Applied Mathematics, 2013, 4, 35-40
Published Online November 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.411A3006
Open Access AM
Canonical and Boundary Representations on Rank
One Para-Hermitian Spaces
Anatoli A. Artemov
G. R. Derzhavin Tambov State University, Tambov, Russia
Email: tria@tsu.tmb.ru
Received July 24, 2013; revised August 24, 2013; accepted September 2, 2013
Copyright © 2013 Anatoli A. Artemov. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This work studies the canonical representations (Berezin representations) for para-Hermitian symmetric spaces of rank
one. These spaces are exhausted up to the covering by spaces GH with
SL ,Gn, . For Her-
mitian symmetric spaces
GL 1,Hn
GK, canonical representations were introduced by Berezin and Vershik-Gelfand-Graev.
They are unitary with respect to some invariant non-local inner product (the Berezin form). We consider canonical rep-
resentations in a wider sense: we give up the condition of unitarity and let these representations act on spaces of distri-
butions. For our spaces GH, the canonical representations turn out to be tensor products of representations of maxi-
mal degenerate series and contragredient representations. We decompose the canonical representations into irreducible
constituents and decompose boundary representations.
Keywords: Para-Hermitian Symmetric Spaces; Overgroups; Canonical Representations; Boundary Representations;
Poisson and Fourier Transforms
1. Introduction
Canonical representations on Hermitian symmetric spac-
es GK were introduced by Vershik-Gelfand-Graev [1]
(for the Lobachevsky plane) and Berezin [2]. They are
unitary with respect to some invariant non-local inner
product (the Berezin form). Molchanov’s idea is that it is
natural to consider canonical representations in a wider
sense: to give up the condition of unitarity and let these
representations act on sufficiently extensive spaces, in
particular, on distributions. Moreover, the notion of ca-
nonical representation (in this wide sense) can be ex-
tended to other classes of semisimple symmetric spaces
GH, in particular, to para-Hermitian symmetric spaces,
see [3]. Moreover, sometimes it is natural to consider se-
veral spaces i
GH together, possibly with different
i
H
, embedded as open -orbits into a compact mani-
fold , where acts, so that is the closure of
these orbits.
G
G
Canonical representations can be constructed as fol-
lows. Let be a group containing (an overgroup),
a series of representations of induced by charac-
ters of some parabolic subgroup associated with
G
G
P
R
G
GH and acting on functions on . The canonical
representations of are restrictions of to G.
R GR
In this talk we carry out this program for para-Hermi-
tian symmetric spaces of rank one. These spaces are ex-
hausted up to the covering by spaces GH with
SL ,Gn,
1,GLHn. For these spaces
GH, an overgroup is the direct product GG
and
canonical representations turn out to be tensor products
of representations of maximal degenerate series and con-
tragredient representations. These tensor products are
studied in [4], see also [5]. So we lean essentially on
these papers [4,5]. We decompose canonical representa-
tions into irreducible constituents and decompose boun-
dary representations. Notice that in our case the inverse
of the Berezin transform ,
Q
can be easily written:
precisely it is the Berezin transform ,n
Q
 .
Canonical and boundary representations for GH in
the case 2n
(then GH is the hyperboloid of one
sheet in ) were studied in [6]. For the two-sheeted
hyperboloid in , it was done in [7].
3
3
In this paper we present only the main results. The de-
tailed theory of canonical and boundary representations,
for example, on a sphere with an action of the generaliz-
ed Lorentz group, can be seen in [8].
A. A. ARTEMOV
36
Let us introduce some notation and agreements.
By we denote
. The sign denotes
the congruence modulo 2.
0,1,2,
For a character of the group we shall
use the following notation

*0
,sgn ,tt
 
t
where ,
*
t
, 0,1
.
For a manifold
M
, let
M
denote the Schwartz
space of compactly supported infinitely differentiable
-valued functions on
M
, with a usual topology, and

M
the space of distributions on
—of anti-li-
near continuous functionals on

M
.
2. The Space GH and the Manifold
We consider the symmetric space GH where
, , .

