q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra U_q(u(n,1))

0.2 MB·English

by  R. M. Asherova

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6(2010), 010, 13 pages
q-Analog of Gelfand–Graev Basis
⋆
for the Noncompact Quantum Algebra U (u(n, 1))
q
Raisa M. ASHEROVA †, Cˇestm´ır BURD´IK ‡, Miloslav HAVL´ICˇEK ‡,
Yuri F. SMIRNOV †§ and Valeriy N. TOLSTOY †‡
† Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia
E-mail: [email protected], [email protected]
0 ‡ Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,
1
Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic
0
2 E-mail: [email protected]fi.cvut.cz, miloslav.havlicek@fjfi.cvut.cz

n § Deceased
a
J Received November 05, 2009, in final form January 15, 2010; Published online January 26, 2010

6 doi:10.3842/SIGMA.2010.010
2

] Abstract. For the quantum algebra U (gl(n + 1)) in its reduction on the subalgeb-
A q
ra U (gl(n)) an explicit description of a Mickelsson–Zhelobenko reduction Z-algebra
q
Q
Z (gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using
q
. this Z-algebra we describe Hermitian irreducible representations of a discrete series for the
h
t noncompact quantum algebra Uq(u(n,1)) which is a real form of Uq(gl(n+1)), namely, an
a
orthonormalGelfand–Graev basis is constructed in an explicit form.
m
[ Key words: quantumalgebra;extremalprojector;reductionalgebra;Shapovalovform;non-

compact quantum algebra; discrete series of representations; Gelfand–Graev basis
2
v 2010 Mathematics Subject Classification: 17B37;81R50
3
0
4
5 1 Introduction
.
2
1
In 1950, I.M. Gelfand and M.L. Tsetlin [1] proposed a formal description of finite-dimensional
9
0 irreducible representations (IR) for the compact Lie algebra u(n). This description is a genera-
: lization of the results for u(2) and u(3) to the u(n) case. It is the following. In the IR space
v
i of u(n) there is a orthonormal basis which is numerated by the following formal schemes:
X
r
m m ... m m
a 1n 2n n−1,n nn
m m ... m
1,n−1 2,n−1 n−1,n−1
 
... ... ... ,
 m m 
 12 22 
 m 
 11 
 
where all numbers m (1 ≤ i ≤ j ≤ n) are nonnegative integers and they satisfy the standard
ij
inequalities, “between conditions”:
m ≥ m ≥ m for 1≤ i ≤ j ≤ n−1.
ij+1 ij i+1j+1
The first line of this scheme is defined by the components of the highest weight of u(n) IR, the
second line is defined by the components of the highest weight of u(n−1) IR and so on.
⋆This paper is a contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Sys-
tems and Quantum Symmetries (June 18–20, 2009, Prague, Czech Republic). The full collection is available at
http://www.emis.de/journals/SIGMA/ISQS2009.html
2 R.M. Asherova, Cˇ. Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
Laterthisbasiswasconstructedinmanypapers(see, e.g.,[2,3,4])byusingone-steplowering
and raising operators.
In 1965, I.M. Gelfand and M.I. Graev [5], using analytic continuation of the results for u(n),
obtained some results for noncompact Lie algebra u(n,m). It was shown that some class of
Hermitian IR of u(n,m) is characterized by an “extremal weight” parametrized by a set of
integers m = (m ,m ,...,m ) (N = n+m) such that m ≥ m ≥ ··· ≥ m , and
N 1N 2N NN 1N 2N NN
by a representation type which is defined by a partition of n in the sum of two nonnegative
integers α and β, n = α+β (also see [6]).
For simplicity we consider the case u(2,1). In this case we have three types of schemes
m m m
13 23 33
m m for (α,β) = (2,0),
12 22
 
