𝐺 we should consider some 
steps as follows. 

 
1. For orbit Ω ⊂ 𝔤∗, we choose a point 𝑓 ∈ Ω. 
2. We consider a polarization Υ. Υ is said to be a polarization of  𝔤 if Υ is a subalgebra of 𝔤 

and it has maximal dimension, namely its codimension equals 
1

2
dim Ω, which is 

subordinate to 𝑓. This means 〈𝑓, [Υ, Υ]〉 = 0. 
3. Let Η ≔ exp Υ be a subgroup and let 𝜓𝑓  be a one dimensional irreducible unitary 

representations of Η given by  
𝜓𝑓 (exp 𝑋) = 𝑒

2𝜋𝑖〈𝑓,𝑋〉,   (𝑓 ∈ Ω, 𝑋 ∈ 𝔤).          (2) 

4. Identifying the coset space 𝐺/Η by M and considering a section  
𝑠: M → G.                  (3) 

5. Solving the master equation  
𝑠(𝑥)𝑔 = ℎ(𝑥, 𝑔)𝑠(𝑥. 𝑔),    (𝑥 ∈ M, 𝑔 ∈ 𝐺, ℎ(𝑥, 𝑔) ∈ H),            (4) 

which is called a master equation, and computing the measure on M,  we have  
𝜋(𝑔)𝑓(𝑥) = 𝜓𝑓 (ℎ(𝑥, 𝑔))𝑓(𝑥. 𝑔).                        (5) 

 
The notion of square-integrable representations can be found in ( [6], [7], [8], and 

[9]). Let (𝜋, 𝜘𝜋) be a irreducible unitary representations locally compact group 𝐺.  The 
irreducible unitary representation (𝜋, 𝜘𝜋) is said to be square-integrable if there exists a 
non zero vector 𝜈 ∈ 𝜘𝜋  such that 

 

∫ |(𝜈|𝜋(𝑔)𝜈)𝜘𝜋 |
2

  𝑑𝑔 < ∞
𝐺

.       (6) 

 
Such vector is called admissible. Furthermore, in the case the representation (𝜋, 𝜘𝜋 ) is 
square-integrable, Duflo-Moore in [6] found  a densely defined and positive self-adjoint 
unique operator given by  
 

𝐾𝜋 : 𝜘𝜋 → 𝜘𝜋.                                                                                      (7) 
 

which satisfies two following conditions: 
1. The neccessary and sufficient conditions for 𝜈 to be admissible are 𝜈  in domain 𝐾𝜋. 
2. If  vectors 𝑓1, 𝑓3 ∈ 𝜘𝜋 and 𝑓2, 𝑓4 ∈ Dom 𝐾𝜋, then we have 

 

∫ (𝑓1|𝜋(𝑥)𝑓2)𝜘𝜋 (𝜋(𝑥)𝑓4|𝑓3)𝜘𝜋    𝑑𝑥𝐺 =
(𝑓1|𝑓3)𝜘𝜋 (𝐾𝜋 (𝑓4)|𝐾𝜋 (𝑓2))𝜘𝜋

.        (8) 

 
 Let 𝔤 be a 4-dimensional standard filiform Lie algebra with basis Δ ≔

{𝑥1, 𝑥2, 𝑥3, 𝑥3} whose Lie group is 𝐺. The non zero brackets of 𝔤 are given by 
 

[𝑥1, 𝑥2] = 𝑥3, [𝑥1, 𝑥3] = 𝑥4       (9) 
 

We mention here that in previous work in [2] and  [10], we found the representation of 𝐺 
is realized on Hilbert space L2(ℝ) as follows: 



On Square-Integrable Representations of A Lie Group of 4 Dimensional Standard Filiform Lie 
Algebra 

Edi Kurniadi 86 

𝜋𝛿,𝛽 (𝑔)𝑓(𝑥) ≔ 𝑒
2𝜋𝑖(𝛽𝑏+𝛿𝑑+𝛿𝑥𝑐+(

1

2
)𝛿𝑥2𝑏+(

1

2
)𝛿𝑥𝑎𝑏+(

1

2
)𝛿𝑎𝑐+(

1

6
)𝛿𝑎2𝑏)

𝑓(𝑥 + 𝑎),     (10) 

 
with 𝑔 ≔ 𝑔(𝑎, 𝑏, 𝑐, 𝑑) ∈ 𝐺 and 𝑓 ∈ L2(ℝ).  The representation (1) corresponds to 2-
dimensional generic coadjoint orbits 
 

