Plancherel Measure for the Quantum Matrix Ball-1
The Plancherel formula is one of the celebrated results of harmonic analysis on semisimple Lie groups and their homogeneous spaces. The main goal of this work is to ¯nd a q-analogue of the Plancherel formula for spherical transform on the unit matrix ball. Here we present an explicit formula for the...
Збережено в:
| Опубліковано в: : | Журнал математической физики, анализа, геометрии |
|---|---|
| Дата: | 2009 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2009
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/106546 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Plancherel Measure for the Quantum Matrix Ball-1 / O. Bershtein, Ye. Kolisnyk // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 315-346. — Бібліогр.: 32 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-106546 |
|---|---|
| record_format |
dspace |
| spelling |
Bershtein, O. Kolisnyk, Ye. 2016-09-30T08:20:40Z 2016-09-30T08:20:40Z 2009 Plancherel Measure for the Quantum Matrix Ball-1 / O. Bershtein, Ye. Kolisnyk // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 315-346. — Бібліогр.: 32 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106546 The Plancherel formula is one of the celebrated results of harmonic analysis on semisimple Lie groups and their homogeneous spaces. The main goal of this work is to ¯nd a q-analogue of the Plancherel formula for spherical transform on the unit matrix ball. Here we present an explicit formula for the radial part of the Plancherel measure. The q-Jacobi polynomials as spherical functions naturally arise on the way. The first author was partially supported by the N. I. Akhiezer fund. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Plancherel Measure for the Quantum Matrix Ball-1 Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Plancherel Measure for the Quantum Matrix Ball-1 |
| spellingShingle |
Plancherel Measure for the Quantum Matrix Ball-1 Bershtein, O. Kolisnyk, Ye. |
| title_short |
Plancherel Measure for the Quantum Matrix Ball-1 |
| title_full |
Plancherel Measure for the Quantum Matrix Ball-1 |
| title_fullStr |
Plancherel Measure for the Quantum Matrix Ball-1 |
| title_full_unstemmed |
Plancherel Measure for the Quantum Matrix Ball-1 |
| title_sort |
plancherel measure for the quantum matrix ball-1 |
| author |
Bershtein, O. Kolisnyk, Ye. |
| author_facet |
Bershtein, O. Kolisnyk, Ye. |
| publishDate |
2009 |
| language |
English |
| container_title |
Журнал математической физики, анализа, геометрии |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
The Plancherel formula is one of the celebrated results of harmonic analysis on semisimple Lie groups and their homogeneous spaces. The main goal of this work is to ¯nd a q-analogue of the Plancherel formula for spherical transform on the unit matrix ball. Here we present an explicit formula for the radial part of the Plancherel measure. The q-Jacobi polynomials as spherical functions naturally arise on the way.
|
| issn |
1812-9471 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106546 |
| citation_txt |
Plancherel Measure for the Quantum Matrix Ball-1 / O. Bershtein, Ye. Kolisnyk // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 315-346. — Бібліогр.: 32 назв. — англ. |
| work_keys_str_mv |
AT bershteino plancherelmeasureforthequantummatrixball1 AT kolisnykye plancherelmeasureforthequantummatrixball1 |
| first_indexed |
2025-11-26T23:36:14Z |
| last_indexed |
2025-11-26T23:36:14Z |
| _version_ |
1850781527837769728 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 4, pp. 315–346
Plancherel Measure for the Quantum Matrix Ball-1
O. Bershtein∗ and Ye. Kolisnyk
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:bershtein@ilt.kharkov.ua
evgen.kolesnik@gmail.com
Received April 7, 2008
The Plancherel formula is one of the celebrated results of harmonic anal-
ysis on semisimple Lie groups and their homogeneous spaces. The main
goal of this work is to find a q-analogue of the Plancherel formula for sphe-
rical transform on the unit matrix ball. Here we present an explicit formula
for the radial part of the Plancherel measure. The q-Jacobi polynomials as
spherical functions naturally arise on the way.
Key words: quantum matrix ball, Plancherel formula, spherical functions,
difference operators.
Mathematics Subject Classification 2000: 81R50, 17B37 (primary); 20G42
(secondary).
1. Introduction
Let us recall one of the most common problems of harmonic analysis on
homogenous spaces. Let G be a real Lie group, K be a closed subgroup and
dν be a G-invariant Haar measure on X = K\G. The representation of G by
right shifts in L2(X, dν)
R(g) : f(x) 7→ f(xg), x ∈ X, g ∈ G,
is strongly continuous and unitary. It is called a quasiregular representation.
The problem is to find a decomposition of R into irreducible representations.
A special case of the Riemannian symmetric space X = K\G and its isometry
group was studied in detail ([9, p. 192], [10, p. 506]). Harmonic analysis for these
spaces was developed by E. Cartan, I. Gelfand, F. Berezin, Harish-Chandra,
S. Gindikin and F. Karpelevich.
∗The first author was partially supported by the N. I. Akhiezer fund.
c© O. Bershtein and Ye. Kolisnyk, 2009
O. Bershtein and Ye. Kolisnyk
The problem of harmonic analysis is closely connected with the following.
Consider the algebra DG(X) of all G-invariant differential operators on X. An im-
portant result of the representation theory is that the decomposition of R can
be obtained by using common eigenfunctions of operators from DG(X). Namely,
the shifts of a common eigenfunction generate an irreducible subrepresentation
of R.
In the case of a Riemannian symmetric space the algebra DG(X) is finitely
generated and commutative ([9, p. 431]). Consider a set of generators of DG(X)
and their restrictions L1,L2, . . . ,Lr onto the subspace of smooth K-invariant
functions on X with compact support. The Plancherel measure is a Borel measure
in r-dimensional space, and the problem is to find this measure.
Let us describe briefly how to find common eigenfunctions. Recall that an ir-
reducible strongly continuous unitary representation T is called a representation
of type I if it contains a nonzero K-invariant vector v. We can assume that
(v, v) = 1. The function f(g) = (T (g)v, v) is called a spherical function. It is
constant on double cosets K\G/K, so it corresponds to a K-invariant function
on X. This function is a common eigenfunction of the operators L1,L2, . . . ,Lr.
The problem of the decomposition of K-biinvariant functions on G in terms of
the spherical functions naturally arises while solving the general decomposition
problem of L2(X, dν).
Consider a more special case. The homogeneous space SUn,n/S(Un × Un) is
a Hermitian symmetric space of noncompact type. It has the standard Harish-
Chandra realization as the unit ball
D = {z ∈ Matn | ||z|| < I}
(in the space of complex n × n-matrices with respect to the operator norm).
It worth to be mentioned that standard generators of DG(X) are well known and
their common eigenfunctions are Jacobi polynomials [7, 11].
Quantum bounded symmetric domains were introduced in 1998 by L. Vaks-
man and S. Sinel’shchikov [24]. L. Vaksman with his collaborators managed to de-
velop the noncommutative complex analysis and representation theory on quan-
tum domains. A series of works were dedicated to quantum matrix balls that are
the simplest examples of quantum bounded symmetric domains [3, 21, 22, 29, 30].
For the case of quantum disk some problems of noncommutative harmonic
analysis are solved [15, 16]. In particular, explicit formulas for the invariant
integral, spherical functions, and the Plansherel measure are obtained.
In this paper we generalize the results mentioned above for the quantum
matrix ball case. Imitating the classical approach, we construct a family of com-
muting ’q-differential’ operators and find the exact formula for their common
eigenfunctions. We use spherical functions which appeare to be q-Jacobi polyno-
mials. We obtain the decomposition of the biinvariant functions in terms of the
316 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
spherical functions and the exact formula for the ’radial part’ of the Plancherel
measure.
The authors are grateful to L. Vaksman for the formulation of the problems
and invaluable discussions. We also thank D. Shklyarov for many important
remarks.
2. The Radial Part of the Uqsl2n-invariant Integral
2.1. Preliminaries on the quantum matrix ball
All results from the next two subsections form the basic notions in the the-
ory of quantum bounded symmetric domains. We refer to [21, 24] for the first
appearance and full consideration of these notions.
Let q ∈ (0, 1). All algebras are assumed to be associative and unital, and C
is the ground field.
Consider the well-known quantum universal enveloping algebra (see e.g. [13])
Uqsl2n corresponding to the Lie algebra sl2n. Recall that Uqsl2n is a Hopf algebra
with the generators {Ei, Fi, Ki, K−1
i }2n−1
i=1 and the relations
KiKj = KjKi, KiK
−1
i = K−1
i Ki = 1;
KiEi = q2EiKi, KiFi = q−2FiKi;
KiEj = q−1EjKi, KiFj = qFjKi, |i− j| = 1;
KiEj = EjKi, KiFj = FjKi, |i− j| > 1;
EiFj − FjEi = δij
Ki −K−1
i
q − q−1
;
E2
i Ej − (q + q−1)EiEjEi + EjE
2
i = 0, |i− j| = 1;
F 2
i Fj − (q + q−1)FiFjFi + FjF
2
i = 0, |i− j| = 1;
EiEj − EjEi = FiFj − FjFi = 0, |i− j| > 1.
