Spherical Principal Series of Quantum Harish-Chandra Modules
The nondegenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmetric domains.
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Bershtein, O. Stolin, A. Vaksman, L. 2016-09-28T19:00:00Z 2016-09-28T19:00:00Z 2007 Spherical Principal Series of Quantum Harish-Chandra Modules / O. Bershtein, A. Stolin, L. Vaksman // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 157-175. — Бібліогр.: 22 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106443 The nondegenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmetric domains. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Spherical Principal Series of Quantum Harish-Chandra Modules Article published earlier |
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The nondegenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmetric domains.
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Spherical Principal Series of Quantum Harish-Chandra Modules / O. Bershtein, A. Stolin, L. Vaksman // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 157-175. — Бібліогр.: 22 назв. — англ. |
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Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 2, pp. 157�175
Spherical Principal Series of Quantum Harish-Chandra
Modules
O. Bershtein
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
A. Stolin
Chalmers University of Technology, G�oteborg, Sweden
E-mail:astolin@math.chalmers.se
L. Vaksman
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:vaksman@ilt.kharkov.ua
Received September 28, 2006
The nondegenerate spherical principal series of quantum Harish�Chandra
modules is constructed. These modules appear in the theory of quantum
bounded symmetric domains.
Key words: quantum groups, spherical principal series.
Mathematics Subject Classi�cation 2000: 17B37, 22E45.
1. Introduction
In [3] the unit disc D = fz 2 C jjzj < 1g is considered as the Poincare model of
hyperbolic plane. The Plancherel formula for D is one of the most profound results
of noncommutative harmonic analysis, and q-analogs of this result are known for
q 2 (0; 1) being the deformation parameter (see, e.g., [19]). It is important to
note that the representations of spherical principal series are the crucial tools in
decomposing quasiregular representation both in a classical and a quantum cases.
The unit disc is the simplest bounded symmetric domain. It is known that
there is a Plancherel formula for any bounded symmetric domain in the classical
setting. On the other hand, it is absent in the quantum case. One of the obstacles
here is that there are some di�culties in producing the nondegenerate spherical
c
O. Bershtein, A. Stolin, and L. Vaksman, 2007
O. Bershtein, A. Stolin, and L. Vaksman
principal series of Harish�Chandra modules over a quantum universal enveloping
algebra. In this paper we overcome these di�culties. A geometrical approach
to the representation theory is used instead of the traditional construction of
principal series, which is inapplicable in the quantum case. Hence, we generalize
the results of [17].
The Casselman theorem claims that any simple Harish�Chandra module can
be embedded in a module of the nondegenerate spherical principal series. Thus,
one has another class of applications for the constructions made in this paper,
beyond the harmonic analysis.
2. A Quantum Analog of the Open K-Orbit in BnG
Let (aij)i;j=1;:::;l be a Cartan matrix of positive type, g the corresponding
simple complex Lie algebra. So the Lie algebra can be de�ned by the generators
ei, fi, hi, i = 1; : : : ; l, and the well-known relations (see [4]). Let h be the
linear span of hi, i = 1; : : : ; l. The simple roots f�i 2 h
�ji = 1; : : : ; lg are
given by �i(hj) = aji. Also, let f$iji = 1; : : : ; lg be the fundamental weights,
hence P =
Ll
i=1 Z$i = f� = (�1; : : : ; �l)j�j 2 Zg is the weight lattice and
P+ =
Ll
i=1 Z+$i = f� = (�1; : : : ; �l)j�j 2 Z+g is the set of integral dominant
weights.
Fix l0 2 f1; : : : ; lg, together with the Lie subalgebra k � g generated by
ei; fi; i 6= l0; hi; i = 1; : : : ; l:
De�ne h0 2 h by
�i(h0) = 0; i 6= l0; �l0(h0) = 2:
We restrict ourselves by Lie algebras g that can be equipped with a Z-grading as
follows:
g = g�1 � g0 � g+1; gj = f� 2 gj [h0; �] = 2j�g: (1)
Let Æ be the maximal root, and Æ =
Pl
i=1 ci�i. (1) holds if and only if cl0 = 1.
In this case g0 = k and the pair (g; k) is called a Hermitian symmetric pair.
Fix a Hermitian symmetric pair (g; k). Let G be a complex algebraic a�ne
group with Lie (G) = g and K � G the connected subgroup with Lie (K) = k.
Consider the Lie subalgebra b � g generated by ei, hi, i = 1; : : : ; l, together with
the corresponding connected subgroup B � G. The homogeneous space G=B has
a unique open K-orbit
� X. It is an a�ne algebraic variety which is a crucial
tool in producing nondegenerate principal series of Harish�Chandra modules (see
[16]).
We introduce a q-analog of the algebra of regular functions on the open orbit.
Recall some background and introduce the notations. First of all, recall some
notions from the quantum group theory [4]. In the sequel the ground �eld is C ,
q 2 (0; 1), and all the algebras are associative and unital.