m
11
 
m m m
13 23 33
m m for (α,β) = (1,1),
12 22
 
m
11
 
m m m
13 23 33
m m for (α,β) = (0,2).
12 22
 
m
11
 
The numbers m of the first scheme satisfy the following inequalities
ij
m ≥ m +1, m +1 ≥ m ≥ m +1, m ≥ m ≥ m .
12 13 13 22 23 12 11 22
The numbers m of the second scheme satisfy the following inequalities
ij
m ≥ m +1, m −1 ≥ m , m ≥ m ≥ m .
12 13 33 22 12 11 22
The numbers of the third scheme satisfy the following inequalities
m −1 ≥ m ≥ m −1, m −1 ≥ m , m ≥ m ≥ m .
23 12 33 33 22 12 11 22
Construction of the Gelfand–Graev basis for u(n,m) in terms of one-step lowering and raising
operators is more complicated than in the compact case u(n+m).
In 1975, T.J. Enright and V.S. Varadarajan [7] obtained a classification of discrete series of
noncompact Lie algebras. Later it was proved by A.I. Molev [8] that in the case of u(n,m) the
Gelfand–Graev modules are part of the Enright–Varadarajan modules and Molev constructed
the Gelfand–Graev basis for u(n,m) in terms of the Mickelsson S-algebra [9].
A goal of this work is to obtain analogous results for the noncompact quantum algebra
U (u(n,m)). Since the general case is very complicated we at first consider the case U (u(n,1)).
q q
The special case U (u(2,1)) was considered in [10, 11]. It should be noted that the principal
q
series representations of U (u(n,1)) were studied in [12] and a classification of unitary highest
q
weight modules of U (u(n,1)) was considered in [13].
q
2 Quantum algebra U (gl(N)) and its noncompact
q
real forms U (u(n,m)) (n + m = N)
q
The quantum algebra Uq(gl(N)) is generated by the Chevalley elements q±eii (i = 1,...,N),
e , e (i =1,2,...,N −1) with the defining relations [14, 15]:
i,i+1 i+1,i
qeiiq−eii = q−eiiqeii = 1, (2.1)
q-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra U (u(n,1)) 3
q
qeiiqejj =qejjqeii, (2.2)
qeiie q−eii = qδij−δike (|j −k| = 1), (2.3)
jk jk
qeii−ei+1,i+1 −qei+1,i+1−eii
[e ,e ] = δ , (2.4)
i,i+1 j+1,j ij q−q−1
[e ,e ] = 0 for |i−j| ≥ 2, (2.5)
i,i+1 j,j+1
[e ,e ] = 0 for |i−j| ≥ 2, (2.6)
i+1,i j+1,j
[[e ,e ] ,e ] = 0 for |i−j| = 1, (2.7)
i,i+1 j,j+1 q j,j+1 q
[[e ,e ] ,e ] = 0 for |i−j| = 1, (2.8)
i+1,i j+1,j q j+1,j q
where [e ,e ] denotes the q-commutator:
β γ q
[e ,e ] := e e −q(β,γ)e e .
β γ q β γ γ β
The definition of a quantum algebra also includes operations of a comultiplication ∆ , an an-
q
tipode S , and a co-unit ǫ . Explicit formulas of these operations will not be used in our later
q q
calculations and they are not given here.
Let ε (i = 1,2,...,N) be a dual basis to the Cartan basis e (i = 1,2,...,N), ε (e ) =
i ii i jj
(ε ,ε ) = δ . In terms of the orthonormal basis vectors ε the positive root system ∆ of gl(N)