Ω𝛿,𝛽 ≔ {𝛿𝑥4
∗ + 𝑠𝑥3

∗ + (𝛽 +
𝑠2

2𝛿
) 𝑥2

∗ + 𝑢𝑥1
∗       ;   𝑠, 𝑢 ∈ ℝ, 𝛿 ≠ 0}.          (11) 

 
 

Contrary to [10] and [2], in this paper we consider the representations of 𝐺 on 
Hilbert space L2(ℝ) with respect to the basis Δ ≔ {𝑥1, 𝑥2, 𝑥3, 𝑥3} of 𝔤. We thought that our 
computations are simpler than previous results and another reason is to attract 
Indonesian young researcher to study representation theory of Lie groups. We shall claim 
two our main proposition as follows: 

 
Proposition 1. Let 𝐺 be a Lie group of a 4-dimensional standard filiform Lie algebra 𝔤 with 
basis 𝛥 ≔ {𝑥1, 𝑥2, 𝑥3, 𝑥3} whose its non zero brackets is given in (9).  Then, the irreducible 
unitary representation of 𝐺 on the Hilbert space L2(ℝ) corresponding to coadjoint orbits 
Ω𝛿,𝛽 in (11)  can be written as follows: 

 
 

𝜋𝜇 (𝑒
𝑎𝑥1 )𝑓(𝑥) = 𝜑(𝑥 + 𝑎),  

𝜋𝜇 (𝑒
𝑏𝑥2 )𝑓(𝑥) = 𝑒

2𝜋𝑖(𝛽𝑏+(
1

2
)𝛿𝑥2𝑏)

𝜑(𝑥),         (12) 

𝜋𝜇 (𝑒
𝑐𝑥3 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑐𝑥 𝜑(𝑥), 

𝜋𝜇 (𝑒
𝑑𝑥4 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑑 𝜑(𝑥),  

 
where  𝜇 ≔ 𝛽𝑥2

∗ + 𝛿𝑥4
∗ ∈ Ω𝛿,𝛽 ⊂ 𝔤

∗  and  𝜑 ∈ L2(ℝ).   

 
Furthermore, we investigate square-integrability of 𝜋𝜇 of 𝐺 and we have. 
 

Proposition 2. The irreducible unitary representation  𝜋𝜇  of 𝐺 as written  in eqs. (12) is 

square-integrable and its Duflo-Moore operator is a multiple of identity. 
  

METHODS  

The method of this research is based on literature survey, specially from [10]. We 
give another approach in construction of irreducible unitary representation of a Lie group 
𝐺 whose Lie algebra is 4-dimensional standard filiform. We compute it with respect to 
basis of 𝔤. Therefore, the computations are more simpler than previous work.  

RESULTS AND DISCUSSION  

In this section, we shall prove our main propositions as already mentioned in 
introduction 
 
Proposition 1. Let 𝐺 be a Lie group of a 4-dimensional standard filiform Lie algebra 𝔤 with 
basis 𝛥 ≔ {𝑥1, 𝑥2, 𝑥3, 𝑥4} whose its non zero brackets is given in (9).  Then, the irreducible 



On Square-Integrable Representations of A Lie Group of 4 Dimensional Standard Filiform Lie 
Algebra 

Edi Kurniadi 87 

unitary representation of 𝐺 on the Hilbert space L2(ℝ) corresponding to coadjoint orbits 
Ω𝛿,𝛽 in (12)  can be written as follows: 

 
𝜋𝜇 (𝑒

𝑎𝑥1 )𝑓(𝑥) = 𝜑(𝑥 + 𝑎),  

𝜋𝜇 (𝑒
𝑏𝑥2 )𝑓(𝑥) = 𝑒

2𝜋𝑖(𝛽𝑏+(
1

2
)𝛿𝑥2𝑏)

𝜑(𝑥),          

𝜋𝜇 (𝑒
𝑐𝑥3 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑐𝑥 𝜑(𝑥), 

𝜋𝜇 (𝑒
𝑑𝑥4 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑑 𝜑(𝑥),  

 
where  𝜇 ≔ 𝛽𝑥2

∗ + 𝛿𝑥4
∗ ∈ Ω𝛿,𝛽 ⊂ 𝔤

∗  and  𝜑 ∈ L2(ℝ).   

 
Proof. 

Let  𝛥∗ ≔ {𝑥1
∗, 𝑥2

∗, 𝑥3
∗, 𝑥4

∗}  be a basis for 𝔤∗. The detail computations for coadjoint 
orbits of 𝐺 can be found in ([10], p. 33—34) or ([2], p. 77—80).  Let 𝑓 = 𝛼𝑥1

∗ + 𝛽𝑥2
∗ +

𝛾𝑥3
∗ + 𝛿𝑥4

∗ ∈ 𝔤∗ . We recall the result of 2-dimensional coadjoint orbits as follows: 
 

 Ω𝛿,𝛽 ≔ {𝛿𝑥4
∗ + 𝑠𝑥3

∗ + (𝛽 +
𝑠2

2𝛿
) 𝑥2

∗ + 𝑢𝑥1
∗       ;   𝑠, 𝑢 ∈ ℝ, 𝛿 ≠ 0}.       

