EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 2092-2105
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Decomposition of the Unitary Representation of SL2(R)
on the Upper Half Plane into Irreducible Components

Fatimah Abdullah Alabbad

Department of Mathematics and Statistics, Faculty of science, King Faisal University,
Saudi Arabia

Abstract. The main purpose of this paper is to find the inversion formula for the covariant
transform Wρk

φ0
. This formula is equivalent to the decomposition of the unitary representation ρk

into irreducible components. We consider an eigenvalue 1 + s2 of the Casimir operator:

dρk(C) = −4v2
(
∂2u + ∂2v

)
, where k = 0.

To find the inversion formula, first we study the representations of SL2(R), ρk and ρτ , induced from
the complex characters of K and N respectively. Then, we find the induced covariant transform
Wρk

φ0
withN -eigenvector to obtain a transform in the space L2(SL2(R)/N). Thereafter, we compute

the contravariant transform with K-eigenvector

Mρτ

ϕ0
: L2(SL2(R)/N) → L2(SL2(R)/K).

2020 Mathematics Subject Classifications: 22D10,22D30,42A38

Key Words and Phrases: Unitary representation, SL2(R) group, Covariant transform, Inversion
formula

1. Introduction

Integral transforms establishes a correspondence between functions on a manifold X
and functions on some manifold M of submanifolds of X. The main problems are in
the description of the images and kernels of these transforms and in the construction of
explicit inversion formulas recovering the original objects from their images. The first book
devoted to this area was by I. M. Gelfand,M. I. Graev and N. Ya. Vilenkin [4]. From the
1940s, one of the main problems in mathematics was to develop an analog of the Fourier
transform for noncommutative Lie group. For the group SL2(C), I. M. Gelfand and M.
A. Naimark constructed a theory in which the role of exponential functions was played by
irreducible infinite-dimensional unitary representations of the SL2(C) group. Obtaining
analogs of the inversion formula and the Plancherel formula for the Fourier transform was

DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5245

Email address: falabbad@kfu.edu.sa (F. A. Alabbad)

https://www.ejpam.com 2092 © 2024 EJPAM All rights reserved.



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2093

the most important result of this theory. Our contribution is to use a new method starting
with the covariant transform to obtain the inversion formula.

Action of SL2(R) by linear-fractional transformation on complex numbers produces
isometrical motions of the Lobachevsky geometry. It is less known that there are related
actions of SL2(R) on dual and double numbers which have the form z = x+ ιy, ι2 = 0 or
ι2 = 1, correspondingly. We write ε and j instead of ι within dual and double numbers,
respectively.
Three possible values −1, 0 and 1 of σ := ι2 will be referred to elliptic, parabolic and
hyperbolic cases, respectively.

A generic cycle [9], § 4.2 is the set of points (u, v) ∈ R2 defined for all values of σ by
the equation

k(u2 − σv2)− 2lu− 2nv +m = 0. (1)

This equation is represented by a point (k, l, n,m) from a projective space P3, since for a
scaling factor λ ̸= 0, the point (λk, λl, λn, λm) defines an equation equivalent to (1). We
call P3 the cycle space and refer to the initial R2 as the point space.
In order to obtain a connection with the Möbius action, we arrange numbers (k, l, n,m)
into the matrix [9], Definition 4.11

Cσ̆ =

(
l + ῐn −m
k −l + ῐn

)
. (2)

The values of σ̆ := ῐ2 are −1, 0 or 1 may be chosen to be independent of the values of σ.

Theorem 1. [9], Theorem 4.13 Let a matrix g=

(
a b
c d

)
∈ SL2(R) defines a Möbius

transformation

g : (u+ ιv) → a(u+ ιv) + b

c(u+ ιv) + d
. (3)

Then the image C̃σ̆ of a cycle Cσ̆ under transformation with g ∈ SL2(R) is given by
similarity of the matrix (2):

C̃σ̆ = gCσ̆g
−1. (4)

Definition 1. [9], Definition 5.11 For two cycles C and C1, define the cycles product by:

⟨C,C1⟩ = − tr(CC̄1), (5)

where tr denotes the trace of a matrix.

We can find the explicit expression of the cycle product (5) with σ = −1, 0 and 1:

⟨C,C1⟩ = km1 + k1m− 2ll1 + 2σnn1, (6)

where C = (k, l, n,m) and C1 = (k1, l1, n1,m1).



