Plancherel Theorem and the Left Ideals of the Group Algebra for the Jacobi Group

0.2 MB·

by  Kahar El-Hussein

Plancherel Theorem and the Left Ideals of the
6 Group Algebra for the Jacobi Group.
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Kahar El-Hussein

n
Department of Mathematics, Faculty of Science,
a
J Al Furat University, Dear El Zore, Syria and

1 Department of Mathematics, Faculty of Arts Science Al Quryyat,
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Al-Jouf University, KSA

] E-mail : [email protected], [email protected]
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R
January 13, 2016
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h
t
a
Abstract
m
[ LetG=SL(2,R)bethe2×2connectedrealsemisimpleLiegroupand
let KAN betheIwasawa decomposition of SL(2,R). Let J =H⋊SL(2,
1 R)betheJacobigroup,whichisthesemidirectproductofthetwogroups
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H withSL(2,R). ItplaysanimportantroleinQuantumMechanics. The
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purposeofthispaperistodefinetheFouriertransform inordertoobtain
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6 thePlanchereltheorem forthegroup J. Tothisendaclassification ofall
2 left ideals of thegroup algebra L1(H⋊AN).
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.
1 Keywords: JacobiGroup, IwasawaDecomposition, FourierTransformand
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Plancherel Theorem, Left Ideals
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AMS 2000 Subject Classification: 43A30&35D 05
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:
v
i 1 Introduction
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r
a
1.1. The Jacobi group the semidirect product of the Heisenberg and the sym-
plecticgroupSL(2,R)playsanimportantroleinquantummechanics. InQuan-
tum optics represent a physical realization of the coherent states associated to
the Jacobi group. The Jacobi group is responsible for the squeezed states and
hasanimportantobjectinquantummechanics,geometricquantization,optics.
Abstractharmonicanalysisisthefieldofthemostmodernbranchesofharmonic
analysis, having its roots in the mid-twentieth century, is analysis on topologi-
cal groups. If the group is neither abelian nor compact, no general satisfactory
theory is currently known.
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1.2. First In this paper I will define the Fourier transform in order to
establishthe Plancherelformula onthe JacobigroupH⋊SL(2,R), where H is
the 3-dimensional Heisenberg group and
a b
SL(2,R)={ X = : detX =1} (1)
c d
(cid:18) (cid:19)
Secondly, I will give classification for all left ideals of the group algebra
L1(H⋊AN),whereAN isthesolvableLiegroupintheIwasawadecomposition
KAN of SL(2,R).
2 Fourier Transform and Plancherel Formula on
SL(2,R)
2.1. InthefollowingandfarawayfromtherepresentationstheoryofLiegroups
we use the Iwasawadecomposition of SL(2,R), to define the Fourier transform
and to demonstrate Plancherel formula on the connected real semisimple Lie
group SL(2,R). Therefore let SL(2,R) be the real Lie group, which is
a b
SL(2,R)={ : (a,b,c,d)∈R4 and ad−bc=1} (2)
c d
(cid:18) (cid:19)
and let SL(2,R)=KNA be the Iwasawa decomposition of SL(2,R), where
cosφ −sinφ
K = { =SO(2): φ∈R }
sinφ cosφ
(cid:18) (cid:19)
1 n
N = { : n∈R }
0 1
(cid:18) (cid:19)
a 0
A = { : a∈R⋆} (3)
0 a−1 +
(cid:18) (cid:19)
Hence every g ∈ SL(2,R) can be written as g = kan ∈ SL(2,R), where
k ∈K, a∈A, n∈R.