The coproduct, the counit, and the antipode are defined as follows:
4Ej = Ej ⊗ 1 + Kj ⊗ Ej , ε(Ej) = 0, S(Ej) = −K−1
j Ej ,
4Fj = Fj ⊗K−1
j + 1⊗ Fj , ε(Fj) = 0, S(Fj) = −FjKj ,
4Kj = Kj ⊗Kj , ε(Kj) = 1, S(Kj) = K−1
j , j = 1, . . . , 2n− 1.
Equip the Hopf algebra Uqsl2n with the involution ∗:
(K±1
j )∗ = K±1
j , E∗
j =
{
KjFj , j 6= n,
−KjFj , j = n,
F ∗
j =
{
EjK
−1
j , j 6= n,
−EjK
−1
j , j = n.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 317
O. Bershtein and Ye. Kolisnyk
Then Uqsun,n
def= (Uqsl2n, ∗) is a ∗-Hopf algebra. It is a quantum analogue of
the algebra Usun,n ⊗R C, where sun,n stands for the Lie algebra of the noncom-
pact real Lie group SUn,n.
Let Uqs(gln× gln) ⊂ Uqsl2n denote the Hopf subalgebra generated by Ej , Fj ,
j 6= n, and Ki,K
−1
i , i = 1, . . . , 2n − 1. The corresponding ∗-Hopf subalgebra in
Uqsun,n is denoted by Uqs(un × un).
Recall an important definition of the weight module. A Uqsl2n-module V is
called a weight one if
V =
⊕
λ∈P
Vλ, Vλ =
{
v ∈ V
∣∣∣ Kiv = qλiv, i = 1, 2, . . . , 2n− 1
}
,
where λ = (λ1, λ2, . . . , λ2n−1) and P is the weight lattice of the Lie algebra sl2n.
Nonzero summand Vλ is called a weight subspace of weight λ.
Further, all Uqsl2n-modules are assumed to be the weight ones what allows
us to introduce the linear operators Hj , j = 1, . . . , 2n− 1, in V such that
Hjv = ηjv, v ∈ Vη.
Therefore, one can formally consider
K±1
i = q±Hi .
We recall a definition of the ∗-algebra Pol(Matn)q from [21]. First, let
C[Matn]q denote the well-known algebra with the generators zα
a , a, α = 1, . . . , n,
and the relations
zα
a zβ
b − qzβ
b zα
a = 0, a = b & α < β, or a < b & α = β,
(1)
zα
a zβ
b − zβ
b zα
a = 0, α < β & a > b, (2)
zα
a zβ
b − zβ
b zα
a − (q − q−1)zβ
a zα
b = 0, α < β & a < b. (3)
The algebra C[Matn]q is called the algebra of holomorphic polynomials on the
quantum n-matrices space (see [13]).
Similarly, let C[Matn]q denote the algebra with the generators (zα
a )∗, a, α =
1, . . . , n, and the relations
(zβ
b )∗(zα
a )∗ − q(zα
a )∗(zβ
b )∗ = 0, a = b & α < β, or a < b & α = β, (4)
(zβ
b )∗(zα
a )∗ − (zα
a )∗(zβ
b )∗ = 0, α < β & a > b, (5)
(zβ
b )∗(zα
a )∗ − (zα
a )∗(zβ
b )∗ − (q − q−1)(zα
b )∗(zβ
a )∗ = 0, α < β & a < b. (6)
318 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
Moreover, let C[Matn ⊕ Matn]q denote the algebra with the generators zα
a ,
(zα
a )∗, a, α = 1, . . . , n, relations (1)–(6), and additional relations
(zβ
b )∗zα
a = q2
n∑
a′,b′=1
n∑
α′,β′=1
R(b, a, b′, a′)R(β, α, β′, α′)zα′
a′
(
zβ′
b′
)∗
+ (1− q2)δabδ
αβ ,
where δab, δαβ are Kronecker symbols and
R(j, i, j′, i′) =
q−1, i 6= j & j = j′ & i = i′,
1, i = j = i′ = j′,
−(q−2 − 1), i = j & i′ = j′ & i′ > i,
0, otherwise.
Finally, let Pol(Matn)q
def= (C[Matn ⊕ Matn]q, ∗) be the ∗-algebra with the
natural involution: ∗ : zα
a 7→ (zα
a )∗. The algebra Pol(Matn)q is called the algebra
of polynomials on the quantum n-matrices space (see [13]).
We now recall an irreducible ∗-representation of Pol(Matn)q in a pre-Hilbert
space. Let H denote the Pol(Matn)q-module with one generator v0 and the
defining relations
(zα
a )∗v0 = 0, a, α = 1, . . . , n.
Let TF denote the representation of Pol(Matn)q which corresponds to H. It is
called the Fock representation. All statements of the following proposition are
proved in [21].
Proposition 1. 1. H = C[Matn]qv0.
2. H is a simple Pol(Matn)q-module.
3. There exists a unique sesquilinear form (·, ·) on H with the following pro-
perties:
i) (v0, v0) = 1; ii) (fv, w) = (v, f∗w) for all v, w ∈ H, f ∈ Pol(Matn)q.
4. The form (·, ·) is positive definite on H.
Also it is proved in [21] that Pol(Matn)q is a Uqsun,n-module algebra∗.
The action of the generators of Uqsun,n is given by the formulae
Hnzα
a =
2zα
a , a = n & α = n,
zα
a , a = n & α 6= n or a 6= n & α = n,
0, otherwise,
∗I.e., the multiplication in Pol(Matn)q is a morphism of Uqsun,n-modules, and the involutions
in Pol(Matn)q and Uqsun,n are compatible.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 319
O. Bershtein and Ye. Kolisnyk
Fnzα
a = q1/2 ·
{
1, a = n & α = n,
0, otherwise,
Enzα
a = −q1/2 ·
q−1zn
a zα
n , a 6= n & α 6= n,
(zn
n)2, a = n & α = n,
zn
nzα
a , otherwise,
for all a = 1, . . . , n; α = 1, . . . , n, and with k 6= n
Hkz
α
a =
zα
a , k < n & a = k or k > n & α = 2n− k,
−zα
a , k < n & a = k + 1 or k > n & α = 2n− k + 1,
0, otherwise,
Fkz
α
a = q1/2 ·
zα
a+1, k < n & a = k,
zα+1
a , k > n & α = 2n− k,
0, otherwise,
Ekz
α
a = q−1/2 ·
zα
a−1, k < n & a = k + 1,
zα−1
a , k > n & α = 2n− k + 1,
0, otherwise.
Let
Λn = {(λ1, λ2, . . . , λn) ∈ Zn
+ | λ1 ≥ λ2 ≥ . . . ≥ λn}
be the set of partitions of the length not larger than n. Similarly to the classical
case, one obtains the decomposition C[Matn]q =
⊕
λ∈Λn
C[Matn]q,λ into a sum
of Uqs(un× un)-isotypic components, where C[Matn]q,λ is a simple Uqs(un× un)-
module with highest weight
(λ1 − λ2, . . . , λn−1 − λn, 2λn, λn−1 − λn, . . . , λ1 − λ2).
This decomposition gives rise to the decomposition
H =
⊕
λ∈Λn
Hλ, Hλ = C[Matn]q,λv0.
Recall a quantum analogue of the Harish-Chandra embedding of the Her-
mitian symmetric space S(Un × Un)\SUn,n ↪→ Matn. Let C[SL2n]q denote the
well-known Hopf algebra with the generators {tij}i,j=1,...,2n and the relations
tαatβb − qtβbtαa = 0, a = b & α < β, or a < b & α = β,
tαatβb − tβbtαa = 0, α < β & a > b,
tαatβb − tβbtαa − (q − q−1)tβatαb = 0, α < β & a < b,
detq t = 1.
320 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
Here detq t is the q-determinant of the matrix t = (tij)i,j=1,...,2n defined by
detq t def=
∑
s∈S2n
(−q)l(s)t1 s(1)t2 s(2) . . . t2n s(2n)
with l(s) = card{(i, j)| i < j & s(i) > s(j)}. The comultiplication ∆, the counit
ε, and the antipode S are defined as follows:
∆(tij) =
∑
k
tik ⊗ tkj , ε(tij) = δij , S(tij) = (−q)i−j detq tji,
where tji is the matrix derived from t by discarding its j-th row and its i-th
column.
We equip C[SL2n]q with the standard Uqsl2n-module algebra structure as
follows (see [21]): for k = 1, . . . , 2n− 1,
Ek · tij = q−1/2
{
ti j−1, k = j − 1,
0, otherwise,
Fk · tij = q1/2
{
ti j+1, k = j,
0, otherwise,
(7)
Kk · tij =
qtij , k = j,
q−1tij , k = j − 1,
tij , otherwise.
(8)
Denote by Uqsl
op
2n the Hopf algebra obtained from Uqsl2n by changing the multi-
plication to the opposite one. We can also equip C[SL2n]q with a Uqsl
op
2n-module
algebra structure as follows: for k = 1, . . . , 2n− 1,
Ek · tij = q−1/2
{
ti+1 j , k = i,
0, otherwise,
Fk · tij = q1/2
{
ti−1 j , k = i + 1,
0, otherwise,
Kk · tij =
qtij , k = i,
q−1tij , k = i + 1,
tij , otherwise.