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2094

Definition 2. [3], Chap. 3, § 1.2 On the hyperbolic plane one can define circles of infinitely
large radius (horocycles), which are the limits of non-Euclidean circles as the center and
the radius of these circles consistently tend to infinity. In the Lobachevsky model, the
horocycles are represented either as Euclidean circles tangent to the real axis or as lines
parallel to the real axis.

The horizontal line v−1 = 0 as a cycle is represented by the matrix

(
− i

2 −1
0 − i

2

)
. This

line is invariant under the subgroup N =

(
1 n
0 1

)
, that is

(
1 n
0 1

)(
− i

2 −1
0 − i

2

)(
1 −n
0 1

)
=

(
− i

2 −1
0 − i

2

)
.

Thus all horocycles obtained by SL2(R) action are parametrized by points of the homo-
geneous space SL2(R)/N .

The image of v − 1 = 0 under the lower triangular matrix

(
ξ1 0
ξ2

1
ξ1

)
∈ SL2(R) is

(
ξ1 0
ξ2

1
ξ1

)(
− i

2 −1
0 − i

2

)( 1
ξ1

0

−ξ2 ξ1

)
=

(
ξ1ξ2 − i

2 −ξ21
ξ22 −ξ1ξ2 − i

2 ,

)
(7)

that is, cycle (ξ22 , ξ1ξ2,
1
2 , ξ

2
1) with the equation

ξ22u
2 + ξ22v

2 − 2ξ1ξ2u− v + ξ21 = 0

⇔ (ξ22u
2 − 2ξ1ξ2u+ ξ21) + ξ22v

2 = v

⇔ (ξ2u− ξ1)
2 + (ξ2v)

2 = v

⇔ |(ξ2u− ξ1) + iξ2v|2 = v

⇔ |ξ2(u+ iv)− ξ1|2 = v

⇔ |ξ2z − ξ1|2 = v, z = u+ iv, (ξ1, ξ2) ∈ R2 \ {0}.

(8)

Therefore, the point (ξ1, ξ2) of the parabolic upper half plane SL2(R)/N parametrizes the
space of horocycles. Denote by h(ξ) = h(ξ1, ξ2) the horocycle given by (8).

Every horocycle has a unique common point with the real axis, which is called the
center of the horocycle. Horocycles with common center are said to be parallel. Note
that a horocycle h(ξ1, ξ2) is tangent to the real axis at the point ξ1

ξ2
, hence every parallel

horocycle is of the form {h(λξ1, λξ2) : 0 < λ < ∞} for some chosen (ξ1, ξ2) [3], Chap. 3,
§ 1.2.
Now, in order to find the invariant distance of a point z in the upper half plane to the
horocycle h(ξ), first we calculate the distance from a point z1 = (u, v) ∈ h(λξ1, λξ2) to a
horocycle h(ξ1, ξ2). The point z2 = (u, λ−2v) is in the horocycle h(ξ1, ξ2):

|λξ2z − λξ1|2 = v ⇒ |ξ2z − ξ1|2 = λ−2v.



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2095

Note that the points z1 and z2 are on the same vertical line, thus the distance between
them is ∣∣∣∣∫ v

vλ−2

1

y
dy

∣∣∣∣ = ∣∣log v − log vλ−2
∣∣

= 2 |log λ| .
(9)

Thus, all points of the horocycle h(λξ1, λξ2) are placed at the same distance 2 |log λ| from
the parallel horocycle h(ξ1, ξ2). Then, the signed distance from a point z ∈ h(λξ) to a
horocycle h(ξ) is

ϱ(z; ξ) = −2 log λ

= log
(
v−1|ξ2z − ξ1|2

)
,

(10)

because λ−2 = v−1|ξ2z − ξ1|2.

2. Induced representations of the group SL2(R)

(i) For the subgroupK =

{(
cos t sin t
− sin t cos t

)
: t ∈ R

}
, the homogeneous space SL2(R)/K

are parametrised by points of the upper half-plane H+. The respective maps are:

p

(
a b
c d

)
=

(
bd+ ac

c2 + d2
,

1

c2 + d2

)
,

s(u, v) =
1√
v

(
v u
0 1

)
,

r

(
a b
c d

)
=

1√
c2 + d2

(
d −c
c d

)
.