2.2. We denote by L1(G) the Banach algebra that consists of all complex
valued functions on the groupG, which are integrable with respect to the Haar
measure dg of G and multiplication is defined by convolution product on G ,
where G=SL(2,R). And denote by L2(G) the Hilbert space of G. So we have
for any f ∈L1(G) and φ∈L1(G)
φ∗f(h)= f(g−1h)φ(g)dg (4)
Z
G
The Haarmeasuredg ona connectedrealsemi-simpleLie groupG=SL(n,R),
canbecalculatedfromtheHaarmeasuresdn,daanddkonN;AandK;respectively,
2
by the formula
f(g)dg = f(ank)dadndk (5)
Z Z Z Z
G A N K
Keeping in mind thata−2ρ is the modulus of the automorphismn→ana−1
of N we get also the following representation of dg
f(g)dg = f(ank)dadndk = f(nak)a−2ρdndadk (6)
Z Z Z Z Z Z Z
G A N K N A K
where
ρ=2−1 m(α)α
α≥0,α6=0
X
and m(α) denotes the multiplicity of the root α see [17] or again ρ = the
dimensionofthenilpotentgroupN.Furthermore,usingtherelation f(g)dg =
G
f(g−1)dg, we receive R
G
R
f(g)dg = f(kan)a2ρdndadk (7)
Z Z Z Z
G K A N
2.3. Let Γ be a connected compact Lie group and let k be the Lie algebra
of Γ. Let (X ,X ,.....,X ) a basis of k , such that the both operators
1 2 m
m
∆= X2 (8)
i
i=1
X
m l
D = − X2 (9)
q i
!
0≤l≤q i=1
X X
are left and right invariant (bi-invariant) on Γ, this basis exist see [2, p.564).
For l ∈ N, let Dl = (1−∆)l, then the family of semi-norms {σ , l ∈ N} such
l
that
σl(f)= Dlf(y) 2dy)12, f ∈C∞(Γ) (10)
ZΓ
(cid:12) (cid:12)
define on C∞(Γ) the same(cid:12)topolog(cid:12)y of the Frechet topology defined by the
semi-normas kXαfk defined as
2
kXαfk = (|Xαf(y)|2dy)12, f ∈C∞(Γ) (11)
2
ZΓ
where α=(α ,.....,α )∈Nm, see [2,p.565]
1 m
Let Γ be the set of all equivilance classes of irreducible unitary representa-
tions of Γ. If γ ∈ Γ, we denote by E the space of representation γ and d its
γ γ
dimensiobn then we get
b
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Definition 2.1. The Fourier transform of a function f ∈C∞(Γ) is defined
as
Tf(γ)= f(x)γ(x−1)dx (12)
Z
Γ
where T is the Fourier transform on Γ
Theorem (A. Cerezo) 2.1. Let f ∈ C∞(Γ), then we have the inversion
of the Fourier transform
f(x)= dγtr[Tf(γ)γ(x) (13)
b
γX∈Γ
f(I )= dγtr[Tf(γ)] (14)
Γ
b
γX∈Γ
and the Plancherel formula
kf(x)k2 = |f(x)|2dx= d kTf(γ)k2 (15)
2 γ H.S
ZΓ γX∈Γb
for any f ∈ L1(Γ), where I is the identity element of Γ and kTf(γ)k2 is
Γ H.S
the Hilbert- Schmidt norm of the operator Tf(γ)
Fourier did not actually assume any underlying group structure or representa-