 
To obtain the irreducible unitary representations of 𝐺 corresponding to generic coadjoint 
orbits Ω𝛿,𝛽 , we consider the following steps 

 
Step 1. We choose a point 𝑓𝛽,𝛿 ≔ 𝛽𝑥2

∗ + 𝛿𝑥4
∗ ∈ Ω𝛿,𝛽.  

 
Step 2. We determine a subalgebra Υ ≔ Span{𝑥2, 𝑥3, 𝑥4} of 𝔤. In this case Υ is commutative 
since [𝑥𝑖 , 𝑥𝑗 ] = 0 for 1 ≤ 𝑖, 𝑗 ≤ 3.  Furthermore, since Υ has maximal dimension i.e. 

Codim Υ =
1

2
dim  Ω𝛿,𝛽 =

1

2
. 2 = 1. On the other hand, Υ is subordinate to 𝑓𝛽,𝛿 since 

〈𝑓𝛽,𝛿 , [Υ, Υ]〉 = 0. Thus, Υ is a polarization of 𝔤.  

 
Step 3. Let Η ≔ exp Υ be subgroup and let 𝜓𝑓  be a 1-dimensional irreducible unitary 

representations of Η given by  
 

𝜓𝑓𝛽,𝛿 (exp(𝑎𝑥1 + 𝑏𝑥2 + 𝑐𝑥3 + 𝑑𝑥4) = 𝑒
2𝜋𝑖〈𝑓𝛽,𝛿,   𝑋〉 = 𝑒2𝜋𝑖(𝛽𝑏+𝛿𝑑),   

(𝑓𝛽,𝛿 ∈ Ω𝛿,𝛽, 𝑋 = 𝑎𝑥1 + 𝑏𝑥2 + 𝑐𝑥3 + 𝑑𝑥4 ∈ 𝔤). 

Step 4.  Identifying the coset space 𝐺/Η with  ℝ  by 
 

ℝ ∋ 𝑥 ↦ Η exp(𝑎𝑥1) ∈ 𝐺/Η,    
 
we have a section given by 

𝑠: 𝐺/Η ≅ ℝ ∋ 𝑥 ↦ exp 𝑥𝑥1 ∈ 𝐺.     
 

Step 5. We now solve  the master equation with respect to (w.r.t) basis  𝛥 ≔
{𝑥1, 𝑥2, 𝑥3, 𝑥4}.  
 

a. W.r.t   𝑒𝑎𝑥1  
exp 𝑥𝑥1 exp 𝑎𝑥1 = exp(𝑥 + 𝑎)𝑥1 . 

 
b. W.r.t   𝑒𝑏𝑥2  



On Square-Integrable Representations of A Lie Group of 4 Dimensional Standard Filiform Lie 
Algebra 

Edi Kurniadi 88 

 
exp 𝑥𝑥1 exp 𝑏𝑥2 = exp(𝑒

ad 𝑥𝑥1 . 𝑎𝑥2) exp 𝑥𝑥1 , 

                                                       = exp(𝑏𝑥2 + 𝑏𝑥𝑥3 + (
1

2
) 𝑥2𝑏𝑥4) exp 𝑥𝑥1. 

 
 

c. W.r.t   𝑒𝑐𝑥3  
 

exp 𝑥𝑥1 exp 𝑐𝑥3 = exp(𝑒
ad 𝑥𝑥1 . 𝑐𝑥3) exp 𝑥𝑥1 , 

                              = exp(𝑐𝑥3 + 𝑐𝑥𝑥4) exp 𝑥𝑥1. 
 
 

d. W.r.t   𝑒𝑑𝑥4  
 

exp 𝑥𝑥1 exp 𝑑𝑥4 = exp(𝑒
ad 𝑥𝑥1 . 𝑑𝑥4) exp 𝑥𝑥1 , 

                = exp(𝑑𝑥4) exp 𝑥𝑥1. 
 

Using induced representation on 1-dimensional irreducible unitary 
representations 𝜓𝑓  of Η, we obtain the explicit formulas for ireducible unitary 

representations of 𝐺 realized on Hilbert space L2(ℝ) written as follows.  
 