(11)

The decomposition defined by the formula g = s(p(g))r(g) takes the form:(
a b
c d

)
=

1

c2 + d2

(
1 bd+ ac
0 c2 + d2

)(
d −c
c d

)
. (12)

The SL2(R)-action defined by the formula g · x = p(g ∗ s(x)) takes the form:(
a b
c d

)
: (u, v) 7→

(
(au+ b)(cu+ d) + cav2

(cu+ d)2 + (cv)2
,

v

(cu+ d)2 + (cv)2

)
. (13)

This map preserves the upper half plane v > 0. We can simplify this map as a
linear-fractional transformation with the complex number unit i2 = −1:(

a b
c d

)
: w 7→ aw + b

cw + d
, wherew = u+ iv. (14)



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2096

The left invariant measure on the upper half plane H+ is equal to

dµ(w) =
dudv

v2
, w = u+ iv. (15)

The character χk

(
cos t sin t
− sin t cos t

)
= e−ikt, k ∈ Z of K, induces a linear representa-

tion ρk on the space of square integrable functions, which is given by the formula:

[ρk(g)f ](w) = χτ (r(g−1 ∗ s(w)))f(g−1 · w), (16)

where g ∈ SL2(R) and w ∈ SL2(R)/K.
By simple calculation we obtain [7], § 8:

[ρk(g)f ](w) =
|a− cw|k

(a− cw)k
f

(
dw − b

a− cw

)
, wherew = u+ iv. (17)

We consider the basis in the Lie algebra sl2(R) :

A =
1

2

(
−1 0
0 1

)
, B =

1

2

(
0 1
1 0

)
, Z =

(
0 1
−1 0

)
. (18)

They generate one-parameter subgroup of SL2(R):

etA =

(
e−

t
2 0

0 e
t
2

)
, etB =

(
cosh t

2 sinh t
2

sinh t
2 cosh t

2

)
, etZ =

(
cos t sin t
− sin t cos t

)
.

The derived representations are:

dρAk = w∂w + w̄∂w̄ (19)

= u∂u + v∂v, (20)

dρBk =
1

4
k(w − w̄) · I − 1

2
(1− w2)∂w − 1

2
(1− w̄2)∂w̄ (21)

=
1

2
kvi · I − 1

2
(1− u2 + v2)∂u + uv∂v, (22)

dρZk = −1

2
k(w − w̄) · I − (1 + w2)∂w − (1 + w̄2)∂w̄ (23)

= −ikv · I − (1 + u2 − v2)∂u − 2uv∂v, (24)

where w = u+ iv, ∂w = 1
2(∂u − i∂v) and ∂w̄ = 1

2(∂u + i∂v).
The Casimir operator is:

dρk(C) = dρZ
2−4A2−4B2

τ

= 4ikv∂u − 4v2(∂2u + ∂2v).
(25)



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2097

(ii) For the subgroup N ′ =

{(
1 0
n 1

)
: n ∈ R

}
, the homogeneous space SL2(R) /N ′ can

be identified with the upper half plane. The respective maps are:

p

(
a b
c d

)
=

(
b

d
,
1

d2

)
,

s(u, v) =
1√
v

(
v u
0 1

)
,

r

(
a b
c d

)
=

(
1 0
c

d
1

)
.

(26)

The maps p and s produce the following decomposition g = s(p(g))r(g):(
a b
c d

)
=

1

d2

(
1 bd
0 d2

)(
d 0
c d

)
, where d ̸= 0. (27)

The action of SL2(R) on SL2(R) /Ń defined by the formula g · x = p(g ∗ s(x)) takes
the form: (

a b
c d

)
: (u, v) 7→

(
au+ b

cu+ d
,

v

(cu+ d)2

)
. (28)

It preserves the upper half plane v > 0. We can rewrite this map as a linear-fractional
transformation with the dual number unit ε2 = 0:(

a b
c d

)
: w 7→ aw + b

cw + d
, wherew = u+ εv. (29)

The complex character χτ of N ′ is:

χτ

(
1 0
n 1

)
= e−2πiτn, where τ ∈ R.