So, C[SL2n]q is a Uqsl
op
2n ⊗ Uqsl2n-module algebra (see [21]). The subalgebra
C[SL2n](Uqs(gln×gln))op⊗Uqs(gln×gln)
q = {f ∈ C[SL2n]q |
(ξ1 ⊗ ξ2)f = ε(ξ1)ε(ξ2)f, ξ1 ∈ Uqs(gln × gln)op, ξ2 ∈ Uqs(gln × gln)} (9)
will be referred as the subalgebra of Uqs(gln × gln)-biinvariants.
Equip C[SL2n]q with the involution given by
t∗ij = sign[(i− n− 1/2)(n− j + 1/2)](−q)j−i detq tij .
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 321
O. Bershtein and Ye. Kolisnyk
It can be proved that C[w0SUn,n]q
def= (C[SL2n]q, ∗) is a Uqsun,n-module ∗-algebra.
It is a q-analogue of the algebra of regular functions on the real affine algebraic
manifold w0SUn,n, where∗
w0 =
(
0 −J
J 0
)
, J =
0 0 ... 0 1
0 0 ... 1 0
...
0 1 ... 0 0
1 0 ... 0 0
.
For any multiindices I = {1 ≤ i1 < i2 < . . . < ik ≤ 2n} and J = {1 ≤ j1 <
j2 < . . . < jk ≤ 2n} we use the following standard notation for the corresponding
q-minor of the matrix t:
t∧k
IJ
def=
∑
s∈Sk
(−q)l(s)ti1js(1)
ti2js(2)
. . . tikjs(k)
.
We now introduce a short notation for the elements
t = t∧n
{1,2,...,n}{n+1,n+2,...,2n}, x = tt∗. (10)
Note that t, t∗, and x quasicommute with all generators tij of C[SL2n]q, and
that C[w0SUn,n]q is an integral domain (see [12]). Let C[w0SUn,n]q,x be the
localization of C[w0SUn,n]q with respect to the multiplicative set xZ+ (see [6]).
The following statements are proved in [21].
Proposition 2. There exists a unique extension of the Uqsun,n-module
∗-algebra structure from C[w0SUn,n]q to C[w0SUn,n]q,x.
Proposition 3. There exists a unique embedding of the Uqsun,n-module
∗-algebras
i : Pol(Matn)q ↪→ C[w0SUn,n]q,x
such that
i(zα
a ) = t−1t∧n
{1,2,...,n}Jaα
,
where Jaα = {n + 1, n + 2, . . . , 2n} \ {2n + 1− α} ∪ {a}.
The last proposition gives us a q-analogue of the Harish-Chandra embedding.
It allows us to identify Pol(Matn)q with its image in C[w0SUn,n]q,x.
∗The matrix w0 corresponds to the longest element of the Weyl group of the Lie algebra sl2n.
322 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
2.2. The algebra of finite functions and the invariant integral
It is well known that in the classical case q = 1 a positive definite
SUn,n-invariant integral can not be defined on the polynomial algebra in the unit
ball D ↪→ Matn. However, it is well defined on the space of smooth functions with
compact support on D. These observations are still applicable for the quantum
case. Here we provide the definition and some basic properties of a q-analogue of
the algebra of finite functions following [19].
Let us consider a Uqsun,n-module ∗-algebra Fun(D)q obtained from Pol(Matn)q
by adding a generator f0 and the relations
f0 = f2
0 = f∗0 ,
(zα
a )∗f0 = 0, f0z
α
a = 0, a, α = 1, 2, . . . , n.
The Uqsun,n-module algebra structure can be extended from Pol(Matn)q
to Fun(D)q as follows:
Hnf0 = 0, Fnf0 = − q1/2
q−2 − 1
f0(zn
n)∗, Enf0 = − q1/2
1− q2
zn
nf0,
Hkf0 = Fkf0 = Ekf0 = 0, k 6= n.
The two-sided ideal D(D)q = Pol(Matn)qf0Pol(Matn)q is a Uqsun,n-module
∗-subalgebra (see [19]). The elements of the two-sided ideal D(D)q will be called
finite functions on the quantum matrix ball D.
The Fock representation TF of Pol(Matn)q can be extended up to the repre-
sentation of Fun(D)q, and so for every finite function f ∈ D(D)q there exists an
operator TF (f), and
TF (D(D)q) = {A ∈ End(H) | A|Hλ
6= 0 for a finite set of indices λ ∈ Λn}.
Consider the gradings
C[Matn]q,k =
⊕
|λ|=k
C[Matn]q,λ, k ∈ Z+,
and
C[Matn]q,−k =
⊕
|λ|=k
C[Matn]q,λ, k ∈ Z+,
where |λ| = λ1 + λ2 + . . . + λn.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 323
O. Bershtein and Ye. Kolisnyk
It is evident that
Lemma 1. The Fock representation TF has a unique extension to a represen-
tation of the ∗-algebra Fun(D)q such that the element f0 maps to the orthogonal
projection onto the vacuum subspace.
Let us keep the same notation TF for this extension.
Proposition 4. The representation TF provides the isomorphism of the
∗-algebra D(D)q and the ∗-algebra of all finite∗ linear operators in H.
P r o o f. TF is a ∗-representation. So, we have to prove that the restriction
of TF on D(D)q is a bijective mapping from D(D)q to the algebra of all finite
linear operators in H.
Let D(D)q,i,j = C[Matn]q,i · f0 · C[Matn]q,−j . If f ∈ D(D)q,i,j , then the linear
operator TF (f) mapsHj toHi and it is equal to zero on
⊕
k 6=j
Hk. We obtain a linear
mapping from D(D)q,i,j to Hom(Hj ,Hi). It is surjective by Proposition 1, and
dimD(D)q,i,j = dim Hom(Hj ,Hi).
Thus the representation TF provides the isomorphism
D(D)q,i,j = C[Matn]q,i f0 C[Matn]q,−j
∼= Hom(Hj ,Hi).
But D(D)q =
∞⊕
i,j=0
D(D)q,i,j , and
∞⊕
i,j=0
Hom(Hj ,Hi) in EndH is the vector space
of finite linear operators.
Proposition 5. The representation TF provides the bijection of the space of
Uqs(gln × gln)-invariants in D(D)q and the space of finite linear operators in H
that are scalars on every Uqs(gln × gln)-isotypic component Hλ, λ ∈ Λn.
P r o o f.
i) If f is a Uqs(gln × gln)-invariant vector, then TF (f) maps a highest vector
of Hλ to a highest vector of a Uqs(gln × gln)-isotypic component with the same
weight.
ii) The action of Uqs(gln × gln) in H is multiplicity free.
iii) Now i) and ii) imply that if f is a Uqs(gln × gln)-invariant vector, then
TF (f)|Hλ
is an endomorphism of the simple Uqs(gln× gln)-module Hλ. So TF (f)
is scalar on Hλ, λ ∈ Λn.
∗A linear operator A in H is called finite if AHj = 0 for all j ∈ Z+ except a finite set.
324 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
Denote the space of Uqs(gln × gln)-invariants in D(D)q by
(D(D)q)Uqs(gln×gln) = {f ∈ D(D)q | ξf = ε(ξ)f, ξ ∈ Uqs(gln × gln)}.
Denote
ρ̌ =
1
2
2n−1∑
j=1
j(2n− j)Hj .
The following proposition is also stated in [19].
Proposition 6. The linear functional
∫
fdν = (1− q2)n2
tr(TF (f)q−2ρ̌), f ∈ D(D)q, (11)
is a positive definite Uqsl2n-invariant integral on D(D)q, i.e.,
∫
ξfdν = ε(ξ)
∫
fdν, ξ ∈ Uqsl2n
and ∫
f∗fdν > 0, for f 6= 0.
For the sketch of the proof refer to [23, §5].
Further we consider a restriction of the invariant integral (11) to the space of
Uqs(gln × gln)-invariants in D(D)q. We will call this restriction the radial part.
2.3. The radial part of the invariant integral
In this subsection we will describe the support of radial part of the invariant
measure dν and find an exact formula for the radial part of the invariant integral.
Consider the elements of C[w0SUn,n]q:
xk = qk(k−1)
∑
I⊂{1,2,...,n}, J⊂{n+1,n+2,...,2n}
card(I)=card(J)=k
q
−2
k∑
m=1
(n−im)
(−q)
k∑
m=1
(jm−im−n)
t∧k
I J t
∧(2n−k)
Ic Jc .
It follows from the results of [4] that xk, k = 1, 2, . . . , n are pairwise commuting
self-adjoint Uqs(un × un)-biinvariants. These elements generate the subalgebra of
all Uqs(un × un)-biinvariant elements in C[w0SUn,n], as follows from the results
of [2] and [4]. So,
C[w0SUn,n](Uqs(un×un))op⊗Uqs(un×un)
q
∼= C[x1, x2, . . . , xn].