i j ij i +
(U (gl(N))) is presented as follows:
q
∆ = {ε −ε |1 ≤ i< j ≤ N},
+ i j
where ε −ε (i = 1,2,...,N −1) are the simple roots.
i i+1
Since for construction of the composite root vectors e := e (|i−j| ≥ 2) of the quantum
ij εi−εj
algebra U (gl(N)) weneed tousethenotation of thenormalorderinginthepositiverootsystem
q
∆ , we recall this notation. We say that the system ∆ is written in normal (convex) ordering,
+ +
∆~ , if each positive composite root ε −ε = (ε −ε )+(ε −ε ) (i ≤ k ≤ j) is located between
+ i j i k k j
its components ε −ε and ε −ε . It means that in the normal ordering system ∆~ we have
i k k j +
either
...,ε −ε ,...,ε −ε ,...,ε −ε ,...,
i k i j k j
or
...,ε −ε ,...,ε −ε ,...,ε −ε ,....
k j i j i k
There are many normal orderings in the root system ∆ = ∆ (gl(N)), more than (N −1)! for
+ +
N > 3. To be definite, we fix the following normal ordering (see [14, 15])
ε −ε ≺ ε −ε ≺ ε −ε ≺ ε −ε ≺ ε −ε ≺ ε −ε ≺ ··· ≺
1 2 1 3 2 3 1 4 2 4 3 4
ε −ε ≺ ε −ε ≺ ··· ≺ ε −ε ≺ ··· ≺ ε −ε ≺ ε −ε ≺ ··· ≺ ε −ε . (2.9)
1 k 2 k k−1 k 1 N 2 N N−1 N
According to this ordering, we determine the composite root vectors e for |i−j| ≥ 2 as follows:
ij
eij := [eik,ekj]q−1, eji := [ejk,eki]q, (2.10)
where 1≤ i < k < j ≤ N. It should be stressed that the structure of the composite root vectors
does not depend on the choice of the index k on the right-hand side of the definition (2.10). In
particular, we have
eij := [ei,i+1,ei+1,j]q−1 = [ei,j−1,ej−1,j]q−1,
e := [e ,e ] = [e ,e ] , (2.11)
ji j,i+1 i+1,i q j,j−1 j−1,i q
where 2 ≤ i+1 < j ≤ N.
4 R.M. Asherova, Cˇ. Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
Using these explicit constructions and defining relations (2.1)–(2.8) for the Chevalley basis it
is not hard to calculate the following relations between the Cartan–Weyl generators e (i,j =
ij
1,2,...,N):
qekke q−ekk = qδki−δkje (1 ≤ i,j,k ≤ N), (2.12)
ij ij
qeii−ejj −qejj−eii
[e ,e ]= (1 ≤ i< j ≤ N), (2.13)
ij ji q−q−1
[eij,ekl]q−1 = δjkeil (1 ≤ i < j ≤ k < l ≤ N), (2.14)
[eik,ejl]q−1 = q−q−1 ejkeil (1 ≤ i< j < k < l ≤ N), (2.15)
[ejk,eil]q−1 = 0(cid:0) (1 ≤(cid:1) i ≤ j <k < l ≤ N), (2.16)
[eik,ejk]q−1 =0 (1 ≤ i< j < k ≤ N), (2.17)
[e ,e ]= 0 (1 ≤ i< j ≤ k < l ≤ N), (2.18)
kl ji
[e ,e ]= 0 (1 ≤ i< j < k < l ≤ N), (2.19)
il kj
[e ,e ] = e qeii−ejj (1 ≤ i < j < l ≤ N), (2.20)
ji il jl
[e ,e ] = e qekk−ell (1 ≤ i < k < l ≤ N), (2.21)
kl li ki
[e ,e ]= q−1−q e e qejj−ekk (1 ≤ i< j < k < l ≤N). (2.22)
jl ki kl ji
If we apply th(cid:0)e Cartan(cid:1) involution (e⋆ = e , q⋆ = q−1) to the formulas (2.12)–(2.22), we get
ij ji
all relations between the elements of the Cartan–Weyl basis.