This character induces a linear representation ρτ on the space of square integrable
functions, which is given by the formula:

[ρτ (g)f ](w) = χτ (r(g−1 ∗ s(w)))f(g−1 · w), (30)

where g ∈ SL2(R) and w ∈ SL2(R)/Ń .
A direct calculation shows that [7], § 8:

[ρτ (g)f ](w) = exp

(
−2πi

τcv

a− cu

)
f

(
dw − b

a− cw

)
, (31)

where w = u+ εv and f ∈ L2(H+, dµ).
This representation is unitary on the space of functions on the upper half plane of



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2098

dual numbers.
The derived representations of the elements A, B and Z (18) of sl2(R) are:

dρAτ = w∂w + w̄∂w̄ (32)

= u∂u + v∂v, (33)

dρBτ = −πivτ · I − 1

2
(1− w2)∂w − 1

2
(1− w̄2)∂w̄ (34)

= −πivτ · I − 1

2
(1− u2)∂u + uv∂v, (35)

dρZτ = 2πivτ · I − (1 + w2)∂w − (1 + w̄2)∂w̄ (36)

= 2πivτ · I − (1 + u2)∂u − 2uv∂v, (37)

where w = u+ εv, ∂w = 1
2(∂u + 1

ε∂v) and ∂w̄ = 1
2(∂u − 1

ε∂v).
The Casimir operator is:

dρτ (C) = dρZ
2−4A2−4B2

τ

= −8πivτ∂u − 4v2∂2v .
(38)

In the following we will find some eigenfunctions, and the special role of them will become
obvious later.

2.1. Joint eigenvector of dρk(C) with dρNk

First, we calculate the eigenvector of the derived representation dρNk :[
dρNk f

]
(w, w̄) = −∂uf(w, w̄) = −(∂w + ∂w̄)f(w, w̄) = 0. (39)

The solution is f(w, w̄) = ϕ(v), where ϕ is an arbitrary function.
Then, we solve the differential equation

dρk(C)ϕ(v) = (1 + s2)ϕ(v), s ∈ R,

where dρk(C) is the Casimir operator(25), and k = 0 for simplicity.
This equation becomes

−4v2
d2ϕ

dv2
(v)− (1 + s2)ϕ(v) = 0. (40)

It is a Cauchy-Euler equation, therefore the solution takes the form ϕ(v) = vm. Differen-

tiating gives d2ϕ
dv2

(v) = m(m− 1)vm−2, and substituting into (40) leeds to

−4m(m− 1)vm − (1 + s2)vm = 0

⇒ −4m(m− 1)− (1 + s2) = 0

⇒ m =
1± is

2
.

Hence, the set of fundamental solution is{
v

1+is
2 , v

1−is
2

}
. (41)



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2099

2.2. Joint eigenvector of dρτ (C) with dρZτ

To begin, we look for an eigenvector of the derived representation dρZτ (37) with τ = 0.
To do that, we solve the equation dρZτ f(w, w̄) = 0 using the method of characteristics:

du

1 + u2
=

dv

2uv
=

df

2πiτvf
.

du

1 + u2
=

dv

2uv
⇒ 2udu

1 + u2
=
dv

v
⇒ 2C1 =

v

1 + u2
.

We need to obtain another integral curve which involves f . Since τ = 0, then

dv

2uv
=

df

2πiτvf
⇒ df

f
= 0 ⇒ C2 = f.

Hence, the general solution is of the form C2 = ψ(C1), that is

f(w, w̄) = ψ

(
v

2(1 + u2)

)
, w = u+ εv, (42)

where ψ is an arbitrary function. To specify this function, we solve the equation

dρτ (C)ψ

(
v

2(1 + u2)

)
= (1 + s2)ψ

(
v

2(1 + u2)

)
, (43)

where dρτ (C) is the Casimir operator(38). This equation turns into

−4
v2

(1 + u2)2
d2ψ

dv2

(
v

2(1 + u2)

)
− (1 + s2)ψ

(
v

2(1 + u2)

)
= 0. (44)

Using the substitution t = v
2(1+u2)

, then we obtain

−4t2
d2ψ

dt2
(t)− (1 + s2)ψ(t) = 0. (45)

It is a Cauchy-Euler equation, so let ψ(t) = tm and substitute in the differential equa-
tion(45), then m = 1+is

2 and m = 1−is
2 are two distinct possible values of m. Therefore,

the set of fundamental solution is
{
t
1+is
2 , t

1−is
2

}
.