158 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Spherical Principal Series of Quantum Harish-Chandra Modules
Denote by di, i = 1; : : : ; l, such positive coprime integers that the matrix
(diaij)i;j=1;:::;l is symmetric. Recall that the quantum universal enveloping algebra
Uqg is a Hopf algebra de�ned by the generators Ki, K
�1
i , Ei, Fi, i = 1; : : : ; l, and
the relations
KiKj = KjKi; KiK
�1
i = K
�1
i Ki = 1;
KiEj = q
aij
i EjKi; KiFj = q
�aij
i FjKi;
EiFj � FjEi = Æij
Ki �K
�1
i
qi � q
�1
i
;
1�aijX
m=0
(�1)m
�
1� aij
m
�
qi
E
1�aij�m
i EjE
m
i = 0;
1�aijX
m=0
(�1)m
�
1� aij
m
�
qi
F
1�aij�m
i FjF
m
i = 0;
where qi = q
di , 1 � i � l, and
h
m
n
i
q
=
[m]q!
[n]q![m� n]q!
; [n]q! = [n]q � : : : � [1]q; [n]q =
q
n � q
�n
q � q�1
:
The comultiplication �, the counit ", and the antipode S are de�ned as follows:
�(Ei) = Ei
1 +Ki
Ei; �(Fi) = Fi
K
�1
i + 1
Fi; �(Ki) = Ki
Ki;
S(Ei) = �K�1
i Ei; S(Fi) = �FiKi; S(Ki) = K
�1
i ;
"(Ei) = "(Fi) = 0; "(Ki) = 1:
A representation � : Uqg ! EndV is called weight (and V is called a weight
module, respectively), if V admits a decomposition into the sum of weight sub-
spaces
V =
M
�
V�; V� = fv 2 V j �(K�1
j )v = q
��j
j v; j = 1; : : : ; lg:
The subspace V� is called a weight subspace of weight �.
It is convenient to de�ne the linear operators Hi in a weight module V by
Hiv = jv i� Kiv = q
j
i v; v 2 V:
Let Uqk � Uqg be a Hopf subalgebra generated by Ei, Fi, i = 1; : : : ; l,
i 6= l0, and K
�1
j , j = 1; : : : ; l. A �nitely generated weight Uqg-module V is called
a quantum Harish�Chandra module if V is a sum of �nite dimensional simple
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 159
O. Bershtein, A. Stolin, and L. Vaksman
Uqk-modules and dimHomUqk(W;V) < 1 for every �nite dimensional simple
Uqk-module W .
We restrict our consideration to quantum Harish�Chandra modules only.
Let � 2 P+. L(�) denotes the simple Uqg-module with a single generator v(�)
and the de�ning relations (see [4])
K
�
j v(�) = q
��jv(�); Ejv(�) = 0; F
�j+1
j v(�) = 0:
Recall the notion of quantum analog of the algebra C [G] of regular functions on
G [4]. Denote by C [G]q � (Uqg)
� the Hopf subalgebra of all matrix coe�cients
of weight �nite dimensional Uqg-representations. Denote by U
op
q g a Hopf algebra
that di�ers from Uqg by the opposite multiplication. Equip C [G]q with a structure
of U
op
q g
Uqg-module algebra in the following way: (�
�)f=(Lreg(�)
Rreg(�))f ,
where
(Rreg(�)f)(�) = f(��); (Lreg(�)f)(�) = f(��); �; � 2 Uqg; f 2 C [G]q :
The algebra C [G]q is called the algebra of regular functions on the quantum
group G.
Introduce special notation for some elements of C [G]q [8]. Consider the �nite
dimensional simple Uqg-module L(�) with highest weight � 2 P+. Equip it with
an invariant scalar product (�; �), given by (v(�); v(�)) = 1 (as usual in the compact
quantum group theory [8]). Choose an orthonormal basis of the weight vectors
fv�;jg 2 L(�)� for all weights �. Let
c
�
�;i;�0;j(�) = (�v�0;j; v�;i):
We will omit the indices i; j, if they do not lead to an ambiguity. Introduce
an auxiliary Uqg-module algebra C [ bX ]q.
Let � 2 P+. For all �
0
; �
00 2 P+
dimHomUqg(L(�
0 + �
00); L(�0)
L(�00)) = 1:
Hence, the following Uqg-morphisms are well-de�ned:
m�0;�00 : L(�
0)
L(�00)! L(�0 + �
00); m�0;�00 : v(�
0)
v(�00) 7! v(�0 + �
00):
Therefore, the vector space
C [ bX ]q
def
=
M
�2P+
L(�)
is equipped with a Uqg-module algebra structure as follows:
f
0 � f 00
def
= m�0;�00(f
0
f
00); f
0 2 L(�0); f 00 2 L(�00):
160 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Spherical Principal Series of Quantum Harish-Chandra Modules
This is a well-known quantum analog of the homogeneous coordinate ring of the
�ag manifold X = BnG.
The Peter�Weyl theorem claims that
C [G]q =
M
�2P+
(L(�)
L(�)�):
Hence, there exists an embedding of Uqg-module algebras
i : C [ bX ]q ,! C [G]q ; i : v(�) 7! c
�
�;�; � 2 P+;
where c��;� are the matrix coe�cients of the representations introduced above, so
we have
Proposition 1. C [ bX ]q is an integral domain.