The explicit formula for the extremal projector for U (gl(N)), corresponding to the fixed
q
normal ordering (2.9), has the form [14, 15]
p(U (gl(N)) = p(U (gl(N −1))(p p ···p p )
q q 1N 2N N−2,N N−1,N
= p (p p )···(p ···p )···(p ···p ), (2.23)
12 13 23 1k k−1,k 1N N−1,N
where the elements p (1 ≤ i< j ≤ N) are given by
ij
∞ (−1)r r −1
p = ϕ er er , ϕ = q−(j−i−1)r [e −e +j −i+s] . (2.24)
ij [r]! ij,r ij ji ij,r ii jj
( )
r=0 s=1
X Y
Here and elsewhere the symbol [x] is given as follows
qx−q−x
[x] = .
q−q−1
The extremal projector p := p(U (gl(N)) satisfies the relations:
q
e p = pe = 0 (1 ≤ i ≤ N −1), p2 = p. (2.25)
i,i+1 i+1,i
The extremal projector p belongs to the Taylor extension TU (gl(N)) of the quantum algebras
q
U (gl(N)). The Taylor extension TU (gl(N)) is an associative algebra generated by formal
q q
Taylor series of the form
C qe11,...,qeNN er˜12er˜13er˜23···er˜N−1,Ner12er13er23···erN−1,N
{r˜},{r} 21 31 32 N,N−1 12 13 23 N−1,N
{r˜},{r}
X (cid:0) (cid:1)
provided that nonnegative integers r˜ ,r˜ ,r˜ ,...,r˜ and r ,r ,r ,...,r are sub-
12 13 23 N−1,N 12 13 23 N−1,N
ject to the constraints
r˜ − r ≤ const
ij ij
(cid:12) (cid:12)
(cid:12)Xi<j Xi<j (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
q-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra U (u(n,1)) 5
q
for each formal series and the coefficients C (qe11,...,qeNN) are rational functions of the
{r˜},{r}
q-Cartan elements qeii. The quantum algebra Uq(gl(N)) is a subalgebra of the Taylor extension
TU (gl(N)), U (gl(N)) ⊂ TU (gl(N)).
q q q
We consider, on the quantum algebra U (gl(N)), two real forms: compact and noncompact.
q
ThecompactquantumalgebraU (u(N))canbeconsideredasthequantumalgebraU (gl(N))
q q
(N = n+m) endowed with the additional Cartan involution ⋆:
h⋆ = h for i = 1,2,...,N, (2.26)
i i
e⋆ = e , e⋆ = e for 1≤ i ≤ N −1, (2.27)
i,i+1 i+1,i i+1,i i,i+1
q⋆ = q or q⋆ = q−1. (2.28)
Thuswehavetwocompactrealforms: withrealq (q⋆ = q)andwithcircularq (q⋆ =q−1). Inthe
caseofthecircularq theCartan–Weylbasise (i,j = 1,2,...,N)constructedbyformulas(2.10)
ij
is ⋆-invariant, i.e. e⋆ =e for all 1≤ i,j ≤ N. In the case of the real q this Cartan–Weyl basis
ij ji
is not ⋆-invariant, since the basis vectors satisfy the relations e⋆ = e′ for |i−j| ≥ 2 where the
ij ji
root vectors e′ are obtained from (2.10) by the replacement q±1 → q∓1.
ji
It is reasonable to consider the real compact form on the Taylor extension TU (gl(N)). In
q
particular, it should be noted that
p⋆ = p for real and circular q. (2.29)
This property is a direct consequence of a uniqueness theorem for the extremal projector, which
states that equations (2.25) have a unique nonzero solution in the space of the Taylor extension