Finally, the set of fundamental solution for the equation(44) is{(
v

2(1 + u2)

) 1+is
2

,

(
v

2(1 + u2)

) 1−is
2

}
. (46)



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2100

3. Induced Covariant transform

Definition 3. [8], § 5.1 Let H be a closed subgroup of G and f ∈ H such that

ρ(h)f = χ(h)f (47)

for some character χ of H where h ∈ H and ρ is a unitary representation of a Lie group
G in a Hilbert space H. For a section s from G/H to G, the induced covariant transform
Wρ

f is a map from the Hilbert space H to a space of function on G/H given as follows:

Wf : υ 7→ υ̃(x) = ⟨υ, ρ(s(x))f⟩, x ∈ G/H. (48)

The map υ 7→ υ̃(x) = υ̃(s(x)) intertwines ρ with the representation ρχ in a certain
function space on G/H induced by the character χ of H. That is,

ρχ ◦Wρ
f = Wρ

f ◦ ρ. (49)

Example 1. We will find the induced wavelet transform with N -eigenvector that inter-
twines respectively the representation ρk (17) where k = 0 with the representation ρτ (31).

We take the fiducial vector φ0(w, w̄) = v
1+is
2 (41) which would be the eigenvector for the

representation ρk

(
1 n
0 1

)
. That is

ρk

(
1 n
0 1

)
φ0 = χτ

(
1 n
0 1

)
φ0. (50)

Then, the corresponding induced covariant transform is:[
Wρk

φ0
f
]
(ξ) = ⟨f, ρk(s(ξ1, ξ2))φ0⟩

=

〈
f, ρk

(
ξ1 0
ξ2

1
ξ1

)
v

1+is
2

〉
=

∫
H+

f(w)

(
v

|ξ1 − ξ2w|2

) 1−is
2 dudv

v2

=

∫
H+

f(w) exp

{
−1− is

2
ϱ(w; ξ)

}
dudv

v2
, w = u+ iv,

(51)

where ϱ(w; ξ) (10) is the signed distance from the point w to the horocycle h(ξ), ξ = (ξ1, ξ2).

4. Contravariant transform

Definition 4. [6], § 5 Let ρ be a unitary square integrable representation of the group
SL2(R) on a Hilbert space H and H be a closed subgroup of SL2(R). Let X = SL2(R)/H
be a homogeneous space with an invariant measure dx. We define the function ws(x) =



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2101

ρ(s(x))w0, where w0 ∈ H and s is a section map. The contravariant transform Mρ
w0 is a

map L2(X) → H defined by

Mρ
w0
f =

∫
X
f(x)ws(x) dx, x ∈ X. (52)

For an admissible vector w0 [1], Definition 8.1.1, the contravariant transform in this setup
is known as a reconstruction formula.

Example 2. For the representation ρτ (31) with τ = 0, we take the K-eigenvector

ϕ0(w, w̄) =
(

v
2+2u2

) 1+is
2

(46). Then, the corresponding contravariant transform is:[
Mρτ

ϕ0
f
]
(w) =

∫
h(ξ)

f(ξ)ρτ (s(ξ1, ξ2))ϕ0 dξ

=

∫
R2

f(ξ)ρτ

((
ξ1 0
ξ2

1
ξ1

))(
v

2(1 + u2)

) 1+is
2

dξ1dξ2

=

∫
R2

f(ξ)

(
vξ21

2ξ21(ξ1 − ξ2u)2 + 2u2

) 1+is
2

dξ1dξ2, w = u+ εv.

(53)

Proposition 1. [6], Prop. 6.4 Contravariant transform Mw0 intertwines left regular rep-
resentation Λ on L2(SL2(R)) and ρ:

Mw0Λ(g) = ρ(g)Mw0 . (54)

Let ρ be an irreducible square integrable representation and φ0 and w0 be admissible
vectors. The covariant transform intertwines ρ and the left regular representation Λ:

Wφ0ρ(g) = Λ(g)Wφ0 .