C [ bX ]q is a P -graded algebra by obvious reasons:
C [ bX ]q =
M
�2P+
C [ bX ]q;�; C [ bX ]q;� = L(�):
A simple weight �nite dimensional Uqg-module L(�) is called spherical, if it
has a nonzero Uqk-invariant vector [3]. This Uqk-invariant vector is unique up to
a nonzero constant, dimL(�)Uqk = 1. Moreover, the set �+ � P+ of all highest
weights of the spherical Uqg-modules L(�) coincides with the similar set in the
classical case, so �+ =
Lr
i=1 Z+�i, with �1; �2; : : : ; �r being the fundamental
spherical weights, and r being the real rank of the bounded symmetric domain D
(see [3]). Choose nonzero elements
i 2 L(�i)
Uqk; i = 1; 2; : : : ; r:
Proposition 2. 1; 2; : : : ; r 2 C [ bX ]q pairwise commute.
P r o o f. i j = const(i; j) j i; with const(i; j) 6= 0, since dim(L(�i +
�j)
Uqk) = 1. Prove that const(i; j) = 1. It follows from Appendix, Lemma 5 that
we can equip C [ bX ]q with an involution ?, and without loss of generality we can
assume j =
?
j for all j = 1; : : : ; r. Therefore, i j = � j i. The involution ?
is a morphism of a continuous vector bundle E� over (0; 1] with the �bers L(�)q
de�ned in Appendix. In the classical case i j = j i, so the same holds in the
quantum case.
Consider a multiplicative subset
= f j1
1
j2
2
� � � jrr j j1; j2; : : : ; jr 2 Z+g
of the algebra C [ bX ]q.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 161
O. Bershtein, A. Stolin, and L. Vaksman
Proposition 3. � C [ bX ]q is an Ore set.
P r o o f. Consider the decomposition of the Uqg-module L(�) into a sum of its
Uqk-isotypic components L(�) =
L
�
L(�)�. Fix a subspace L(�)� and a nonzero
Uqk-invariant vector 2 L(�
0). For all j 2 N,
L(�+ j�
0)� � L(�+ (j � 1)�0)� :
It follows from Prop. 1 that L(�+ j�0)� = L(�+(j�1)�0)� for all large enough
j 2 N, since
dimHomUqk(L(k; �); L(�)) � dimL(k; �) (2)
for any simple weight �nite dimensional Uqk-module L(k; �) and any simple weight
�nite dimensional Uqg-module L(�). The inequality (2) is known in the classical
case [7, p. 206], and the quantum case follows from the classical case. Therefore,
j
L(�)� � L(�+ (j � 1)�0)�
for all large enough j 2 N, since j L(�)� � L(� + j�
0)�. Similarly, L(�)�
j �
L(�+ (j � 1)�0)� for all large enough j 2 N. Hence, for all f 2 C [ bX ]q, 2
f \ C [ bX ]q 6= ?; f \ C [ bX ]q 6= ?;
which is just the Ore condition for .
Consider the localization C [ bX ]q; of the algebra C [ bX ]q with respect to the
multiplicative set .
The P -grading can be extended to C [ bX ]q; , since the elements of are
homogeneous. The subalgebra
C [
]q = ff 2 C [ bX ]q; j deg f = 0g
is a quantum analog of the algebra C [
] of regular functions on the open K-orbit
� X = BnG.
3. Uqg-Module Algebra C [cX ]q;
In this section we equip C [ bX ]q; with a Uqg-module algebra structure. Start
with some auxiliary facts.
Consider the vector space L of sequences fxngn2Z+
xn =
X
i;j2Z+
aij�
n
i n
j
; (3)
162 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Spherical Principal Series of Quantum Harish-Chandra Modules
where the number of nonzero terms is �nite, aij 2 C [ bX ]q; , �i are nonzero and
pairwise di�erent. Sequences x0n and x00n are called asymptotically equal, x0n =
as
x
00
n,
if there exists N 2 Z+ such that for all n � N one has x0n = x
00
n.
Lemma 1. Let fxngn2Z+ 2 L and xn =
as
0. Then xn = 0 for all n 2 Z+.
P r o o f. Let xn =
as
0 and xn 6� 0. Without loss of generality, one assumes
that x0 6= 0, since L is invariant under T , with
T : fu0; u1; u2; : : :g 7! fu1; u2; : : :g; uj 2 C [ bX ]q; :
Let M be the smallest T -invariant subspace containing all nonzero terms
faij�
n
i n
jgn2Z+ from (3). Then dimM < 1 and 0 62 spec(T jM). However,
(T jM)N (xn) = 0 for all large enough N , since xn =
as
0. So, spec(T jM) = f0g.
This contradiction completes the proof.
Lemma 2. Let k 2 Z+ and �0 2 C . The sequence�
d
k
d�k
(1 + �+ �
2 + � � �+ �
n)j�=�0
�
n2Z+
(4)
belongs to L.
P r o o f. Let �0 = 1. (4) is a sequence of values of polynomials at n 2 Z+,
since
n�1X
j=0
j
k�1 =
Bk(n)�Bk(0)
k
;
with Bk(z) being the Bernoulli polynomials [14, Ch. 3]. For �0 6= 1,
1 + �+ �
2 + � � � + �
n�1 =
�
n � 1
�� 1
;
and the statement is now obvious.