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 325
O. Bershtein and Ye. Kolisnyk
Denote by T the ∗-representation of the ∗-algebra C[w0SUn,n]q corresponding
to the permutation
(
1 2 . . . n-1 n n+1 n+2 . . . 2n-1 2n
n+1 n+2 . . . 2n-1 2n 1 2 . . . n-1 n
)
(see [21, §4]). This representation admits a unique extension to the representation
of C[w0SUn,n]q,x, where x is defined in (10). It is proved in [21, §4, 5] that the
representation TF is unitary equivalent to the restriction of the representation T
to Pol(Matn)q. Consider the short notation
qµ = (qµ1 , qµ2 , . . . , qµn) ∈ Cn, µ ∈ Cn.
It is also proved in [4] that
T (xk)|Hλ
= qk(k−1)ek(q−2(λ+δ)), k = 1, 2, . . . , n, λ ∈ Λn,
where δ = (n−1, n−2, . . . , 1, 0) ∈ Λn, ek is the elementary symmetric polynomial
in n variables of degree k. So, the set of common eigenvalues of the operators
T (x1), T (x2), . . . , T (xn) is
ΣD = {(e1(q−2(λ+δ)), q2e2(q−2(λ+δ)), . . . , qn(n−1)en(q−2(λ+δ))) | λ ∈ Λn}.
Thus the algebra C[w0SUn,n](Uqs(un×un))op⊗Uqs(un×un)
q can be identified with
the algebra of polynomial functions on ΣD. Following Propositions 4, 5 the
algebra D(D)Uqs(gln×gln)
q can be identified with the algebra D(ΣD) of functions
f(x1, x2, . . . , xn) with finite support on ΣD.
Lemma 2. The mapping
Λn → ΣD,
λ = (λ1, λ2, . . . , λn) 7→ (e1(q−2(λ+δ)), q2e2(q−2(λ+δ)), . . . , qn(n−1)en(q−2(λ+δ)))
is a bijection.
P r o o f. The surjectivity follows from the definition of ΣD. Let us prove the
injectivity. The function q−l is strictly increasing as l ∈ [0, +∞), so the mapping
Λn → Rn, λ 7→ q−2(λ+δ)
is an injection. Due to the Viet theorem, the mapping
q−2(λ+δ) 7→ (e1(q−2(λ+δ)), q2e2(q−2(λ+δ)), . . . , qn(n−1)en(q−2(λ+δ)))
is also an injection since q−2(λ1+n−1) > q−2(λ2+n−2) > . . . > q−2λn for any λ ∈ Λn.
Now we have the injectivity of the composition.
326 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
Consider the algebra C[u1, u2, . . . , un] and the injection
C[x1, x2, . . . , xn] ↪→ C[u1, u2, . . . , un], xk 7→ qk(k−1)ek(u1, . . . , un), (12)
where ek are the elementary symmetric polynomials in n variables. This injection
allows one to identify the subalgebra C[w0SUn,n](Uqs(un×un))op⊗Uqs(un×un)
q with the
algebra of all symmetric polynomials in variables u1, u2, . . . , un.
Specify
∆D = {q−2(λ+δ) | λ ∈ Λn}.
Let also D(∆D) be the algebra of functions f(u1, u2, . . . , un) with finite support
on the set ∆D. Then
D(∆D) ∼= D(ΣD).
More exactly, the bijection is as follows:
D(ΣD) → D(∆D) : f(x1, x2, . . . , xn) 7→ f(e1(u), q2e2(u), . . . , qn(n−1)en(u)).
Thereby,
D(∆D) ∼= D(ΣD) ∼= D(D)Uqs(gln×gln)
q . (13)
In the sequel we do not distinguish between D(∆D) and D(D)Uqs(gln×gln)
q .
Recall the definition of a multiple Jackson integral with ’base’ q−2 (see [27]):
∞∫
q−2(n−1)
q2un∫
q−2(n−2)
. . .
q2u2∫
1
φ(u)dq−2u1 . . . dq−2un
def= (1−q2)n
∑
λ∈Λn
φ(q−2(λ+δ))q−2|λ+δ|.
(14)
Proposition 7. The restriction of the invariant integral (11) to the
space D(D)Uqs(gln×gln)
q is
∫
f(x1, x2, . . . , xn) dν
=N
∞∫
q−2(n−1)
q2un∫
q−2(n−2)
. . .
q2u2∫
1
f(e1(u), q2e2(u), . . . , qn(n−1)en(u))
×∆(u)2 dq−2u1dq−2u2 . . . dq−2un,
where ∆(u) =
∏
1≤i<j≤n
(ui−uj), N = (1− q2)n(n−1)qn(n−1)∆(q−2δ)−2 is a positive
constant.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 327
O. Bershtein and Ye. Kolisnyk
The constant N can be found easily by calculating the integral for the ele-
ment f0: ∫
f0 dν = (1− q2)n2
= N (1− q2)n∆(q−2δ)2q−2|δ|.
So,
N = (1− q2)n(n−1)qn(n−1)∆(q−2δ)−2. (15)
P r o o f. Consider the integral
η̃ : f 7→
∞∫
q−2(n−1)
q2un∫
q−2(n−2)
. . .
q2u2∫
1
f(e1(u), q2e2(u), . . . , qn(n−1)en(u))
×∆(u)2 dq−2u1dq−2u2 . . . dq−2un.
Let us show that the integrals η and η̃ are equal up to a multiplicative constant
on the space D(D)Uqs(gln×gln)
q (the normalizing constant is calculated in (15)).
Let us compute η̃(f):
η̃(f) = const
∞∫
q−2(n−1)
. . .
q2u2∫
1
f(e1(u), q2e2(u), . . . , qn(n−1)en(u))∆(u)2 dq−2u1 . . . dq−2un
= const
∑
λ∈Λn
f(e1(q−2(λ+δ)), q2e2(q−2(λ+δ)), . . . , qn(n−1)en(q−2(λ+δ)))
×∆(q−2(λ+δ))2 q−2|λ+δ|.
Let us also compute η(f):
η(f) = const tr(TF (f)q−2ρ̌) = const
∑
λ∈Λn
tr ( TF (f)|Hλ
q−2ρ̌|Hλ
)
= const
∑
λ∈Λn
dλ f(e1(q−2(λ+δ)), q2e2(q−2(λ+δ)), . . . , qn(n−1)en(q−2(λ+δ))),
where dλ = tr ( q−2ρ̌|Hλ
). In the last computation we essentially use the fact
that the operators TF (f), f ∈ D(D)Uqs(gln×gln)
q are scalar on each Hλ.
Introduce the notation
H0 = nHn +
n−1∑
j=1
jHj +
n−1∑
j=1
jH2n−j ,
328 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
then
−2ρ̌ = −nH0 −
n−1∑
j=1
j(n− j)Hj −
n−1∑
j=1
j(n− j)H2n−j .
Consider the subalgebra in Uqs(gln × gln) generated by {Ej , Fj ,K
±1
j }j 6=n. It
is isomorphic to Uqsln⊗Uqsln. The restriction of the representation of Uqs(gln×
gln) in Hλ to the subalgebra Uqsln ⊗ Uqsln is equivalent to the representation
π £ π, where π is the irreducible representation of Uqsln with highest weight
(λ1 − λ2, λ2 − λ3, . . . , λn−1 − λn). Consequently (see [13, §7.1.4]),
dλ = tr ( q−nH0 |Hλ
) ( tr(π(q−2ρ̌(n)
)|H(n)
λ
) )2 = q−2|λ|Sλ(q−2δ)2,
where ρ̌(n) =
n−1∑
j=1
j(n− j)Hj , and
Sλ(z1, z2, . . . , zn) =
det(zλj+j−1
i )i,j=1,2,...,n
det(zj−1
i )i,j=1,2,...,n
is the Schur polynomial [17, §1.3]. So Sλ(q−2δ) = ∆(q−2(λ+δ))
∆(q−2δ)
, and
η(f) = const
∑
λ∈Λn
dλ f(e1(q−2(λ+δ)), q2e2(q−2(λ+δ)), . . . , qn(n−1)en(q−2(λ+δ)))
= const
∑
λ∈Λn
q−2|λ| ∆(q−2(λ+δ))2
×f(e1(q−2(λ+δ)), q2e2(q−2(λ+δ)), . . . , qn(n−1)en(q−2(λ+δ))).
Now it is obvious that the integrals η and η̃ are equal up to a multiplier.
3. Spherical Functions on Quantum Grassmanian
Consider the involution ? in Uqsl2n determined by
(K±1
j )? = K±1
j , E?
j = KjFj , F ?
j = EjK
−1
j .
Then Uqsu2n = (Uqsl2n, ?) is a ∗-Hopf algebra. It is a quantum analogue of
Usu2n ⊗R C.
Consider also the involution ? in C[SL2n]q determined by
t?ij = (−q)j−it∧2n−1
{1,2,...,2n}\{i},{1,2,...,2n}\{j}.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 329
O. Bershtein and Ye. Kolisnyk
The ∗-Hopf algebra C[SU2n]q
def= (C[SL2n]q, ?) is a Uqsu2n-module ∗-Hopf algebra.