Combining with (54), we see that the composition Mw0 ◦ WF intertwines ρ with itself.
That is,

(Mw0 ◦Wφ0) ◦ ρ(g) = ρ(g) ◦ (Mw0 ◦Wφ0). (55)

Thus, from the Schur’s lemma we have the relation

Mw0 ◦Wφ0 = kI, (56)

for some constant k ∈ C.
On the other hand, and from the orthogonality relations [1], § 8.2:

⟨Wφ1f1,Wφ2f2⟩ = ⟨f1, f2⟩⟨Cφ2, Cφ1⟩, (57)

where C is a unique positive, self adjoint and invertible operator in the Hilbert space.
This operator is known as Duflo-Moore operator.
If f1, f2 ∈ H, we have

⟨Mw0 ◦Wφ0f1, f2⟩ = ⟨Wφ0f1,Ww0f2⟩
= ⟨f1, f2⟩⟨Cw0, Cφ0⟩
= ⟨⟨Cφ0, Cw0⟩f1, f2⟩.

(58)



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2102

Thus
Mw0 ◦Wφ0 = ⟨Cφ0, Cw0⟩I. (59)

And for non-orthogonal vectors w0 and φ0, we get ⟨Cφ0, Cw0⟩ = k ̸= 0.

4.1. Inversion formula

We will find the inversion formula for the covariant transform (51) from the relation
(59) with the contravariant transform Mρτ

ϕ0
(53):

f(w) =
1

⟨Cφ0, Cϕ0⟩

[
Mρτ

ϕ0

(
Wρk

φ0
f
)]

(w)

=
1

⟨Cφ0, Cϕ0⟩

∫
R2

Wρk
φ0
f(ξ)

(
vξ21

2ξ21(ξ1 − ξ2u)2 + 2u2

) 1+is
2

dξ1dξ2.

(60)

The function h(s) = ⟨Cφ0, Cϕ0⟩ must be explicitly identified.
The following result is an inversion formula similar to that in Gelfand’s book [3], Chap. 3,
Theorem 3.2, but with a difference in the eigenvector and with a different method.

Theorem 2. For f ∈ L2(R2
+, dµ), we have the inversion formula

f(w) =
1

2π2

∫
R
s tanh

πs

2

(∫
R2

Wρk
φ0
f(ξ)ρτ (s(ξ1, ξ2))ϕ0(w) dξ1dξ2

)
ds, (61)

where

ρτ (s(ξ1, ξ2))ϕ0(w) =

(
vξ21

2ξ21(ξ1 − ξ2u)2 + 2u2

) 1+is
2

,

and Wρk
φ0f is the covariant transform (51).

Proof. To find the inversion formula for the covariant transform (51), we need to
identify ⟨Cφ0, Cw0⟩ in (60):

⟨Cφ0, Cw0⟩ =
1

f(w)

[
Mρτ

ϕ0

(
Wρk

φ0
f
)]

(w)

=
1

f(w)

∫
R2

Wρk
φ0
f(ξ)

(
vξ21

2ξ21(ξ1 − ξ2u)2 + 2u2

) 1+is
2

dξ1dξ2.

(62)

To identify this function, it is enough to compute the composition of the covariant trans-
form and the contravariant transform for one particular function. Let

f0(w) =
vis+

1
2 (1 + v)−

is+1
2

1 + u2
∈ L2(R2

+, dµ(w)), 0 < ℜ(is) < 1. (63)

We compute the covariant transform for the function f0:

Wρk
φ0
f0(ξ) =

∫ +∞

0

∫ +∞

−∞

vis+
1
2 (1 + v)−

is+1
2

1 + u2
v

1−is
2

(
1

|ξ1 − ξ2w|2

) 1−is
2 dudv

v2
. (64)



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2103

And for ξ = (ξ1, 0), this value becomes

Wρk
φ0
f0(ξ) =

∫ +∞

0

∫ +∞

−∞

vis+
1
2 (1 + v)−

is+1
2

1 + u2
v

1−is
2

−2
(
ξ21
) is−1

2 dudv

=
(
ξ21
) is−1

2

∫ +∞

0
v

is
2
−1(1 + v)−

is+1
2

(∫ +∞

−∞

1

1 + u2
du

)
dv

=
(
ξ21
) is−1

2

∫ +∞

0
v

is
2
−1(1 + v)−

is+1
2 π dv

= πe
is−1
2

ϱ(i;ξ)B(
is

2
,
1

2
),

(65)

where ϱ(i; ξ) is the distance from the point i to the horocycle h(ξ) and B is the Beta
function.
Then, we find the function (62) with (ξ1, ξ2) = (ξ1, 1) and f(w) = f0(i):