Let 0 =
rQ
j=1
j : Note that for any f 2 C [ bX ]q; , � 2 Uqg, the element �(f n
0
)
is already de�ned and belongs to C [ bX ]q for large enough n 2 Z+.
Proposition 4. For any � 2 Uqg there exists a unique linear operator R� in
C [ bX ]q; such that for any f 2 C [ bX ]q; :
1. R�(f
n
0
) =
as
�(f n
0
);
2. the sequence xn = R�(f
n
0
) � �n
0
belongs to L.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 163
O. Bershtein, A. Stolin, and L. Vaksman
P r o o f. The uniqueness follows from Lem. 1. Prove the existence.
Consider a set of all � 2 Uqg such that there exists R� with the required
properties. The set forms a subalgebra. Indeed,
�(f n
0
) =
as
R�(f
n
0
) =
0
@ X
i;j2Z+
aij�
n
i n
j
1
A
n
0
;
�(aij
n
0
) =
as
R�(aij
n
0
) =
0
@ X
k;l2Z+
bijkl�
n
kn
l
1
A
n
0
;
so
��(f n
0
) =
as
0
@ X
i;j;k;l2Z+
bijkl(�i�k)
n
n
(j+l)
1
A
n
0
;
and one can put R��(f) =
P
i;k bi0k0. From Lemma 1, R��(f) is a linear operator
in C [ bX ]q; . The sequence R��(f
n
0
) has all required properties.
Now we have to construct the linear operators
RK�1
i
; REi
; RKiFi ; i = 1; 2; : : : ; l;
that satisfy the conditions of Prop. 4. The case � = K
�1
i is trivial, while the two
others are very similar.
Prove the existence of REi
. Let N 2 Z+ be the smallest number such that
f
N
0
2 C [ bX ]q. For any n 2 Z+
Ei
�
f
n+N
0
�
�(n+N)
0
= Ei
�
f
N
0
�
�N
0
+
�
Ki
�
f
N
0
�
�N
0
��
N
0
Ei (
n
0
)
�(n+N)
0
�
:
It is enough to prove that xn =
N
0
� Ei(
n
0
) �
�(n+N)
0
looks like (3), since then
one can put
Ei(f
n
0
) =
as
0
@ X
i;j2Z+
aij�
n
i n
j
1
A
n
0
;
and REi
(f) = a00. The linearity of REi
and all required properties reduce to the
special case f 2 C [ bX ]q by Lem. 1, as before.
Turn back to the proof. Let A be the linear operator in C [ bX ]q; , de�ned by
Af = 0f
�1
0
; f 2 C [ bX ]q; :
164 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Spherical Principal Series of Quantum Harish-Chandra Modules
The key observation is that the linear span of fAk(Ei 0gk2Z+ is �nite dimen-
sional. Indeed, all vectors Ak(Ei 0) belong to the same homogeneous component
of the P -graded vector space C [ bX ]q; , and, moreover, to the same its Uqk-isotypic
component that corresponds to the highest weight � =
rP
j=1
j�j + �i. But the di-
mension of intersection E of these components is at most than dimL(k; �)2 by (2).
Now one can use the equality
xn =
N
0
0
@n�1X
j=0
j
0
(Ei 0)
n�1�j
0
1
A
�(n+N)
0
=
N
0
0
@n�1X
j=0
(AjE )
j (Ei 0)
1
A
�(N+1)
0
and Lem. 2 to compute the function 1+z+z2+� � �+zn�1 of the �nite dimensional
linear operator AjE reduced to the Jordan canonical form [13].
Corollary 1. R� provides a Uqg-module structure in C [ bX ]q; which extends
the Uqg-module structure from C [ bX ]q to C [ bX ]q; .
Now prove that C [ bX ]q; is a Uqg-module algebra. Let eL be the vector space
of functions on Z2 that takes values in C [ bX ]q; , such that
x(m;n) =
X
i0;i00;j0;j002Z+
ai0i00j0j00�
n
i0 n
j0
�
m
i00 m
j00
;
where the sum is �nite, ai0i00j0j00 2 C [ bX ]q; , and �i0 ; �i00 2 C . Suppose that all
�i0 , �i00 are pairwise di�erent and nonzero. One can easily expand Lem. 1 to the
functions x(m;n) 2 eL. Namely, if x(m;n) = 0 for any m � M , n � N , then
x(m;n) = 0 for all m;n 2 Z.
It is important to note that either in the statement or in the proof of Prop. 4
one can replace the conditions on R� by the following conditions:
1. R�(
m
0
f
n
0
) = �( m
0
f
n
0
), if m;n 2 Z+ and m+ n is large enough;
2. the function x(m;n) =
�m
0
R�(
m
0
f
n
0
) �n
0
belongs to eL.
One gets the same representation R� of Uqg in the vector space C [ bX ]q; .