SL ,Gn

GL 1,Hn
G3n
Mat n
The group acts on the space by
,
1.
x
gxg
Let us write matrices in in block form
according to the partition
Mat ,n

1nn
1
 of n. Let us
take the matrix
000
.
01
x


The subgroup
H
is just the stabilizer of this point 0
x
,
this subgroup consists of block diagonal matrices:

1
0,GL1,, det
0
hn

.




Thus, our space GH is the -orbit of G0
x
, it con-
sists of matrices of rank one and trace one.
Equip with the standard inner product
n
,
x
y, let
12
,
x
xx. Let be the sphere S1s. Let d
s
be
the Euclidean measure on . The group acts on
by
SG S
s
sg sg.
Let be a cone in
,nMat consisting of matri-
ces of rank one. Therefore, the space
0xGH is
the section of by the hyperplane
tr 1
x
.
Introduce a norm
x
in by

Mat ,n


12
tr ,xxx
where the prime denotes matrix transposition.
Let be the section of by
1x.
Define a map by
SS

,.
s
tts
u
It is a two-fold covering. The measure d
s
defines a
measure on by
du
 
1
ddd,
2SS
The action of the group on gives the follow-
ing action of on
G S
G
:
1
1.
g
ug
u
g
ug
In particular, the subgroup

SO
K
n, a maximal
compact subgroup, acts on by translations:
1.ukuk
Let us consider on
the function
,, .ppu stuts
  (1)
The action on
has three orbits: namely, two open
orbits (of dimension 22n
): and
0p

0p
 and one orbit of dimension 2n3
:
0p
. The orbit
is a Stiefel manifold, it is the
boundary of
. Denote 
 . Each of orbits
can be identified with the space GH. The map is
constructed by means of generating lines of the cone .
3. Maximal D e ge n e r at e S er i e s
Representations
Recall [4] maximal degenerate series representations
,
,
, 0,1
, of the group . Let G
S
be
the subspace of
S consisting of functions
of
parity
:

1
s
s

. The representations
,
act on
S
by



,,
sg
g
ss
sg g


 





1
1
,1
=|
''
'
sg
gs sg
sg |.







4. Representations of Associated with G
GH
Recall [5] a series of representations ,
T
of the group
associated with the space
GGH.
Denote by
the space of functions
in
of parity 0,1
:

1.
 

The representation ,
T
acts on by


1
1
,1.
gg
Tg gg
gg

 




(2)
Let ,
denote the following sesqui-linear form

,d.

(3)
Define an operator ,
A
on by

1,n

.
f
uu ftsstuts







,trd .A



Open Access AM
A. A. ARTEMOV 37
It intertwines ,
T
and 1,n
T
 . The operator ,
A
is a
meromorphic function of
. Let us normalize this
operator (multiplying it by a function of
) such that
the normalized operator ,
A
is an entire non-vanishing
function of
.
There are three series of unitarizable irreducible
representations. The continuous series consists of ,
T
with

12ni
,
, the inner product is (3).
The complementary series consists of ,
T
with

22n
2n, the inner product is ,,A


with a factor. The discrete series consists of the repre-
sentations

,m
T
where
22mn
 m
,
m, 0,1
, which are factor representations of

,m
T
on the quotient spaces


,
Ker m
A
. The
representations

,m
T
with the same and different
m
are equivalent. It is convenient to take
m


1m
where for odd and for
even . The inner product is induced by the form

0m
n

n

m

,mm

,A

.
5. Canonical Representations
We define canonical representations ,
R
,
,
0,1
, of the group as tensor products: G
,.
nn
R
 


 

They can be realized on : let denote the
subspace of consisting of functions of
parity


f
:

1
f
uf
u, then the representation
,
R
acts on by a formula similar to (2):


1
1
,1.
n
gug
Rgfufgug
gug






The inner product

,fhfuhu u
d (4)
is invariant with respect to the pair

,
,n
RR,

 , i.e.