⟨Cφ0, Cw0⟩ =
1

f0(i)

∫
R
Wρk

φ0
f0((ξ1, 1))

(
ξ21

2ξ21(ξ1)
2

) 1+is
2

dξ1

= 2
is+1
2 πB(

is

2
,
1

2
)

∫
R
e

is−1
2

ϱ(i;(ξ1,1))
(
2ξ21
)− 1+is

2 dξ1

= πB(
is

2
,
1

2
)

∫
R
(ξ21 + 1)

is−1
2
(
ξ21
)− 1+is

2 dξ1.

(66)

Put u = (ξ21 + 1)−1, then (66) becomes

⟨Cφ0, Cw0⟩ = πB(
is

2
,
1

2
)

∫ 1

0
u

1
2
−1(1− u)−

is
2
−1 du

= πB(
is

2
,
1

2
)B(− is

2
,
1

2
)

= π
Γ( is2 )Γ(

1
2)

Γ( is+1
2 )

Γ(−is
2 )Γ(12)

Γ(−is+1
2 )

= π2
∣∣∣∣Γ( is

2

)∣∣∣∣2 ∣∣∣∣Γ( is+ 1

2

)∣∣∣∣−2

= π2
2

s
coth

πs

2
.

(67)

Substituting this value in (60), we obtain

f(w) =
1

2π2
s tanh

πs

2

∫
R2

Wρk
φ0
f(ξ)

(
vξ21

2ξ21(ξ1 − ξ2u)2 + 2u2

) 1+is
2

dξ1dξ2. (68)

And for s ∈ R, we get the inversion formula



F. A. Alabbad / Eur. J. Pure Appl. Math, 17 (3) (2024), 2092-2105 2104

f(w) =
1

2π2

∫
R
s tanh

πs

2

(∫
R2

Wρk
φ0
f(ξ)ρτ (s(ξ1, ξ2))ϕ0(w) dξ1dξ2

)
ds, (69)

where

ρτ (s(ξ1, ξ2))ϕ0(w) =

(
vξ21

2ξ21(ξ1 − ξ2u)2 + 2u2

) 1+is
2

. (70)

The inversion formula is equivalent to the decomposition of the unitary representation
ρk, k = 0 (17) into irreducible components. We will describe the irreducible invariant
subspaces Hs. Consider the eigenspace{

f ∈ L2(H+) : dρk(C)f = (1 + s2)f
}
. (71)

This space is spanned by the functions (70). Thus, the elements of this eigenspace can be
presented as a continuous linear combination over a set of such functions, that is

fs(w) =

∫
R2

Wρk
φ0
f(ξ)

(
vξ21

2ξ21(ξ1 − ξ2u)2 + 2u2

) 1+is
2

dξ1dξ2, (72)

where fs belongs to the space Hs ⊂ L2(H+).
Introduce the projection operator Ps : L2(H+) → Hs by

Psf = fs.

Thus, the problem of decomposing the space L2(H+) into irreducible subspaces consists
in expanding the functions f ∈ L2(H+) in their projections fs.
The solution of this problem is given by (69), since this formula can be written as follows:

f(w) =
1

2π2

∫
R
s tanh

πs

2
fsds, fs = Psf. (73)

5. Conclusion

In this paper, we obtain the covariant and contravariant transforms using the repre-
sentation itself like in Gelfand’s approach [5], but the eigenvectors are selected by the
derived representation as in Bargmann’s works [2]. we use the relation between these
transforms to find the inversion formula. Thus, the original contribution is using the co-
variant transform to find the inversion formula with eigenvectors selected by the derived
representation. This new method will be easier to adopt for problems of decomposing a
system into elementary bits in theoretical physics. Also, it is not restricted to SL2(R), it
can be successfully used for many other cases.

Acknowledgements

I would like to express my deepest gratitude to Dr. Vladimir Kisil, for his guidance,
feedback and support.



REFERENCES 2105

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