𝜋𝜇 (𝑒
𝑎𝑥1 )𝑓(𝑥) = 𝜑(𝑥 + 𝑎),  

𝜋𝜇 (𝑒
𝑏𝑥2 )𝑓(𝑥) = 𝑒

2𝜋𝑖(𝛽𝑏+(
1

2
)𝛿𝑥2𝑏)

𝜑(𝑥),          

𝜋𝜇 (𝑒
𝑐𝑥3 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑐𝑥 𝜑(𝑥), 

𝜋𝜇 (𝑒
𝑑𝑥4 )𝑓(𝑥) = 𝑒 2𝜋𝑖𝛿𝑑 𝜑(𝑥),  

 
where  𝜇 ≔ 𝛽𝑥2

∗ + 𝛿𝑥4
∗ ∈ Ω𝛿,𝛽 ⊂ 𝔤

∗  and  𝜑 ∈ L2(ℝ).   

∎ 
 
 
Proposition 2. The irreducible unitary representation  𝜋𝜇  of 𝐺 as written  in eqs. (12) is 

square-integrable and its Duflo-Moore operator is a multiple of identity. 
  
Proof. 
 
Let 𝜑1, 𝜑2  be elements of L

2(ℝ). For these functions we shall compute the integral 
 

∫ |(𝜑1|𝜋(𝑔)𝜑2)L2(ℝ)|
2

𝐺
 𝑑𝑔 .  

 
Let us put 𝑔 = 𝑔′𝑒𝑎𝑥1  where 𝑔′ = 𝑒𝑝𝑥4 𝑒𝑐3 𝑒𝑏𝑥2 . We have 
 

(𝜑1|𝜋(𝑔)𝜑2)L2(ℝ) = ∫ 𝜑1(𝑥)𝑒
2𝜋𝑖(𝛿𝑝+𝛿𝑐𝑥+𝛽𝑏+(

1

2
)𝛿𝑥2𝑏)

𝜋(𝑒𝑎𝑥1 )𝜑(𝑥)    𝑑𝑥

ℝ

 

By Plancherel’s  Theorem and Fubini’s Theorem of integral we have 
 



On Square-Integrable Representations of A Lie Group of 4 Dimensional Standard Filiform Lie 
Algebra 

Edi Kurniadi 89 

∫ |(𝜑1|𝜋(𝑔)𝜑2)L2(ℝ)|
2

ℝ4

 𝑑𝑝𝑑𝑐𝑑𝑏𝑑𝑎 = ∫ |𝜑1(𝑥)|
2{∫ |𝜑2(𝑥 + 𝑎)|

2 𝑑𝑎}𝑑𝑥

ℝℝ

 

                                                                = ∫ |𝜑1(𝑥)|
2 𝑑𝑥 ∫ |𝜑2(𝑎′)|

2 𝑑𝑎′

ℝℝ

 

 
(we put 𝑎′ = 𝑥 + 𝑎 ) 

 
                                            = ‖𝜑1‖L2(ℝ)

2 ‖𝜑2‖L2(ℝ)
2 .  

 
The latter equation gives information that 𝜋𝜇  is a square-integrable representation. 

Therefore, its Duflo-Moore operator is equal to scalar multiple of identity which scalar 
equals one.  

∎ 
 
We mention here some results in [8] regarding unimodular groups as follows. 

1. If  𝐺 is unimodular then the Duflo-Moore operator 𝐾𝜋 in (8) is the scalar multiple of 
identity. Therefore, the equation (8) is of the form 
 

∫ (𝑓1|𝜋(𝑥)𝑓2)𝜘𝜋 (𝜋(𝑥)𝑓4|𝑓3)𝜘𝜋    𝑑𝑥𝐺 = 𝜆
(𝑓1|𝑓3)𝜘𝜋 (𝑓4|𝑓2)𝜘𝜋 .       (13) 

 
2. If 𝐺 is unimodular then all vectors in 𝜘𝜋 are admissible.  

 
In fact, since tr ∘ ad = 0 then the Lie group 𝐺 of 4-dimensional standard filiform 

Lie algebra 𝔤 is unimodular. Therefore, the computations is immediately true as desired. 
Namely, the irreducible unitary representation 𝜋𝜇  is square-integrable and its Duflo-

Moore operator is equal to the scalar 𝜆 multiple with 𝜆 = 1. 
 

CONCLUSION 

We conclude that irreducible unitary representation 𝜋𝜇  of Lie group of 4-

dimensional filiform Lie algebra with respect to its basis is obtained.  Furthermore,  this 
representation  𝜋𝜇  is square-integrable  and we obtain the Duflo-Moore operator is a 

scalar multiple of identity and its scalar  equals one.  

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On Square-Integrable Representations of A Lie Group of 4 Dimensional Standard Filiform Lie 
Algebra 

Edi Kurniadi 90 

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