1
,,
,, . (5)
n
Rgfh fRgh
 

Consider an operator ,
Q
on defined by

 


,
,,tr dQfu cuvfvv



,
It turns out that the composition ,,n
QQ

 is equal to
the identity operator up to a factor. We can take
E
,c
such that
,, ,
n
QQ E


namely,
 
2
121
,
coscos .
22
nn n
c
nn
 



1
 
 
 
 
With the form (4) the operator ,
Q
interacts as fol-
lows:
,
,,Qfh fQh
 
,
.
(6)
This operator ,
Q
intertwines the representations ,
R
and ,n
R
 , i.e.

,,,,
,.
n
RgQQRgg
 
 G
Let us call it the Berezin transform.
Let
be the space of distributions on
of
parity
. We extend ,
R
and ,
Q
to
by (5)
and (6) respectively and retain their names and the
notation.
Let us introduce the following Hermitian form
,
,fh
on
:

,
,
1
,,
2
f
hQf


h
Let us call this form the Berezin form.
6. Boundary Representations
The canonical representation ,
R
gives rise to two re-
presentations ,
L
and ,
M
associated with the boun-
dary
of the manifolds (boundary representa-
tions). The first one acts on distributions concentrated at
, the second one acts on jets orthogonal to
.
We can introduce “polar coordinates” on
corre-
sponding to the foliation of into
K
-orbits. The
K
-
orbits are level surfaces of the function , see (1). For
p
1p1
 the
K
-orbits are diffeomorphic to
. In
these coordinates the measure on is
du


32
2
d1 dd
n
up p,

where d
is the measure on .
Let
f
be a function in . Consider it as a
function of polar coordinates. Consider its Taylor series

2
01 2
aa app
 in powers of . Here p
aaf
mm are functions in . Denote by

k
k,
, 0,1
, the space of distributions in
k
, having the form



0
,
m
m
m
p

where
mm
,
is the Dirac delta function on
the real line,

m
its derivatives. Let
k

 .
Denote by
m
af
* Taylor coefficients of the function
Open Access AM
A. A. ARTEMOV
38



32
2
1n
pf
u. The distribution acts



mp

on a function as follows:

f


  
*
,1!,. (7)
m
mm
pfma f
 

Denote by ,
L
the restriction of ,
R
to
.
This representation is written as a upper triangular matrix
with the diagonal , .
1,nmm
T

 
k
m
Distributions in can be extended in a natural


way to a space wider than
. Namely, let
k
be the space of functions
f
of class C on
and
of parity
and having the Taylor decomposition of
order :
k


01 ,
kk
k
f
uaapapop 
where . Then (7) keeps for

m
a
k
f
with . mk
Let denote the column of Taylor coefficients
m. The representation

af
f
a,
M
acts on these co-
lumns:

,,
.
M
gafaRgf
 
It is written as a lower triangular matrix with the di-
agonal , .
,nmm
The boundary representations
T

 m
,
L
and ,
M
are in
a duality.
7. Poisson and Fourier Transforms
Let us write operators ,; ,
P

and ,; ,
F

intertwining
representations ,
R
and ,
T
. We call them Poisson
and Fourier transforms associated with canonical
representations.
The Poisson transform ,; ,
P

is a map
given by
 
C

 


,
,
,; ,trd .
n
Pupu

 
  

 
It intertwines 1,n
T
 with ,
R
. Here we consider
,
R
as the restriction to
C
of the representation
,
R
acting on distributions in .


For a -finite function
K
and

12n
  the Poisson transform has the
following decomposition in powers of :
p






,
,; ,,,
0
1,
,,
0
,
n
k
k
k
k
k
PupC p
pD
 
 
 



 
 

k
p
where has polar coordinates
u,p
. Here ,,k
C
and
,,k
D
are certain operators acting on
. The
factors ,n
p

 and 1,
p

 give poles of the
Poisson transform in
depending on
:
,1nk ,l

 (8)
where ,kl
and k
, l
. If a pole 
hen the pobelongs one of se tle is simple,
and if a pole belongs to both series (8), then
only to ries (8),
12n
 and the pole is of the second or first
or
Let
der.
k