It is a well-known quantum analogue of the algebra of regular functions on the
Lie group SU2n (see [31, 32]).
It is well known that in the classical case the Cartan duality between compact
and noncompact Hermitian symmetric spaces allows one to predict some results
of harmonic analysis in the noncompact case using the easier compact case. In
this subsection we explore this observation. We construct a family of difference
operators for the quantum Grassmanians. These operators are obtained using
the action of the center of Uqsl2n. Afterwards, our construction allows us to
introduce difference operators in the case of quantum matrix ball.
3.1. Spherical functions
It is well known that for any finite-dimensional irreducible Uqsl2n-module V
dimV Uqs(gln×gln) ≤ 1.
Hence (Uqsl2n, Uqs(gln × gln)) is a ”quantum Gelfand pair”. As in the classi-
cal case, let us define a simple finite-dimensional weight Uqsl2n-module to be
spherical, if dimV Uqs(gln×gln) = 1.
Remark 1. It is well known ([25, Th. 4.4.1]; [26]) that a simple finite-
dimensional weight Uqsl2n-module is Uqs(gln × gln)-spherical if and only if its
highest weight has the following form:
λ̂ = (λ1−λ2, λ2−λ3, . . . , λn−1−λn, 2λn, λn−1−λn, . . . , λ2−λ3, λ1−λ2), λ ∈ Λn.
We will denote by Lλ the Uqsl2n-module with highest weight λ̂.
A scalar product∗ (·, ·) in V is called Uqsu2n-invariant if for any ξ ∈ Uqsl2n
and for any v1, v2 ∈ V
(ξv1, v2) = (v1, ξ
?v2).
Any spherical Uqsl2n-module V can be equipped with a Uqsu2n-invariant scalar
product. Fix v ∈ V Uqs(gln×gln) by the requirement (v, v) = 1. Recall
(see [13, §11.6.4]) that the matrix element ϕV (ξ) = (ξv, v) corresponding to the
Uqs(gln × gln)-invariant vector is called the spherical function on the quantum
group SU2n corresponding to V .
Thus ϕV is a Uqs(gln × gln)-biinvariant element of C[SU2n]q such that
ϕV (1) = 1.
The lemma below follows from the results of [14].
∗A sesquilinear positive definite Hermitian symmetric form.
330 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
Lemma 3. (ϕV )? = ϕV .
It follows from Proposition 7 of [4] and Lemma 1 of [2] that the subalgebra
of Uqs(gln × gln)-biinvariant functions in C[SU2n]q is generated by the pairwise
commuting elements x1, x2, . . . , xn. In particular, every spherical function ϕV is a
polynomial in x1, x2, . . . , xn. Denote by ϕλ(x1, x2, . . . , xn) the spherical function
corresponding to the module Lλ. In this section we will find an exact formula
for ϕλ(x1, x2, . . . , xn).
3.1.1. Little q-Jacobi polynomials
We will use the following partial order on Λn
η ≤ λ
def⇐⇒
k∑
j=1
ηj ≤
k∑
j=1
λj , k = 1, 2, . . . , n.
As usual, η < λ
def⇐⇒ η ≤ λ & η 6= λ.
Introduce the short notation 1k = (1, . . . , 1︸ ︷︷ ︸
k
, 0, . . . , 0). Let us denote by mλ
the monic symmetric polynomial
mλ(z1, z2, . . . , zn) =
∑
w∈Sn
zλ1
w(1)z
λ2
w(2) . . . zλn
w(n).
Let Pλ be a unique symmetric polynomial which satisfies the following two
conditions:
1) Pλ(z) = mλ(z) +
∑
η<λ
dλ,ηmη(z), dλ,η ∈ R,
2)
∫ q2
0 . . .
∫ q2z2
0 Pλ(z)mη(z)∆(z)2dq2z1 . . . dq2zn = 0, η < λ,
where the multiple Jackson integral (cf. (14)) is defined as
q2∫
0
. . .
q2z2∫
0
φ(z)dq2z1 . . . dq2zn = (1− q2)n
∑
λ∈Λn
φ(q2(λ+δ+1n))q2|λ+δ+1n|.
Remark 2. It is easy to see that
Pλ(z) = Pλ(z; 0, 0; q2),
where Pλ(z; a, b; q) are Little q-Jacobi polynomials (see [27]).
Let P̃λ be a polynomial such that
Pλ(z) = P̃λ(e1(z), q2e2(z), . . . , qn(n−1)en(z)).
From the results of Subsection 2.3 and [25, Th. 4.7.5], [26], one can deduce the
following theorem:
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 331
O. Bershtein and Ye. Kolisnyk
Theorem 1. The spherical function ϕλ is equal (up to a multiplicative con-
stant) to
P̃λ(x1, x2, . . . , xn).
Denote the fundamental spherical weights by
µk = 1̂k, k ∈ {1, 2, . . . , n},
and denote by
P spher
+ =
n⊕
k=1
Z+µk = {λ̂ | λ ∈ Λn}
the set of positive spherical weights, and by
P spher =
n⊕
k=1
Zµk = {λ̂ | λ ∈ Zn} (16)
the set of all spherical weights.
Stokman proved the following formula in [27, Prop. 5.9]:
Pλ(z1, z2, . . . , zn) = ∆(z)−1
∑
w∈Sn
sign(w)
n∏
i=1
P(λ+δ)w(i)
(zi),
where Pm(z) are Little q-Jacobi polynomials in one variable.
Recall the ’coordinates’ u1, u2, . . . , un appeared in (12).
Corollary 1. Let λ ∈ Λn. Then
ϕλ(u) = const Pλ(u) = const∆(u)−1
∑
w∈Sn
sign(w)
n∏
i=1
Pd(λ,w,i)(ui),
where d(λ, w, i) = (λ + δ)w(i) ∈ Z.
3.2. Difference operators and the action of the center of U ext
q sl2n
Let aij be the Cartan matrix of the Lie algebra sl2n. Denote by αi, i =
1, 2, . . . , 2n−1, the simple roots such that αi(Hj) = aji and by Φ the root system
of the Lie algebra sl2n.
In this subsection we will consider the action of the center of Uqsl2n in weight
modules. Note that it is more convenient to use the center of the extended quan-
tum universal enveloping algebra U ext
q sl2n. Essentially, U ext
q sl2n can be obtained
from Uqsl2n by adding the elements
Kλ = Ka1
1 Ka2
2 . . . K
a2n−1
2n−1 , λ =
2n−1∑
i=1
aiαi,
332 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
for all λ in the weight lattice P . In particular, the action of Uqsl2n in any
weight module admits a unique extension to the action of U ext
q sl2n. Denote by
Z(U ext
q sl2n) the center of the extended universal enveloping algebra.
Recall some definitions, cf. [5]. Consider the real linear span h∗R of all simple
roots of the Lie algebra sl2n. It is well known that there is a positive definite
scalar product (·, ·) in h∗R. Denote by (h∗R)− ⊂ h∗R the real subspace spanned by
the strictly orthogonal noncompact positive roots
γk = αk + αk+1 + . . . + α2n−k−1 + α2n−k, k ∈ {1, 2, . . . , n},
and by (h∗R)+ ⊂ h∗R its orthogonal complement. It is known that the orthogonal
projection of the root system Φ to (h∗R)− is a root system of type Cn and it
is called the system of restricted roots Φres. The Weyl group W res of the root
system Φres is called the restricted Weyl group.
Let C[P spher]q be an algebra generated by the following functions on P spher:
λ 7→ q(η,λ), η ∈ P spher.
This algebra is naturally isomorphic to the group algebra of the lattice P spher.
Denote by C[P spher]W
res
q the subalgebra of W res-invariants in C[P spher]q:
C[P spher]W
res
q = {f ∈ C[P spher]q | f(wλ) = f(λ) for allw ∈ W res, λ ∈ P spher}.
Here we provide a well-known description of the image of the center Z(U ext
q sl2n)
under the Harish-Chandra homomorphism γspher : Z(U ext
q sl2n) → C[P spher]q (see
[1]).
Proposition 8. The image of Z(U ext
q sl2n) under the Harish-Chandra homo-
morphism is the subalgebra C[P spher]W
res
q .
Set for λ ∈ Cn
a(λ + δ) def= (a(λ1 + n− 1), a(λ2 + n− 2), . . . , a(λn)),
where
a(l) =
(1− q−2l)(1− q2l+2)
(1− q2)2
, l ∈ C. (17)
Proposition 9. There are the elements Ck ∈ Z(U ext
q sl2n), k = 1, 2, . . . , n,
such that
Ckϕλ = ek(a(λ + δ))ϕλ, λ ∈ Λn. (18)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 333
O. Bershtein and Ye. Kolisnyk
P r o o f. Consider the mapping
Λn → Rn, λ 7→ η(λ) = (λ1 − 2n− 1
2
, λ2 − 2n− 3
2
, . . . , λn−1 − 3
2
, λn − 1
2
).