TU (gl(N)) and this solution does not depend on the choice of normal ordering and on the
q
replacement q±1 → q∓1 in formulas (2.10).
The noncompact quantum algebra U (u(n,m)) can be considered as the quantum algebra
q
U (gl(N)) (N = n+m) endowed with the additional Cartan involution ∗:
q
h∗ = h for i = 1,2,...,N, (2.30)
i i
e∗ = e , e∗ = e for 1≤ i ≤ N −1, i 6= n, (2.31)
i,i+1 i+1,i i+1,i i,i+1
e∗ =−e , e∗ = −e , (2.32)
n,n+1 n+1,n n+1,n n,n+1
q∗ = q or q∗ = q−1. (2.33)
We also have two noncompact real forms: with real q (q⋆ = q) and with circular q (q⋆ = q−1).
Below we will consider the real form U (u(n,1)), i.e. the case N = n+1.
q
3 The reduction algebra Z (gl(n + 1),gl(n))
q
In the linear space TU (gl(n +1)) we separate out a subspace of “two-sided highest vectors”
q
with respect to the subalgebra U (gl(n)) ⊂ U (gl(n+1)), i.e.
q q
Z˜ (gl(n+1),gl(n)) = x ∈TU (gl(n+1)) e x = xe = 0, 1≤ i ≤ n−1 .
q q i,i+1 i+1,i
It is evident that if x ∈ Z˜q(g(cid:8)l(n+1),gl(n)) then(cid:12)(cid:12) (cid:9)
x = pxp,
where p := p(U (gl(n)). Again, using the annihilation properties of the projection operator p
q
we have that any vector x ∈ Z˜ (gl(n+1),gl(n)) can be presented in the form of a formal Taylor
q
series on the following monomials
per1′ ···ern′ ern ···er1 p. (3.1)
n+1,1 n+1,n n,n+1 1,n+1
6 R.M. Asherova, Cˇ. Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
Itis evident thatZ˜ (gl(n+1),gl(n)) is asubalgebrainTU (gl(n+1)). We considera subalgebra
q q
Z (gl(n+1),gl(n)) generated by finite series on monomials (3.1).
q
We set
z := p, z := pe p, z := pe p (i = 1,2,...,n).
0 i i,n+1 −i n+1,i
Theorem 1. The elements z (i = 0,±1,±2,...,±n) generate the unital associative algebra
i
Z (gl(n+1),gl(n)) and satisfy the following relations
q
z z = z z = z for i = 0,±1,±2,...,±n, (3.2)
0 i i 0 i
z z = z z for 1≤ i,j ≤ n, i6= j, (3.3)
i −j −j i
[ϕ +1]
ij
z z = z z for 1 ≤ i< j ≤ n, (3.4)
i j j i
[ϕ ]
ij
[ϕ ]
ij
z z = z z for 1 ≤ i < j ≤ n, (3.5)
−i −j −j −i
[ϕ +1]
ij
and
n
z z = B z z +γ z for i= 1,2,...,n, (3.6)
i −i ij −j j i 0
j=1
X
where
b−b+
B = − i j , γ = [ϕ −1]b−, (3.7)
ij [ϕ −1] i i,n+1 i
ij
n
[ϕ ±1]
b± = is , ϕ = e −e +j −i. (3.8)
i [ϕ ] ij ii jj
is
s=i+1
Y
Remark 1. The relations (3.2) state that the element z is an algebraic unit in Z (gl(n+1),
0 q
gl(n)).
A proof of the theorem can be obtained by direct calculations using the explicit form of
extremal projector (2.23), (2.24).
It should be noted that the theorem was proved by V.N.T. as early as 1989 but it has not
been published up to now, however, the results of the theorem were used for construction of the
Gelfand–Tsetlin basis for the compact quantum algebra U (u(n)) [14, 15].
q