, k
, be simple. The
the pole
residue ,,
ˆk
P

of ,; ,
P

at this ole is an operator p
k
. Denote the image of this operator by  
,,k
V
.
The Fourier ,; ,
F

transform is a map

 given by
 
tr
,
 

,
,; ,d .
F
fupf


  


uu
It intertwines

,
R
with ,
T
.
n trThe Fourier Poissoan
ea
andsforms are conjugate to
ch other:
,; ,,; ,
,,
n
Ff fP
   

 .
Poles in
of the Fourier transform are situated at
po
,l
ints
,1nk

 (9)
where ,kl
and k
, l

(9), then
. If a pole
belongs the pole is
simple, and if a pole belongs to both series (9), then
only to one of ries the se
12n
  and the pole is of the second or first
or
Let
der.
the pole nk

 , k
, be simple.
The residue ,,
ˆ
F
nk
 

perator
of at th

,;
F

,is pole is a
“boundary” o
,,k
B

 
, k
:
.
The operator ,,k
B
is dyl-
efficients
e terms of Tafined inor co
m
a is a linear combination of functions
f
: it
,,nk k
Daf

 . Therefore, we may consider
rator ,
B
*
mm
the following ope
acting on columns
012
,,,aaaa of functio

k
a
: this ope-
rator to any col
ns
umn a assigns the column
,,,0,,,,2
,,,Ba B aBB a

of functio
1
ans in the
same space
—by the same formulas without
f
.
This operator ,
B
is given by a lower triangular m-
trix.
8. D
a
ecomposition of Boundary
Th ure of the Poisson and Fourier
Representations
e meromorphic struct
transforms is a basis for decompositions of boundary re-
presentations ,
L
and ,
M
.
Let the polek

the Poisson transform is
si
of
mple, in particupens when

lar, it hap12n
 .
Open Access AM
A. A. ARTEMOV 39
Then the boundary representation L,
is diagonaliz -
able which means that

decoses into the di- mpo
rect sum of ,, ,
k
Vk

the restriction of ,
L
, and
to ,,k
V
is eto 1,nkk
T

  (by meansf quivalent o
If a pole is of the second order, thee decomposition
of
,, k

).
ˆ
P
n th
,
L
contains a finite number of Jordan blocks, this
num depends on ber
.
Let the pole nk


rticular, when
of turier transform
is
he Fo
simple, in pa
12n
. Then
the matrix ,
M
is diagonalins that
1
,,,
BMB
zablch meae whi

a diagonal matrix. Its diagonal is is
If a pole is of the second order, thee decomposition
of
,nkk
T

 , k.
n th
,
M
contains a finite number of Jordan blocks, this
numpends on ber de
.
9. Decomposition of Canonical
Leon ofpresentations.
Representations
t us write decompositi canonical re
We restrict ourselves to a generic case:
lies in strips
11nn 

:Re ,.
22
k
Ik kk


 


Case (A):
0
I
.
1 Let Theorem 0
I
. The canonical
re en th
presentation ,
R
decomposesas the quasiregular
representation into irreducible unitary representa-
tions of continuous and discrete series with multiplicity
one. Namely , let us assign to a function
[5]—
fD
 the
family of its Fourier components ,; ,
F
f,

  
12ni
 ,
, 0,1
, and
h responde

,;1,nmm
F

 , m. Tis cornce is G-
equivariant. There is an inversion formula:



,;1, ,;12
,; ,,;1
0
,d
nni
mmm n
m
fPf
PF f
  
 

,
,mm
F

 
 


(10)
and a “Plancherel formula” for the Berezin form:
  

 

 
,
,; ,,;1,12
0
,;1,,; ,
,,,;,
,
,; ,
,.
nni
m
m
nmm mm
fh
FfF h
mm
FfFh

 
 
 
d
 
,


 



(11)
Here and m
stand for the Plancherel mea-
sure for GH,
ula:
see [5 the factoris given by fol-
lowing form
],
 
 


 
,; ,
111cosπ
1sinπ
sin2 π1sin 2π
co π
n
n
nn

 
.
s 21cos 2πnn
 
 