tiontheory butwe typically associatehis workwith the case ofthe circlegroup
in the following form using complex exponentials
∞ ∞
f(x)= Tf(m)eixm = c eixm, m∈Z (16)
n
n=−∞ m=−∞
X X
where
c =Tf(m)= f(x)e−ixmdx (17)
m
Z
SO(2)
The group is SO(2)=S1or R/Z and the multiplicative characters are eixn,
grouphomomorphismsfromthecircleK =SO(2)tothemultiplicativegroupof
non-zerocomplexnumbers. Fourieractuallypreferredtoexpressthecoefficients
using what is now known as the Plancherel formula
∞ ∞
kf(x)k2 = |f(x)|2dx= |c |2 = |Tf(m)|2 (18)
2 m
ZSO(2) n=−∞ n=−∞
X X
where
cosφ −sinφ
S1 =SO(2)={ :φ∈R } (19)
sinφ cosφ
(cid:18) (cid:19)
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Definition 2.2. For any function f ∈ D(G), we can define a function
Υ(f)on G×K =G×SO(2) by
Υ(f)(g,k )=Υ(f)(kna,k )=f(gk )=f(knak ) (20)
1 1 1 1
for g =kna∈G, and k ∈K . The restriction of Υ(f)∗ψ(g,k ) on K(G) is
1 1
Υ(f)∗ψ(g,k ) ↓ = f(nak ) = f(g) ∈ D(G), and Υ(f)(g,k ) ↓ = f(kna)
1 K(G) 1 1 K
∈D(G)
Remark 2.1. Υ(f) is invariant in the following sense
Υ(f)(gh,h−1k )=Υ(f)(g,k ) (21)
1 1
Definition 2.3. If f and ψ are two functions belong to D(G), then we can
define the convolution of Υ(f) and ψ on G ×K =G×S1 =G×SO(2) as
Υ(f)∗ψ(g,k ) = Υ(f)(gg−1,k )ψ(g )dg
1 2 1 2 2
Z
G
= Υ(f)(knaa−1n−1k−1k )ψ(k n a )dk dn da
2 2 1 2 2 2 2 2 2
Z Z Z
SO(2) N A
So we get
Υ(f)∗ψ(g,k ) ↓ =Υ(f)∗ψ(I na,k )
1 K(G) K 1
= f(naa−1n−1k−1k )ψ(k n a )dk dn da
2 2 1 2 2 2 2 2 2
Z Z Z
SO(2) N A
= Υ(f)∗ψ(na,k )
1
where g =k n a
2 2 2 2
Definition 2.4. If f ∈D(G) and let Υ(f) be the associated function to f ,
we define the Fourier transform of Υ(f)(g,k ) by
1
FΥ(f))(IS1,ξ,λ,γ,IS1)=FΥ(f)(IS1,ξ,λ,IS1)
∞
= [ TΥ(f)(kna,k )e−ilkdk]a−iλe−ihξ,ni e−imk1dadndk
1 1
ZS1ZNZA l=−∞ZS1
X
= [Υ(f)(IS1na,k1)]a−iλe−ihξ,ni e−imk1dadndk1 (22)
ZS1ZNZA
where F is the Fourier transform on AN and T is the Fourier transform on
SO(2), and IS1 is the identity element of S1 =SO(2)
Plancherel’s Theorem on the Group G 2.2. For any function f ∈
L1(G)∩ L2(G),we get
∞
|f(g)|2dg = |f(kna)|2dadndk = kTFf(λ,ξ,m)k2dλdξ
2
ZG AZ NZ SZ1 mX=−∞ZR ZR
(23)
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∞ ∞
f(IAINIS1)= TFf((λ,ξ,m)]dλdξ = TFf(λ,ξ,m)dλdξ
NZ AZ mX=−∞ mX=−∞ZR ZR
(24)
where I ,I , and I are the identity elements of A, N and K respectively,
A N K
where F is the Fourier transform on AN and T is the Fourier transform on K,
and I is the identity element of K