ForconstructionandstudyofthediscreteseriesofthenoncompactquantumalgebraU (u(n,1)
q
we need other relations than (3.6). The system (3.6) expresses the elements z z in terms of
i −i
the elements z z (i = 1,2,...,n) but we would like to express the elements z z ,...,z z ,
−i i −1 1 −α α
z z ,...,z z in terms of the elements z z ,...,z z ,z z ,...,z z for α =
α+1 −α−1 n −n 1 −1 α −α −α−1 α+1 −n n
0,1,...,n.1,2 These relations are given by the proposition.
Proposition 1. The elements z z ,...,z z ,z z ,...,z z are expressed in terms of
−1 1 −α α α+1 −α−1 n −n
the elements z z ,...,z z ,z z ,...,z z by the formulas
1 −1 α −α −α−1 α+1 −n n
α n
(α) (α) (α)
z z = B z z + B z z +γ z (1 ≤ i ≤ α), (3.9)
−i i ij j −j il −l l i 0
j=1 l=α+1
X X
α n
(α) (α) (α)
z z = B z z + B z z +γ z (α+1 ≤ k ≤ n). (3.10)
k −k kj j −j kl −l l k 0
j=1 l=α+1
X X
1In the case α=0 we haverelations (3.6) and for α=n we obtain thesystem inverseto (3.6).
2In Section 5 theparameter α will characterize a representation typeof thediscrete series.
q-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra U (u(n,1)) 7
q
Here
b(α)+b(α)− b(α)+b(α)+
B(α) = i j , B(α) = i l ,
ij [ϕ +1] il [ϕ ]
ij il
(α) (α)+
γ = −[ϕ −α]b for 1≤ i,j ≤ α < l ≤ n, (3.11)
i i,n+1 i
b(α)−b(α)− b(α)−b(α)+
B(α) =− k j , B(α) = − k l ,
kj [ϕ ] kl [ϕ −1]
kj kl
(α) (α)−
γ = [ϕ −α−1]b for 1 ≤ j ≤ α < k,l ≤ n, (3.12)
k k,n+1 k
where
i−1 n
[ϕ ±1] [ϕ ]
(α)± is is
b = (1 ≤ i≤ α), (3.13)
i [ϕ ] [ϕ ±1]
is ! is !
s=1 s=α+1
Y Y
α n
[ϕ ] [ϕ ±1]
(α)± ls ls
b = (α+1≤ l ≤n). (3.14)
l [ϕ ±1] [ϕ ]
ls ! ls !
s=1 s=l+1
Y Y
Scheme of proof. The relations (3.9) and (3.10) with the coefficients (3.11)–(3.14) can be
proved by induction on α. For α = 0 they coincide with the relations (3.6)–(3.8)3. Next
we assume that relations (3.9)–(3.14) are valid for α ≥ 1 and we extract from (3.10) the
relation with k = α + 1 and express in it the term z z in terms of the elements
−α−1 α+1
z z ,...,z z ,z z ,..., z z ; then this expression is substituted in the right
1 −1 α+1 −α−1 −α−2 α+2 −n n
side of the rest relations (3.9) and (3.10) and after some algebraic manipulations we obtain
the relations (3.9)–(3.14) where α should be replaced by α+1. (cid:4)
Using (3.3)–(3.5) and (3.9)–(3.14) we can prove some power relations.
Proposition 2. The following power relations are valid
zrzs = zs zr for 1 ≤ i,j ≤ n, i 6= j and r,s ∈N, (3.15)
i −j −j i
[ϕ +r]![ϕ −s]!
zrzs = zszr ij ij for 1≤ i < j ≤ n and r,s ∈ N, (3.16)
i j j i [ϕ ]![ϕ +r−s]!
ij ij
[ϕ ]![ϕ −r+s]!
zr zs = zs zr ij ij for 1 ≤ i< j ≤ n and r,s ∈ N, (3.17)
−i −j −j −i[ϕ −r]![ϕ +s]!
ij ij
α n
z zr = zr−1 B(α)(r)z z + B(α)(r)z z +γ(α)(r)z (1 ≤ i ≤ α), (3.18)
−i i i  ij j −j il −l l i 0
j=1 l=α+1
X X
 