Proposition 5. Let � 2 Uqg and 4� =
P
i
�
0
i
�
00
i . Then for any f1; f2 2
C [ bX ]q; one has
�(f1f2) =
X
i
(�0if1)(�
00
i f2):
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 165
O. Bershtein, A. Stolin, and L. Vaksman
P r o o f. Let
x
0(m;n) =
�m
0
�( m
0
f1f2
n
0
) �n
0
;
and
x
00(m;n) =
X
i
�m
0
�
0
i(
m
0
f1)�
00
i (f2
n
0
) �n
0
:
They are equal for large enough m;n, since m
0
f1; f2
n
0
2 C [ bX ]q, and C [ bX ]q
is a Uqg-module algebra. Also, both belong to eL. Therefore, x0(m;n) = x
00(m;n)
for any m;n 2 Z: Now put m = n = 0 in the last equality.
R e m a r k 1. It should be noted that V. Lunts and A. Rosenberg described
another approach to the extension of the Uqg-module algebra structure in [11, 10].
Their approach is more general but more intricate.
4. Degenerate and Nondegenerate Spherical Principal Series
For simplicity and more clear presentation, start with the certain degenerate
spherical principal series. The same approach is used in producing nondegenerate
spherical principal series.
Fix k 2 f1; 2; : : : ; rg. Consider
C [ bXk ]q
def
=
M
j2Z+
L(j�k):
As in the previous section, one equips C [ bXk ]q with a Uqg-module algebra structure.
C [ bXk ]q naturally embeds in the Uqg-module algebra C [ bX ]q and has a Z+-grading:
deg f = j; i� f 2 L(j�k):
It follows from Prop. 3 that k =
Z+
k is an Ore set, and the localization C [ bXk ]q; k
is a Z-graded Uqg-module algebra. Consider the subalgebra
C [
k ]q = ff 2 C [ bXk ]q; k
j deg(f) = 0g:
Evidentially, C [
k ]q � C [
]q is a Uqg-module algebra.
For u 2 Z denote
�k;u(�)f
def
= � (f uk ) �
�u
k ; � 2 Uqg; f 2 C [
k ]q: (5)
The representations �k;u are the representations of degenerate spherical principal
series. Now we are going to introduce �k;u for arbitrary u 2 C . We need some
auxiliary constructions.
166 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Spherical Principal Series of Quantum Harish-Chandra Modules
Let
C [
k ]q =
M
�2PS+
C [
k ]q;�
be the decomposition of C [
k ]q into a sum of its Uqk-isotypic components.
P
S
+
denotes the set of all integral dominant weights of k. By considerations from
Appendix, C [
k ]q;� are �bers of a continuous vector bundle F� over (0; 1] that is
analytic on (0; 1). We identify the morphisms of such vector bundles over (0; 1]
with the corresponding continuous in (0; 1] and analytic in (0; 1) "operator valued
functions".
It is easy to prove that the operator valued function
Ak;�(q) : C [
k ]q;� ! C [
k ]q;�; Ak;�(q) : f 7! kf
�1
k
is well-de�ned, invertible, continuous in (0; 1] and analytic in (0; 1).
Lemma 3. All eigenvalues of Ak;�(q) are positive, and, moreover, rational
powers of q.
P r o o f. Let a be an eigenvalue of Ak;�(q), thus there exists a nonzero
f 2 L(j�k) � C [ bXk ]q, such that kf = a f k.
It is easy to show that aq�j(�k;�k) is an eigenvalue of the linear operator
RL(j�k)L(�k) corresponding to the universal R-matrix of Uqg. Here (�; �) is �xed
by (�i; �j) = diaij .
It remains to prove that the eigenvalues of RL(�);L(�0) are rational powers of
q for all �; �0 2 P+. There exists a suitable basis of tensor products of weight
vectors such that the matrix of RL(�);L(�0) is upper-triangular and its diagonal
elements belong to the set fq�(�
0;�00) j �0; �00 2 Pg, hence, they are rational powers
of q.
�k;u(K
�1
i ); �k;u(Ei); �k;u(Fi) are de�ned for u 2 Z. We are going to extend
these operator valued functions to the complex plane. Evidentially,
�k;u(K
�1
i )f = K
�1
i (f uk )
�u
k = K
�1
i fK
�1
i ( uk )
�u
k ;
�k;u(Ei)f = Eif +KifEi(
u
k )
�u
k ;
�k;u(KiFi)f = KiFif +KifKiFi(
u
k )
�u
k ; f 2 C [
k ]q:
Denote by P� a projection in C [
k ]q with ImP� = C [
k ]q;�, KerP� =L
�0 6=� C [
k ]q;�0 : In the sequel we deal with the operator valued functions
P�2�k;u(K
�1
i )jC [
k ]q;�1
; P�2�k;u(Ei)jC [
k ]q;�1
; P�2�k;u(KiFi)jC [
k ]q;�1
:
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 167
O. Bershtein, A. Stolin, and L. Vaksman
Consider a sequence faugu2Z+
au = Ei
u
k =
0
@u�1X
j=0
A
j
k(Ei k)
1
A
u�1
k ; u 2 Z+;
where Ak : C [
k ]q ! C [
k ]q and AkjC [
k ]q;�
= Ak;�(q):
As in the previous section, show that fEi
u
kgu 2 L.