K
∨
Proof: First let f be the function defined by
∨
f(kna)=f((kna)−1)=f(a−1n−1k−1) (25)
Then we have
|f(g)|2dg
ZG
∨
= Υ(f)∗f(IS1INIA,IS1)
∨
= Υ(f)(IS1INIA(g2−1),IS1)f(g2)dg2
Z
G
∨
= Υ(f)(a−21n−21k2−1,IS1)f(k2n2a2)da2dn2dk2
Z Z ZS1
A N
= f(a−1n−1k−1)f((k n a )−1)da dn dk
2 2 2 2 2 2 2 2 2
Z Z ZS1
A N
= |f(a n k )|2da dn dk (26)
2 2 2 2 2 2
Z Z ZS1
A N
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Secondly
∨
Υ(f)∗f(IS1INIA,IS1)
∨
= F(Υ(f)∗f)(IS1,λ,ξ,IS1)dλdξ
ZR ZR
∞ ∞
∨
= Υ(f)∗f(kna,k )e−ilkdka−iλe−ihξ,ni e−imk1dadndk dλdξ
1 1
ZS1ZR ZR AZ NZ mX=−∞l=X−∞ZS1
∞
∨
= Υ(f)∗f(IS1na,k1)e−ilkdka−iλe−ihξ,ni e−imk1dadndk1dλdξ
mX=−∞ZS1ZR ZR AZ NZ
∞
∨
= Υ(f)(IS1naa−21n−21k2−1,k1)f(k2n2a2)e−imk1dk1
ZR ZR AZ NZ ZS1AZ NZ mX=−∞ZS1
dndadk dn da a−iλe−ihξ,nidλdξ
2 2 2
∞
∨
= f(naa−1n−1k−1k )f(k n a )e−imk1dk dk
2 2 2 1 2 2 2 1 2
ZR ZR AZ NZ ZS1AZ NZ mX=−∞ZS1
a−iλe−ihξ,nidndadn da dλdξ
2 2
where
e−ihξ,ni =e−iξn (27)
Using the fact that
f(kna)dadndk = f(kan)a2dndadk (28)
Z Z ZS1 Z Z ZS1
A N N A
and
f(kna)e−ihξ,nidadndkddξ
ZR AZ NZ ZS1
= f(kan)e−ihξ,an1a−1 ia2dadndkdξ
ZR AZ NZ ZS1
= f(kan)e−ihaξa−1,nia2dadndkdξ
ZR AZ NZ ZS1
= f(kan)e−ihξ,nidadndkdξ (29)
ZR AZ NZ ZS1
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Then we get
∨
Υ(f)∗f(IS1INIA,IS1)
∞
∨
= f(naa−1n−1k−1,k )f(k n a )e−imk1dk dk
2 2 2 1 2 2 2 1 2
ZR ZR AZ NZ ZS1AZ NZ mX=−∞ZS1
a−iλe−ihξ,nidndadn da dλdξ
2 2
∞
∨
= f(aa−1nn−1k−1,k )f(k n a )e−imk1dk dk
2 2 2 1 2 2 2 1 2
ZR ZR AZ NZ AZ NZ mX=−∞ZS1ZS1
a−iλe−ihξ,nidndadn da dλdξ
2 2
∞
∨
= f(ank−1,k )f(k n a )e−imk1dk dk
2 1 2 2 2 1 2
ZR ZR AZ NZ AZ NZ mX=−∞ZS1ZS1
a−iλe−ihξ,nidndadn da dλdξ
2 2
∞
∨
= f(ank−1k )f(k n a )e−imk1dk dk
2 1 2 2 2 1 2
ZR ZR AZ NZ AZ NZ mX=−∞ZS1ZS1
a−iλe−ihξ,nidndadn da dλdξ
2 2
∞
∨
= f(ank−1)f(k n a )e−imk1e−imk2dk dk
1 2 2 2 1 2
ZR ZR AZ NZ AZ NZ mX=−∞ZS1ZS1
a−iλa−iλe−ihξ,n+n2idndadn da dλdξ
2 2 2
∞
= f(ank−1)f(a −1n−1k−1)e−imk1e−imk2dk dk
1 2 2 2 1 2
ZR ZR AZ NZ AZ NZ mX=−∞ZS1ZS1
a−iλe−ihξ,nia−iλe−ihξ,n2idndadn da dλdξ
2 2 2
∞
= f(ank−1)f(a n k )e−imk2e−imk1dk dk
1 2 2 2 1 2
ZR ZR AZ NZ AZ NZ mX=−∞ZS1ZS1
a−iλe−ihξ,nia−iλeihξ,n2idndadn da dλdξ
2 2 2
∞
= f(ank−1)f(a n k )e−imk2e−imk1dk dk
1 2 2 2 1 2
ZR ZR AZ NZ AZ NZ mX=−∞ZS1ZS1
a−iλe−ihξ,nia−iλe−ihξ,n2idndadn da dλdξ
2 2 2
∞ ∞
= TFf(λ,ξ,m)TFf(λ,ξ,m)dλdξ = |TF(f)(λ,ξ,m)|2dλdξ
ZR ZR mX=−∞ ZR ZR mX=−∞
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3 Fourier Transform and Plancherel Formula H.