 

 
 
Case (B): 1,
k
Ik
.
Here we continue decomposition (10) analytically in
to 1k
I
, k
from 0
I
. Some poles in
of the
integrating line—the line integrand intersect the Re
12n. They are poles m

 and 1n
m
of the
give additional summands to the right hand side. So after
the continuation we obtain:
(12)
whn and the ser a
Poisson transform with 0mk. They

,, ,
k
m
ff

 

00
mm


ere the itegralies mean the sames in (10)
and
,,,, ,;1
ˆ
,,
mm
LmP F
 


,,
 

nmm
,,Lm

are some numbers.
Similarly, the continuation of (11) gives
 
,00
,,
k
mm
f
,;
1,,; ,
,,
nm
m mm
,hM




 

(13)
where the integral and the series mean the same as in (11)
and
m
FfFh
  
  
,,
M
m

he ope
are some numbers.
Trators ,,m

, mk, can be extended from
to the space
k
and therefore to the sum

.
kk


Then these operators ,,m

turn out to be
operators ontothere are some “or-
(1 Theo
projection
,,m
V

. Moreover,
thogonality relations” for them. Decomposition (13) can
also be extended to the space

k
. This decompo-
sitionthagorean theorem” is a “Pyfor decomposition
2). rem 2 Let 1k
I
, k. Then the space
the represe
has to On this space
ntation
be completed to
,
R

k
.
splits int
s as o the sum of two terms:
the first one decompose,
R
does in Case (A), the
second one decomposes into the sum of 1k irreducible
representations 1,nmm
T

 , 0,1, ,mk. Namely, let
us assign to any
k
f
th


e family
,; ,,,
,;1,
,,
m
nmm
F
fFf f
  


Open Access AM
A. A. ARTEMOV
Open Access AM
40

12ni
 , n
,
ivariant.
where This
co -equ
formula rel form
0,1, ,mk.
T
rrespondence is Ghere is an inverse
, see (12), and a Plancheula”, see (13).
Case (C): 1k
Ik,

.
Now we continue decompion (10) analytically in osit
from 0
I
to 1k
I
. Here poles nm

   and
1m
, m
 , mk
, of the integrand (they
are poles of the Fform) give addourier transitional terms.
We obtain

,
k
m,,
00mm
f
f
 



(14)
where the integral and the series mean the same as in (10
)
an
m
somebers. Tperat
of e op
ere are
Now we continue decomposition (11) from
d
,,,,,; 1,mm m
NP B
 

,,m

numhe oors ,,m

can be ex-
tended to the space

k
, mk. Denote by ,,m

the image
,,
m

N
,,m

them th
,,

are projection
some
m. It turn
operat
“orthogon
s ou
ors onto
ality relatio
t that th
n
erators
and for
,,m

s”.
0
I
to
1k
I
. Poles of the integrand which intersect the inte-
grating line

Re 12n
 and give additionalrms
e poles of both Fourier transforms) tu or-
tely to be of the first order, since at these po
te
ut fo
ints th
(they ar
tuna
functi
rn
e
on ,; ,

as a function of
o
has
ntinuan we
zero of the
first order. After the cotiobtain:
 
 
,00
,,,
k
mm
fhK m
 


,,
,,
nm m m
AB
fBh


 

where the inties mean the same as in (11),

,,
egral and the ser
(15)
K
m

some numbers. It is a “Pythagorean the-
orem” for decomposition (14).
Theore et 1
k
Im 3 L
, k Then the repre-
sentation ,
R.
considered on the space

k
splits
into the sum of two termse a
space of functions f such that their Taylor coefficients
m
af are equal to 0 for mk and decomposes as
,
R
does in Case (ce decomposes into
the direct sum of 1k
A), the onseond
irreducible representations
,nm m
T

 , mk
acting on the sum of the spaces
,,m

. There is annversion formula, see (14), and a
herel formula for the Bn form, see (15).
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
2,SLR Where R Is a Rin
i
p
“Q
ra
Planc
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erezi
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