Then
η̂(λ) = λ̂− ρ,
where
λ̂ = (λ1 − λ2, λ2 − λ3, . . . , λn−1 − λn, 2λn, λn−1 − λn, . . . , λ2 − λ3, λ1 − λ2) ∈ P.
We need the following functions on P spher:
ψk : λ̂ 7→ ek(a(η(λ) + δ)), k ∈ {1, 2, . . . , n}. (19)
λ ∈ Zn is uniquely defined by the spherical weight λ̂ ∈ P spher, see (16).
Due to Proposition 8 we only need to check the W res-invariance of the func-
tions ψk.
It is easy to see that
ek(a(η(λ) + δ)) = (1− q2)−2k
×ek((1−q−2λ1+1)(1−q2λ1+1), (1−q−2λ2+1)(1−q2λ2+1), . . . , (1−q−2λn+1)(1−q2λn+1)).
Besides,
λ̂ = λ1γ1 + λ2γ2 + . . . + λnγn.
As the group W res acts on γk by permutations and sign changes, the function
(19) is W res-invariant.
Let Lk be the linear operator in C[SL2n]q defined by Lkf = Ckf .
The action of U ext
q sl2n in the space of Uqs(gln × gln)-biinvariant functions
determines the homomorphism
Z(U ext
q sl2n) → End(C[u1, u2, . . . , un]Sn),
as
C[u1, u2, . . . , un]Sn ∼= C[x1, x2, . . . , xn] ∼= C[SL2n](Uqs(gln×gln))op⊗Uqs(gln×gln)
q ,
(20)
(see Subsect. 2.3). Here we will describe the action of the linear operators L1,
L2, . . . ,Ln in the space (20).
Let us define the difference operator 2ui in the space C[u1, u2, . . . , un] with
2uif(u1, . . . , un) = Duiui(1− q−1ui)Duif(u1, . . . , un), (21)
where Duif(u1, . . . , un) = f(u1,...,ui−1,q−1ui,ui+1,...,un)−f(u1,...,ui−1,qui,ui+1,...,un)
q−1ui−qui
.
334 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
Proposition 10.
Lk|C[u1,u2,...,un]Sn =
1
∆(u)
ek(2u1 , . . . ,2un) ∆(u). (22)
P r o o f. In Subsection 4.1.3 it will be shown that in the case of one variable
2u ϕl(u) = a(l)ϕl(u).
From (17) and the determinant decomposition described in Corollary 1 it
follows that
1
∆(u)
ek(2u1 , . . . , 2un) ∆(u) ϕλ(u) = ek(a(λ + δ))ϕλ(u), λ ∈ Λn.
Equality (22) follows from Proposition 9, as the set {ϕλ}λ∈Λn is a basis of the
vector space C[SL2n](Uqs(gln×gln))op⊗Uqs(gln×gln)
q .
4. Plancherel Measure for the Quantum Matrix Ball
4.1. The Plancherel measure for a family of the operators Lradial
1 ,
Lradial
2 , . . ., Lradial
n
4.1.1. Linear operators Lradial
1 ,Lradial
2 , . . . ,Lradial
n in the space L2(∆D, dνq)
Let us consider the elements
C1, C2, . . . , Cn ∈ Z(U ext
q sl2n)
defined in (18). Let also Lk be the linear operator in D(D)q defined by
Lkf = Ckf.
Now we describe the restriction of the linear operator Lk, k = 1, 2, . . . , n to
the space D(D)Uqs(gln×gln)
q of Uqs(gln × gln)-invariants in D(D)q.
Let us introduce the short notation Lradial
k for the restriction of Lk to
D(D)Uqs(gln×gln)
q .
Proposition 11.
Lradial
k =
1
∆(u)
ek(2u1 , . . . ,2un) ∆(u), (23)
where 2uj are the difference operators in the vector space (13) defined by the
same formula as in (21).
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 335
O. Bershtein and Ye. Kolisnyk
P r o o f. Following Subsection 2.3, the vector space of Uqs(gln × gln)-
invariants in D(D)q can be identified with the space D(ΣD) of functions on ΣD
with finite support. Using Lemma 2 one can obtain that the vector space of
Uqs(gln×gln)-invariants in D(D)q is canonically isomorphic to the space D(∆D) of
functions on ∆D with finite support. Consider the pointwise convergence topology
on ∆D.
The space of symmetric polynomials in ∆D is dense in the topological space
F(∆D) of functions on ∆D, and equation (23) takes place for symmetric polyno-
mials following (22) and (20).
The linear operators in both parts of equation (23) can be extended contin-
uously from the space of symmetric polynomials in ∆D to F(∆D), and equation
(23) takes place for the whole space F(∆D).
Now we recall the measure on ∆D:
dνq(u) = N ∆(u)2dq−2u1dq−2u2 . . . dq−2un, (24)
where N is defined in (15). It is the restriction of the invariant measure to the
space D(D)Uqs(gln×gln)
q , which we already identified with D(∆D) (see Prop. 7).
Let us introduce the Hilbert space L2(∆D, dνq) of functions on the set ∆D
which satisfy ∫
∆D
|f(u)|2dνq(u) < ∞,
where
(f, g) =
∫
∆D
g(u)f(u)dνq(u).
It will be proved in the sequel (Lemma 7) that the linear operators Lradial
1 ,
Lradial
2 , . . . ,Lradial
n can be continuously extended to bounded pairwise commuting
selfadjoint operators in L2(∆D, dνq).
Our goal is to find a Plancherel measure dΣ on the joint spectrum of commut-
ing selfadjoint linear operators Lradial
1 , Lradial
2 , . . . ,Lradial
n and a unitary operator
F : L2(∆D, dνq) → L2(dΣ) which provides a unitary equivalence between the
operators Lradial
1 , Lradial
2 , . . . ,Lradial
n and the operators of multiplication by inde-
pendent variable, such as Ff0 = 1.
The element f0 ∈ L2(∆D, dνq) is a cyclic vector under the action of Lradial
1 ,
Lradial
2 , . . . ,Lradial
n (one can prove it explicitly, see Subsect. 4.1.2). However, it
follows from the isometry of the operator F and Remarks 8 and 9.
The considered problems are typical for the theory of commutative operator
∗-algebras with a cyclic vector [18, p. 570, 571], [28, p. 103].
336 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
4.1.2. The cyclic vector f0
Here we discuss the fact that the element f0 ∈ L2(∆D, dνq) is a cyclic vector
under the action of the operators Lradial
1 , Lradial
2 , . . . ,Lradial
n .
By direct computation we obtain the following lemma.
Lemma 4. In the case of quantum disk
2uf0(q2ku) = ck,−1f0(q2k−2u) + ck,0f0(q2ku) + ck,1f0(q2k+2u), k ∈ Z+, (25)
where ck,−1, ck,0, ck,1 are nonzero constants.
For example,
¤uf0(u) = Duu(1− q−1u)Duf0(u) =
f0(u)
1− q2
− q2f0(q2u)
1− q2
.
Here f0(q−2u) = 0 for u ∈ q−2Z+ , so the first term in (25) vanishes.
The lemma below follows from the previous one by induction.
Lemma 5.
Lradial
i f0(q2(λ+δ)u) =
∑
cdf0(q2(λ+δ+d)u),
where d ∈ {−1, 0, 1}n, card{j|dj 6= 0} ≤ i and cd 6= 0.
Lemma 6. The linear span of the action of Lradial
1 , Lradial
2 , . . . ,Lradial
n on f0
contains the set of finite functions on D(∆D).
Sketch of the proof. Lemma 5 implies that the linear span of the action of
Lradial
1 , Lradial
2 , . . . ,Lradial
n on f0 contains the set
SD = {f0(q2(λ+δ)u) | λ ∈ Λn}
of characteristic functions of points of ∆D.
The last lemma implies that f0 is cyclic as the set of finite functions D(∆D)
is dense in L2(∆D, dνq).
4.1.3. Example: the quantum disk
In this subsection we recall the Plancherel measure dσ for the quantum disk
found in [20].
Consider the Hilbert space L2(q−2Z+) of functions on the geometric series
q−2Z+ which satisfy the condition
∞∫
1
|f(u)|2dq−2u < ∞
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 337
O. Bershtein and Ye. Kolisnyk
with the scalar product
(f, g) =
∞∫
1
g(u)f(u)dq−2u.
Recall the notation for the difference operator 2u which acts in the space of
functions on geometric series q−2Z+ by
2uf(u) = Duu(1− q−1u)Duf(u),
where
Duf(u) 7→ f(q−1u)− f(qu)
q−1u− qu
.
Then Lradial
1 = 2u.
Let us describe the eigenfunctions of the difference operator 2u. Introduce
the notation
Φl(x) = 3Φ2
(
q−2l, q2(l+1), x
q2, 0
; q2, q2
)
, l ∈ C,
for the basic hypergeometric function (see [8]).
Proposition 12. ([20, §8]).
2uΦl(u) = a(l)Φl(u),
where a(l) is defined in (17):
a(l) =
(1− q−2l)(1− q2l+2)
(1− q2)2
.