It is clear that Ei k belongs to V = ��2MC [
k ]q;�, with M � P
S
+
being
a �nite set. Consider the restriction Ak to V , AkV � V . Using the Jordan
canonical form of AkjV and the equality
Ei
u
k =
0
@u�1X
j=0
A
j
k(Ei k)
1
A
u�1
k ; u 2 Z+;
one has fEi
u
kgu 2 L.
So one can extend the operator valued function �k;u(Ei) to the complex plane,
since the eigenvalues of Ak;�(q) are positive (Lem. 3). Similarly, one can extend
the operator valued function �k;u(KiFi). The extensions of the operator valued
functions �k;u(K
�1
i ), i = 1; 2; : : : ; l, exist by obvious reasons. At last,
�k;u(Fi) = �k;u(K
�1
i )�k;u(KiFi); i = 1; 2; : : : ; l:
Now we have to check whether the map
Ei 7! �k;u(Ei); Fi 7! �k;u(Fi); Ki 7! �k;u(Ki)
can be extended to an algebra homomorphism.
Introduce an auxiliary algebra of analytic functions F (u; q) on C � (0; 1)
that take values in the space of linear operators in C [
k ]q =
L
�2PS+
C [
k ]q;�.
In other words, we assume the analyticity of all operator valued functions
P�2F (u; q)jC [
k ]q;�1
, where �1; �2 2 P
S
+
.
The vector bundle F� is equipped with a Hermitian metric, so the operator
norm kF (u; q)k is well-de�ned. Consider a subalgebra of operator valued functions
satisfying the following condition:
kF (u; q)k � a
F
(q) exp(b
F
(q)juj); (6)
for some a
F
(q) > 0 and b
F
(q) > 0 such that lim
q!1
b
F
(q) = 0. Note that the
subalgebra does not depend on the choice of metrics.
The operator valued functions
�k;u(Ei); �k;u(Fi); �k;u(K
�1
i ); i = 1; 2; : : : ; l;
168 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Spherical Principal Series of Quantum Harish-Chandra Modules
are analytic and satisfy (6). Prove it. Consider AkjV , where V is the �nite
dimensional Uqk-invariant subspace and Ei k 2 V . One has
k(AkjV )
u(Ei k)k = k
X
i;j2Z+
aij�
u
i u
jk:
The number of terms in the r.h.s. of expression is at most (dimV )2 (obviously,
0 � i; j � dimV � 1). Hence,
k(AkjV )
u(Ei k)k = k
X
i;j2Z+
aij�
u
i u
jk � (dimV )2max kaijkmax j�ij
u
u
j
:
Proposition 6. ([2]). Let f(z) be continuous in fz 2 C jRe z � 0g and
holomorphic in fz 2 C jRe z > 0g. Assume also that
1. jf(iy)j � const � expf(� � ")jyjg, y 2 R,
2. jf(z)j �M exp(ajzj), Re z � 0,
for some real M;a and " > 0.
If f(n) = 0 for all n 2 N, then f(z) � 0.
Corollary 2. The extension of �k;u(Ei); �k;u(Fi); �k;u(Ki) is unique.
Using (6), one can prove easily that the Drinfeld�Jimbo relations hold for
these operator valued functions. So �k;u is a representation. It is a q-analog
of a representation of degenerate spherical principal series of Harish�Chandra
modules.
Now turn back to the nondegenerate spherical principal series. For u =
(u1; u2; : : : ; ur) 2 Zr de�ne (Cf. (5))
�u(�)f
def
= �
0
@f rY
j=1
uj
j
1
A �
rY
j=1
�uj
j ; � 2 Uqg; f 2 C [
]q :
We describe the extension of �u to C r . As before, consider the decomposition
of C [
]q into a sum of its Uqk-isotypic components
C [
]q =
M
�2PS+
C [
]q;�
and operator valued functions
Aj;�(q) : C [
]q;� ! C [
]q;� ; Aj;�(q) : f 7! jf
�1
j :
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 169
O. Bershtein, A. Stolin, and L. Vaksman
The construction of
�u(Ei); �u(Fi); �u(K
�1
i ); i = 1; 2; : : : ; l; (7)
essentially reduces to an analytic continuation of the vector-valued functions
uk�1X
j=1
Aj
k;�(Ei k); k = 1; 2; : : : ; r:
The Drinfeld�Jimbo relations for the operator valued functions (7) can be
proved in the same way. Namely, consider an algebra of analytic operator valued
functions F (u1; u2; : : : ; ur; q) such that
kF (u1; u2; : : : ; ur; q)k � a
F
(q) exp(b
F
(q)
rX
k=1
jukj)
(Cf. (6)). Now one can prove the uniqueness of the interpolation of �u in this
subalgebra.
5. Appendix
We present some auxiliary statements on certain vector bundles over (0; 1].