3.1. Let H be the real Heisenberg group of dimension 2n+1 which consists of
all matrices of the form
1 x z
0 I y (30)
 
0 0 1
where x∈Rn, y ∈Rn, z ∈R and I is the identity matrix of order n.
Let H = Rn+1⋊ Rn be the group of the semi-direct product of the group
ι
Rn+1 and Rn, via the group homomorphism ι : Rn → Aut(Rn+1), which is
defined by:
ι(x)(z,y)=(z+xy,y)=x(z,y) (31)
for any x = (x ,x , ... ,x ) ∈ Rn , y = (y ,y , ... ,y ) ∈ Rn , z ∈ R , and
1 2 n 1 2 n
n
xy = x y , where Aut(Rn+1) is the group of all automorphism of Rn+1
i i
i=1
3.2P. LetC∞(H),D(H),D′(H),E′(G) respectivelythespaceofC∞-func-
tions , C∞ with compact support, distribution and distribution with compact
supportonG. TheSchwartsspaceS(G)ofGcanbeconsideredastheSchwarts
spaceS(R2n+1)ofthevectorgroupR2n+1. TheactionιofthegroupRnonRn+1
definesanaturalactionιonthedual(Rn)∗ ofthegroupRn+1((Rn+1)∗ ≃Rn+1)
which is given by :
x(η,λ)=(η,ηx+λ)
for any λ∈Rn, x∈Rn and η ∈R ,where ;
x(η,λ)=ι(x)(η,λ)
and
n
ηx= ηx
i
i=1
X
Definition 3.1. For every f ∈ S(G), one can define its Fourier transform Ff
by :
Ff(ξ)= f(X) e−ihξ,Xi dX (32)
Z
G
where X =((z,y);x)∈G,ξ =((η,λ);µ)∈G,and dX =dzdydx theLebesgue
measure on G
n n
hξ,Xi=zη+yλ+xµ=zη+ λ y + x µ
i i i i
i=1 i=1
X X
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It is clear that the function Ff ∈ S(G) and the mapping f 7→ Ff is a
topological isomorphism vector space S(G) onto it self.
Theorem 3.1. The Fourier transform F satisfies :
∨
g∗f(0)= Ff(ξ) Fg(ξ) dξ (33)
Z
G
∨
for every f ∈ S(G) and g ∈ S(G), where g(X) = g(X−1), ξ = ((η,λ);µ),
dξ = dηdλdµ, is the Lebesgue measure on G = R2n+1, and ∗ denotes the con-
volution product on G
Proof : By the classical Fourier transform, we have:
∨ ∨
g∗f(0)= F(g∗f)(ξ) dξ
Z
G
= ∨g∗f(X) e−ihξ,Xi dX dξ
Z Z
G G
= f(Y−1X)g(Y−1) e−ihξ,Xi dY dX dξ
Z Z Z
G G G
= f(YX)g(Y) e−ihξ,Xi dY dX dξ (34)
Z Z Z
G G G
By change of variable YX = X′,with X′ = ((z,y);x) and Y = ((z′,y′);x′)
we get :
X = Y−1X′ =((−x′(−z′,−y′))−x′)((z,y);x)
= ((−x′(z−z′,y−y′));x−x′)
this gives us :
e−ihξ,Xi
= e−ihξ,Y−1X′i
= e−ih(−x′(η,λ);µ);((z−z′,y−y′);x−x′)i
= e−ih((η,−ηx′+λ);µ),((z−z′,y−y′);x−x′)i (35)
By the invariant of the Lebesgue measures dη, dλ, and dµ we obtain,
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