Remark 3. Φl(1) = 1.
Remark 4. Φl(u) is equal up to a multiplicative constant to ϕl(u).
Let
c(l) =
Γq2(2l + 1)
(Γq2(l + 1))2
be a q-analogue of the Harish-Chandra c-function. Here Γq2(x) =
(q2,q2)∞
(q2x,q2)∞ (1− q2)1−x is a well-known q-analogue of the Gamma function Γ(x).
Let us consider the measure
dσ(ρ) =
1
2π
· h
1− q2
· dρ
c(−1
2 + iρ)c(−1
2 − iρ)
on the interval [0, π/h], where h = −2 ln q.
338 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
Consider the operator
F : f 7→ f̂(ρ) =
∞∫
1
Φ− 1
2
+iρ(u)f(x)dq−2u
defined in the space of finite functions on the geometric series q−2Z+ . It is shown
in [20, Th. 9.2] that this operator can be extended to a unitary operator
F : L2(q−2Z+) → L2([0, π/h], dσ) such as
F 2u f = a(−1
2
+ iρ)Ff, f ∈ L2([0, π/h], dσ),
where a(l) is defined in (17). The inverse operator F−1 has the form
f̂(ρ) 7→
π/h∫
0
f̂(ρ)Φ− 1
2
+iρ(u)dσ(ρ).
4.1.4. The quantum matrix ball
We will call the eigenfunction of a difference operator a generalized one if it
does not belong to L2. These functions are used in the sequel for the construction
of the operator F .
Consider the isometric linear operator∗
I : L2(∆D, dνq) → L2(q−2Zn
+),
I : f(u)7→∆(u)f̃(u), (26)
where f̃ is defined in the following way: for every u = (u1, . . . , un) with ui 6= uj for
i 6= j there exists a unique permutation w ∈ Sn such as uw1 > uw2 > . . . > uwn .
Then
f̃(u) =
{
1√
n!
f(uw1 , . . . , uwn), ui 6= uj , i 6= j,
0, otherwise,
u1, u2, . . . , un ∈ q−2Z+ .
∗L2(q−2Zn
+)q is a short notation for L2(q−2Z+ × . . .× q−2Z+
︸ ︷︷ ︸
n
)q with the product measure mul-
tiplicated by N .
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 339
O. Bershtein and Ye. Kolisnyk
Consider the notation
L̃k = ek(2u1 ,2u2 , . . . , 2un), k = 1, 2, . . . , n, (27)
for the difference operators in L2(q−2Zn
+)q. Then the following diagram is com-
mutative:
L2(∆D, dνq)
Lradial
k
²²
I // L2(q−2Zn
+)
L̃k
²²
L2(∆D, dνq)
I // L2(q−2Zn
+).
Lemma 7. The operators Lradial
1 ,Lradial
2 , . . . ,Lradial
n in the Hilbert space
L2(∆D, dνq) are bounded selfadjoint and pairwise commuting.
P r o o f. The explicit formulas of Subsection 4.1.3 imply that the operators
2ui are bounded for all 1 ≤ i ≤ n (unlike in the classical case), so the same holds
for L̃1, L̃2, . . . , L̃n. Moreover, it is easy to see that the operators 2ui and 2uj
commute for 1 ≤ i < j ≤ n (as they act in different variables), so the operators
L̃i, L̃j commute for 1 ≤ i < j ≤ n, too.
Also, the operators 2ui , i = 1, 2, . . . , n are symmetric, so they are bounded
selfadjoint operators in L2(q−2Zn
+)q. Thus, L̃i, i = 1, 2, . . . , n are pairwise com-
muting bounded selfadjoint linear operators in L2(q−2Zn
+)q. As the mapping I
is isometric, the operators Lradial
1 ,Lradial
2 , . . . ,Lradial
n are also bounded selfadjoint
and pairwise commuting.
Using Proposition 12, one can easily show that the functions Φl1(u1)Φl2(u2) . . .
Φln(un) on q−2Zn
+ are common generalized eigenfunctions of the operators (27).
We will need the common eigenfunctions which are in the image of the operator I.
It is easy to see that
φ̃l1,l2,...,ln(u1, u2, . . . , un) =
∑
σ∈Sn
sign(σ) Φl1(uσ1)Φl2(uσ2) . . .Φln(uσn) ∈ ImI
are common generalized eigenfunctions. Let
R = {(ρ1, ρ2, . . . , ρn) ∈ [0, π/h]n, ρ1 > ρ2 . . . > ρn}.
Lemma 8. The pairwise commuting bounded selfadjoint operators L̃k,
k = 1, 2, . . . , n are unitary equivalent to the operators of multiplication by
ek(a(−1
2
+ iρ1), a(−1
2
+ iρ2), . . . , a(−1
2
+ iρn)), k = 1, 2, . . . , n,
340 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
(respectively) in the Hilbert space L2(R, (n!)N (dσ)n|R). The unitary equivalence
is provided by the mapping
Ũ : Im I → L2(R, (n!)N (dσ)n|R),
Ũ : f(u1, u2, . . . , un) 7→ f̂(ρ1, ρ2, . . . , ρn)
= N
∞∫
1
. . .
∞∫
1
φ̃− 1
2
+iρ1,− 1
2
+iρ2,...,− 1
2
+iρn
(u)f(u)d
q−2u1 . . . d
q−2un.
The inverse operator is
Ũ−1 : f̂(ρ1, ρ2, . . . , ρn) 7→
N
∫
. . .
∫
︸ ︷︷ ︸
R
f̂(ρ1, ρ2, . . . , ρn)φ̃− 1
2
+iρ1,− 1
2
+iρ2,...,− 1
2
+iρn
(u) (n!)dσ(ρ1) . . . dσ(ρn).
P r o o f. This lemma follows from the results of subsection 4.1.3 and the
explicit formulas for the operators L̃1, . . . , L̃n.
Remark 5. The last equalities define Ũ on a dense linear manifold of the
functions with finite support on the set q−2Zn
+.
Let us introduce the notation
Φl1,l2,...,ln(u) =
∑
σ∈Sn
sign(σ) Φl1(uσ1)Φl2(uσ2) . . .Φln(uσn)
∆(u)
. (28)
Remark 6. (See Corollary 1 and Remark 4). The spherical function ϕλ(u),
λ ∈ Λn is equal up to a multiplicative constant to Φl1,l2,...,ln(u), where
li = (λ + δ)i ∈ Z.
Using this lemma and the definition (26) of the operator I, one can easily
obtain the following lemma.
Lemma 9. The pairwise commuting bounded selfadjoint operators Lradial
k ,
k = 1, 2, . . . , n, are unitary equivalent to the operators of multiplication by
ek(a(−1
2
+ iρ1), a(−1
2
+ iρ2), . . . , a(−1
2
+ iρn)), k = 1, 2, . . . , n,
(respectively) in the Hilbert space L2(R, (n!)N (dσ)n|R). The unitary equivalence
is provided by the mapping
U : L2(∆D, dνq) → L2(R, (n!)N (dσ)n|R),
U : f(u) 7→ f̂(ρ1, ρ2, . . . , ρn) =
∫
∆D
Φ− 1
2
+iρ1,− 1
2
+iρ2,...,− 1
2
+iρn
(u)f(u)dνq(u),
where the measure dνq(u) is defined in (24).
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 341
O. Bershtein and Ye. Kolisnyk
The inverse operator is
U−1 : f̂(ρ1, ρ2, . . . , ρn)
7→
∫
R
f̂(ρ1, ρ2, . . . , ρn)Φ− 1
2
+iρ1,− 1
2
+iρ2,...,− 1
2
+iρn
(u) (n!)N dσ(ρ1) . . . dσ(ρn).
Remark 7. The last equalities define U on a dense linear manifold of the
functions with finite support on the set ∆D.
Lemma 10.
Uf0 = N∆(q−2δ)−1
n−1∏
j=0
(q−2j ; q2)j
(q2; q2)2j
q(j+1)2−1
∏
1≤k<j≤n
(q−2iρj +q2iρj−q−2iρk−q2iρk),
(29)
where the constant N is defined in (15).
P r o o f.
(Uf0)(ρ1, ρ2, . . . , ρn) = NΦ− 1
2
+iρ1,− 1
2
+iρ2,...,− 1
2
+iρn
(1, q−2, . . . , q−2(n−1))
= N∆(q−2δ)−1
∑
σ∈Sn
sign(σ) Φ− 1
2
+iρ1
(1)Φ− 1
2
+iρ2
(q−2) . . .Φ− 1
2
+iρn
(q−2(n−1)).