Start with the well-known facts on Verma modules. Let Uqb
+ be a Hopf subalge-
bra generated by Ei;K
�1
i . Let � 2 P+, and C � be a one dimensional Uqb
+-module
de�ned by its generator 1� and the relations
K
�1
i 1� = q
��i
i 1�; Ei1� = 0; i = 1; 2; : : : ; l:
As usual, a Verma module over Uqg can be de�ned as follows
M(�)q
def
= Uqg
Uqb+ C � :
Fix v� = 1
1�. It is known that v� is a generator, and M(�)q can be de�ned
by the relations
Eiv� = 0; K
�1
i v� = q
��i
i v�; i = 1; 2; : : : ; l:
Recall that the Weyl group W acts on the root system R of Lie algebra g and
is generated by simple re�ections si(�j) = �j � aij�i. Fix the reduced expression
of the longest element w0 = si1 � si2 � : : : � siM 2W . One can associate to it a total
order on the set of positive roots of g, and then a basis in the vector space Uqg.
We use the following total order on the set of positive roots:
�1 = �1; �2 = si1(�i2); �3 = si1si2(�i3); : : : �M = si1 : : : siM�1
(�iM ):
170 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Spherical Principal Series of Quantum Harish-Chandra Modules
Following G. Lusztig [12, 15, 1], introduce the elements E�s ; F�s 2 Uqg for
s = 1; : : : ;M . As a direct consequence of de�nitions, E�s , (resp. F�s), is a linear
combination of Em1
j1
� : : : � Eml
jl
(resp. F n1
i1
� : : : � F nk
ik
) with the coe�cients in the
expansion being rational functions of q without poles in (0; 1].
Proposition 7. The set fF jM
�M
� F
jM�1
�M�1
� : : : � F j1
�1
� Ki1
1
� Ki2
2
� : : : � Kil
l � E
j1
�1
�
E
j2
�2
� : : : � EjM
�M
jk1; k2; : : : ; kM ; j1; j2; : : : ; jM 2 Z+ ; i1; i2; : : : ; il 2 Zg is a basis in
the vector space Uqg.
Hence, the weight vectors
vJ(�) = F
jM
�M
F
jM�1
�M�1
: : : F
j1
�1
v�; j1; j2; : : : ; jM 2 Z+; (8)
form a basis of M(�)q.
Equip Uqg with a �-Hopf algebra structure as follows:
(K�1
j )? = K
�1
j ; E
?
j = KjFj ; F
?
j = EjK
�1
j ; j = 1; 2; : : : ; l:
Lemma 4. There exists a unique Hermitian form in M(�)q such that:
� (�v0; v00) = (v0; �?v00); v
0
; v
00 2M(�)q; � 2 Uqg;
� (v�; v�) = 1.
The kernel K(�)q of the form (�; �) is the largest proper submodule of M(�)q.
In this section we write L(�)q instead of L(�) to make the dependence on q
explicit.
Proposition 8. 1. L(�)q 'M(�)q=K(�)q.
2. The form (�; �) is nondegenerate in L(�)q.
Introduce a morphism p� :M(�)q ! L(�)q, v� 7! v(�).
The �rst statement on special vector bundles over (0; 1] is as follows. Let
� 2 P+. There exists the continuous over (0; 1] and analytic in (0; 1) vector
bundle E� with �bers isomorphic to L(�)q. Of course, Ei, Fi, Hi, i = 1; 2; : : : ; l,
are the endomorphisms of E�.
�
Describe the construction of E�. Recall that to any reduced expression of w0
we assign the basis of M(�)q (see (8)).
Fix q0 2 (0; 1]. Choose a subset fvjgj=1;:::;dimL(�) of one of the mentioned
bases in such a way that the matrix ((vi; vj))i;j=1;:::;dimL(�) is nondegenerate. It is
nondegenerate for all q that are close enough to q0. Hence, in a neighborhood
�I.e. Ei, Fi, Hi correspond to continuous in (0; 1] and analytic in (0; 1) operator valued
functions.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 171
O. Bershtein, A. Stolin, and L. Vaksman
of q0, the set fp�(vj)gj=1;:::;dimL(�) is a basis, since dimL(�)q does not depend on
q 2 (0; 1]. One gets a trivial vector bundle with the required properties over the
neighborhood of q0.
The elements of the matrix ((vi; vj)) are the continuous in (0; 1] and analytic
in (0; 1) functions. Therefore, the matrices of Ei, Fi, Hi in the basis
fp�(vj)gj=1;:::;dimL(�) are continuous in (0; 1] and analytic in (0; 1). Indeed, any
function
(p�(E
m1
j1
� : : : � Eml
jl
F
n1
i1
� : : : � F nk
ik
v�); p�(v�))
is continuous in (0; 1] and analytic in (0; 1), since to calculate the value one should
just use the commutation relations, which are �well-dependent� on q. Therefore,
the functions
(Eip�(F
n1
i1
� : : : � F nk
ik
v�); p�(F
m1
j1
� : : : � Fmk
jk
v�))
and
(Eip�(F
n1
�1
� ::: � F nM
�M
v�); p�(F
m1
�1
� : : : � FmM
�M
v�))
are �well-dependent� on q. Hence, the matrix elements of the operator Ei are
�well-dependent� on q, since the matrix is nondegenerate in the neighborhood of
q0. The same holds for Fi;Hi, and the transition matrices de�ned on intersection
of the neighborhoods. Finally, the vector bundle over (0; 1] which we obtain in
this way does not depend on the choices made above. So, the vector bundle E� is
constructed.