(30)
It can be verified that the last expression is a polynomial in the variables
qiρ1 + q−iρ1 , . . . , qiρn + q−iρn . It is antisymmetric, so
∏
1≤k<j≤n
(q−2iρj + q2iρj − q−2iρk − q2iρk) (31)
is a factor of (30). One can compare the degrees of the polynomials in
the right-hand side of (30) and (31) as the elements of the graded algebra
C[qiρ1 + q−iρ1 , qiρ2 + q−iρ2 , . . . , qiρn + q−iρn ]. The degree of the polynomial (31)
is n(n−1)
2 . Since
Φ− 1
2
+iρ(q
−2k) = 3Φ2
(
q1+iρ, q1−iρ, q−2k
q2, 0
; q2, q2
)
=
k∑
j=0
(q1+iρ; q2)j (q1−iρ; q2)j (q−2k; q2)j q2j
(q2; q2)2j
,
then the degree of Uf0 is n(n−1)
2 , and it proves (29) up to a constant. This
constant can be found by comparing the highest monomial coefficients in the
lexicographic order.
342 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
Denote
κ(ρ1, ρ2, . . . , ρn)
= N∆(q−2δ)−1
n−1∏
j=0
(q−2j ; q2)j
(q2; q2)2j
q(j+1)2−1
∏
1≤k<j≤n
(q−2iρj +q2iρj−q−2iρk−q2iρk).
(32)
Notice that the function κ(ρ1, ρ2, . . . , ρn) is positive on R. Consider the ope-
rator
F =
1
κ(ρ1, ρ2, . . . , ρn)
U
and the measure
dΣ(ρ1, ρ2, . . . , ρn) = κ(ρ1, ρ2, . . . , ρn)2(n!)N (dσ(ρ1) . . . dσ(ρn))|R (33)
on the set R (the constant N is defined in (15)).
The following proposition is the consequence of Lemmas 9 and 10.
Proposition 13. The pairwise commuting bounded selfadjoint operators
Lradial
k , k = 1, 2, . . . , n, are unitary equivalent to the operators of multiplication
by
ek(a(−1
2 + iρ1), a(−1
2 + iρ2), . . . , a(−1
2 + iρn))
κ(ρ1, ρ2, . . . , ρn)
, k = 1, 2, . . . , n,
(respectively) in the Hilbert space L2(R, dΣ). The unitary equivalence is provided
by the mapping
F : L2(∆D, dνq) → L2(R, dΣ),
F : f(u) 7→ f̂(ρ1, ρ2, . . . , ρn)
=
1
κ(ρ1, ρ2, . . . , ρn)
∫
∆D
Φ− 1
2
+iρ1,− 1
2
+iρ2,...,− 1
2
+iρn
(u)f(u)dνq(u), (34)
where Φl1,l2,...,ln(u) are defined in (28), and the measure dνq(u) is defined in (24).
The inverse mapping is
F−1 : f̂(ρ1, ρ2, . . . , ρn)
7→
∫
R
f̂(ρ1, ρ2, . . . , ρn)Φ− 1
2
+iρ1,− 1
2
+iρ2,...,− 1
2
+iρn
(u) dΣ(ρ1, ρ2, . . . , ρn).
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 343
O. Bershtein and Ye. Kolisnyk
Remark 8. The cyclic vector f0 ∈ L2(∆D, dνq) is mapped into 1 ∈ L2(R, dΣ)
by F .
Remark 9. For the convenience we use the variables ρ1, ρ2, . . . , ρn in the
image of F . Notice that if we change the variables
zk =
ek(a(−1
2 + iρ1), a(−1
2 + iρ2), . . . , a(−1
2 + iρn))
κ(ρ1, ρ2, . . . , ρn)
,
we get that F maps Lradial
k into the operator of multiplication by the independent
variable zk.
The measure dΣ on R defined in (33) is a sought-for radial part of the
Plancherel measure.
References
[1] W. Baldoni and P.M. Frajria, The Harish-Chandra Homomorphism for a Quantized
Classical Hermitean Symmetric Pair. — Ann. Inst. Fourier 49 (1999), 1179–1214.
[2] O. Bershtein, On a q-Analog of a Sahi Result. — J. Math. Phys. 48 (2007), No. 4,
1917–1924.
[3] O. Bershtein, Degenerate Principal Series of Quantum Harish-Chandra Modules.
— J. Math. Phys. 45 (2004), No. 10, 3800–3827.
[4] O. Bershtein, Y. Kolisnyk, and L. Vaksman, On a q-Analog of the Wallach-
Okounkov Formula. — Lett. Math. Phys. 78 (2006), 97–109.
[5] N. Bourbaki, Lie Groups and Lie Algebras. Chap. 4–6. Springer, Berlin, Heidelberg,
New York, 2004.
[6] K.A. Brown and K.R. Goodearl, Lectures on Algebraic Quantum Groups.
Birkhäuser, Basel, Boston, Berlin, 2002.
[7] J.F. Van Diejen, Commuting Difference Operators with Polynomial Eigenfunctions.
— Comp. Math. 95 (1995), No. 2, 183–233.
[8] G. Gasper and M. Rahman, Basic Hypergeometric Series. Cambridge Univ. Press,
Cambridge, 2004.
[9] S. Helgason, Differential Geometry and Symmetric Spaces. Mir, Moscow, 1964.
(Russian)
[10] S. Helgason, Groups and Geometryc Analysis. Mir, Moscow, 1987. (Russian)
[11] B. Hoogenboom, Spherical Functions and Invariant Differential Operators on Com-
plex Grassmann Manifolds. — Arkiv Mat. 20 (1982), No. 1, 2, 69–85.
344 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Plancherel Measure for the Quantum Matrix Ball-1
[12] A. Joseph, Quantum Groups and Their Primitive Ideals. Springer–Verlag, Berlin,
Heidelberg, 1995.
[13] A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations.
Springer, Berlin, 1997.
[14] L.I. Korogodski and Y.S. Soibelman, Algebra of Functions on Quantum Groups.
Part I. Amer. Math. Soc., Providence, RI, 1998.
[15] L.I. Korogodski and L.L. Vaksman, Spherical Functions on Quantum Group
SU(1, 1) and q-Analogue of the Meller–Fock Formula. — Funct. Anal. and Appl.
25 (1991), No. 1, 60–62. (Russian)
[16] L.I. Korogodski and L.L. Vaksman, On Harmonic Analysis on Quantum Group
SU(1, 1). In: 14th School on Operator Theory in Functional Spaces. N. Novgorod,
1989. (Russian)
[17] I.G. Macdonald, Symmetric Functions and Hall Polynomials. Clarendon Press,
Oxford, 1995.
[18] M.A. Nainmark, Normalized Rings. Nauka, Moscow, 1968. (Russian)
[19] D. Shklyarov, S. Sinel’shchikov, and L. Vaksman, q-Analogues of Some Bounded
Symmetric Domains. — Czehoslovak J. Phys. 50 (2000), No. 1, 175–180.
[20] D.L. Shklyarov and L.L. Vaksman, Intergral Representations of Functions in the
Quantum Disk. I. — Mat. Fiz., Anal., Geom. 4 (1997), 286–308. (Russian)
[21] D. Shklyarov, S. Sinel’shchikov, and L. Vaksman, Fock Representations and Quan-
tum Matrices. — Intern. J. Math. 15 (2004), No. 9, 1–40.
[22] D. Shklyarov, S. Sinel’shchikov, and L. Vaksman, Hidden Symmetry of Some Alge-
bras of q-Differential Operators. In: Noncommutative Structures in Mathematics
and Physics. (S. Duplij and J. Wess, Eds.) NATO Sci. Ser., Kluwer, Dordrecht,
Boston, London, 2000.
[23] D. Shklyarov, S. Sinel’shchikov, and L. Vaksman, On Function Theory in the Quan-
tum Matrix Ball: an Invariant Integral. Preprint Math-QA/9803110, 1998.
[24] S. Sinel’shchikov and L. Vaksman, On q-Analogues of Bounded Symmetric Domains
and Dolbeault Complexes. — Math. Phys., Anal., Geom. 1 (1998), No. 1, 75–100.
[25] J.V. Stokman, Multivariable Orthogonal Polynomials and Quantum Grassmanians.
Universal Press, Veenendaal, 1998.
[26] J.V. Stokman and M.S. Dijkhuizen, Some Limit Transitions between BC Type
Orthogonal Polynomials Interpreted on Quantum Complex Grassmanians. — Publ.
Res. Inst. Math. Sci. 35 (1999), 451–500.
[27] J.V. Stokman, Multivariable Big and Little q-Jacobi Polynomials. — SIAMM J.
Math. Anal. 28 (1997), 452–480.
[28] M. Takesaki, Theory of Operator Algebras I. Springer, Berlin, New York, 1979.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 345
O. Bershtein and Ye. Kolisnyk
[29] L.L. Vaksman, Quantum Matrix Ball: the Cauchy-Szegö Kernel and the Shilov
Boundary. — Mat. Fiz., Anal., Geom. 8 (2001), 366–384. (Russian)
[30] L.L. Vaksman, Maximum Principle for ’Holomorfic Functions’ in Quantum Ball. —
Mat. Fiz., Anal., Geom. 10 (2003), 12–28. (Russian)
[31] S.L. Woronowicz, Compact Matrix Pseudogroups. — Commun. Math. Phys. 111
(1987), No. 4, 613–665.
[32] S.L. Woronowicz, A Remark on Compact Matrix Quantum Groups. — Lett. Math.
Phys. 21 (1991), 35–39.
346 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
|