Now proceed to the construction of a subbundle of E� corresponding to the
�xed Uqk-type �. Consider the decomposition of L(�)q =
L
� L(�)q;� into a sum
of its Uqk-isotypic components. Then one has the following statement on special
vector bundles. For any � 2 P+ and � 2 P
S
+
, L(�)q;� is a �ber of a continuous
vector bundle over (0; 1], analytic in (0; 1).
Fix � 2 P
S
+
and consider E�� = f(f; q)jf 2 L(�)q;�g. Prove that it de�nes
a subbundle of E�. Indeed, consider a �ber L(�)q together with its decomposition
L(�)q =
L
�2PS+
L(�)q;� . Note that the sum consists of �nite number of terms.
Hence, there exists cq 2 Z(Uqk), that is polynomial in q, and
cqjL(�)q;� = 1; cqjL(�)q;� = 0; � 6= �;
see [4, p. 125�126]. Hence cq de�nes the morphism of vector bundles
E�
j
����
��
��
��
cq �� E�
��
(0; 1]
:
172 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Spherical Principal Series of Quantum Harish-Chandra Modules
It is an orthogonal projection onto L(�)q;� in any �ber L(�)q. Since rank cq is
constant, the image of cq is a vector subbundle.
Now we can construct the last required vector bundle. C [
k ]q;� are the �bers
of a continuous vector bundle F� over (0; 1], analytic in (0; 1).
Consider a map
q : C [ bXk ]q; k
! C [ bXk ]q; k
; f 7! kf:
It is easy to prove that q is an invertible, continuous operator valued function,
analytic in (0; 1). Using q one can carry the vector bundle structure to F� from
f(f; q)jf 2 C [
k ]q;�
N
k ; q 2 (0; 1]g for large enough N .
R e m a r k 2. Note that the matrix elements of all operator valued functions
belong to Q(q1=s) with s = card(P=Q), and Q being the root lattice.
The next considerations are related to the self-adjointness. Consider an auxi-
liary algebra
C [ bXspher ]q =
M
�2�+
L(�)q:
Equip Uqg and C [ bXspher ]q with a �complex conjugation�. Recall that dimL(�)
Uqk
q
= 1 for any � 2 �+. Consider the antilinear involutive automorphism �� of Uqg
de�ned by
�Ei = Ei;
�Fi = Fi;
�K�1
i = K
�1
i ; i = 1; 2; : : : ; l:
There exists a unique antilinear involutive operator: ��
L(�)q ! L(�)q; �v(�) 7! ��v(�); � 2 Uqg:
(Indeed, the uniqueness is obvious, while the existence follows from the de�nition
of L(�)q.)
It is easy to see that L(�)
Uqk
q = L(�)
Uqk
q . Hence, there exists a nonzero vector
w� 2 L(�)
Uqk
q such that w� = �w�. Let l(�) = Rw� .
Lemma 5. There exists a unique involution ? of C [ bXspher ]q such that
C [ bXspher ]q is a (Uqg; ?)-module algebra, and ?jl(�) = id.
P r o o f. Firstly prove that
L(�)�q � L(�)q; � 2 �+: (9)
Following [20], introduce a system of strongly orthogonal roots
1 >
2 > : : : >
r
with
1 being the maximal root. It is easy to prove that �w0(
j) =
j
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 173
O. Bershtein, A. Stolin, and L. Vaksman
for j = 1; 2; : : : ; r; where w0 2 W is the longest element. Hence, (9) follows
from the fact that the fundamental spherical weights belong to the linear span of
1;
2; : : : ;
r (see [3]).
An involution ? on L(�)q such that
(�f)? = (S(�))?f?; � 2 Uqg; f 2 L(�)q;
is unique up to �1. The uniqueness of the involution now follows from the fact
that w� = �w�.
Turn to the proof of existence of ?. Uqg is equipped with the involution as
follows:
(K�1
j )? = K
�1
j ; E
?
j = KjFj ; F
?
j = EjK
�1
j ; j = 1; 2; : : : ; l:
Consider the �-algebra (C [G]q ; ?) with the involution ? given by
f
?(�)
def
= f((S(�))?); � 2 Uqg; f 2 C [G]q :
Let
F = ff 2 C [G]q j Lreg(�)f = "(�)f; � 2 Uqkg:
It follows from the Peter�Weyl expansion that F � ��2�+L(�)q as a Uqg-
module. One can consider �+ with a natural partial order �, and F can be
equipped with a (Uqg; ?)-invariant �ltration F =
S
�2�+
F�; F� =
L
��� L(�)q.
Consider the zonal spherical functions �� related to the Uqg-modules L(�)q.
In the associated Zr-graded algebra GrF their images pairwise commute. Indeed,
the quasicommutativity is evident, while the commutativity follows from the self-
adjointness of �� and the commutativity in the classical case. Now, it follows
from [22, Cor. 2.6] that GrF is naturally isomorphic to C [ bXspher ]q. Note that
l(�) corresponds to R�� . Carry the involution ? from F to GrF and C [ bXspher ]q.
It can be veri�ed easily that ? is a morphism of the vector bundle with �bers
L(�)q.
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