α n
z zr = zr−1 B(α)(r)z z + B(α)(r)z z +γ(α)(r)z
k −k −k  kj j −j kl −l l k 0
j=1 l=α+1
X X
(α+1 ≤ k ≤ n).  (3.19)
Here
[r] [r]
(α) (α)+ (α)− (α) (α)+ (α)+
B (r) = b (r)b , B (r)= b (r)b ,
ij [ϕ +r] i j il [ϕ +r−1] i l
ij il
γ(α)(r)= −[r][ϕ −α+r−1]b(α)+(r) (1 ≤ i,j ≤ α< l ≤ n; r ∈ N), (3.20)
i i,n+1 i
3Inthiscase,therelations(3.9)areabsentand,moreover,thefirstsumintherightsideoftherelations(3.10)
is equal to 0 for α=0.
8 R.M. Asherova, Cˇ. Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
[r] [r]
(α) (α)− (α)− (α) (α)− (α)+
B (r) = − b (r)b , B = − b (r)b ,
kj [ϕ −r+1] k j kl [ϕ −r] k l
kj kl
γ(α)(r)= [r][ϕ −α−r]b(α)−(r) (1 ≤ j ≤ α < k,l ≤ n; r ∈ N), (3.21)
k k,n+1 k
where
i−1 n
[ϕ +r] [ϕ +r−1]
(α)+ is is
b (r)= (1 ≤ i ≤ α), (3.22)
i [ϕ +r−1] [ϕ +r]
is ! is !
s=1 s=α+1
Y Y
j−1 n
[ϕ −1] [ϕ ]
(α)− js js
b = (1 ≤ j ≤ α), (3.23)
j [ϕ ] [ϕ −1]
js ! js !
s=1 s=α+1
Y Y
α n
[ϕ −r+1] [ϕ −r]
(α)− ks ks
b (r)= (α+1 ≤ k ≤ n), (3.24)
k [ϕ −r] [ϕ −r+1]
ks ! ks !
s=1 s=k+1
Y Y
α n
[ϕ ] [ϕ +1]
(α)+ ls ls
b = (α+1≤ l ≤n). (3.25)
l [ϕ +1] [ϕ ]
ls ! ls !
s=1 s=l+1
Y Y
Here in (3.16), (3.17) and thoughtout in Section 4 we use the short notation of the q-factorial
[x+n]! = [x+n][x+n−1]···[x+1][x]! instead the q-Gamma function, [x+n]! ≡ Γ ([x+n+1]).
q
Sketch of proof. The relations (3.15)–(3.17) are a direct consequence of the relations (3.3)–
(3.5). Relations (3.18) and (3.19) with the coefficients (3.20)–(3.25) are proved by induction on
r using the initial relations (3.9) and (3.10) with the coefficients (3.11)–(3.14) for r = 1. (cid:4)
4 Shapovalov forms on Z (gl(n + 1),gl(n))
q
Let us consider properties of the Z-algebra Z (gl(n+1),gl(n)) with respect to the involutions ⋆
q
(2.26)–(2.28), and ∗ (2.30)–(2.33).
Proposition 3. The Z-algebra Z (gl(n+1),gl(n)) is invariant with respect to the involutions ⋆
q
and ∗, besides
z⋆ = z , z⋆ = z for i = ±1,±2,...,±n, (4.1)
0 0 i −i
and
z∗ = z , z∗ = −z for i = ±1,±2,...,±n. (4.2)
0 0 i −i
Proof. Because the extremal projector p = p(gl(n)) is >-invariant, p> = p for > = ⋆,∗ (see the
formula (2.29)), it turns out that
> > > >
z = pe p, z = pe p for i= 1,2,...,n.
i i,n+1 −i n+1,i
> >
If q is circular, then e = ±e , e = ±e , where the plus belongs to the compact
i,n+1 n+1,i n+1,i i,n+1
case and the minus belongs to the noncompact case, and we obtain the formulas (4.1) and (4.2).
If q is real, then e> = ±e′ , e> = ±e′ where the root vectors e′ and e′
n+1,i i,n+1 i,n+1 n+1,i i,n+1 n+1,i
areobtainedfrom(2.10)bythereplacementq±1 → q∓1. Letusconsiderthedifferencez> −z =
−i i
p(e′ −e )p for 1 ≤ i≤ n). Substituting here (see the formulas (2.11))
i,n+1 i,n+1
e′ = e′ e −q−1e e′ , e = e e −qe e , (4.3)
i,n+1 in n,n+1 n,n+1 in i,n+1 in n,n+1 n,n+1 in
q-Analog of Gelfand–Graev Basis for the Noncompact Quantum Algebra U (u(n,1)) 9
q
and using the annihilation properties of the projector p (see (2.25)) we obtain z> −z = p(e′ −
−i i in
e )e p. In a similar way, usingexplicit formulas of type (4.3) for the generators e′ and e ,
in n,n+1 in in
we obtain z> −z = p(e′ −e )e e p. By proceeding as above, we have z> −
−i i i,n−1 i,n−1 n−1,n n,n+1 −i
z = p(e − e )e e ···e e p = 0. In a similar way, we prove that
i i,i+1 i,i+1 i+1,i+2 i+2,i+3 n−1,n n,n+1
z>−z = 0. (cid:4)
i −i
TheZ-algebraZ (gl(n+1),gl(n))with theinvolution ⋆is called thecompact real formand is
q
denoted by the symbol Z(c)(gl(n+1),gl(n)). The noncompact real form on Z (gl(n+1),gl(n))
q q
(nc)
is defined by the involution ∗ and is denoted by the symbol Z (gl(n+1),gl(n)).
q
Let p(α) be an extremal projector for Z (gl(n+1),gl(n)) satisfying the relations
q
z p(α) = p(α)z for i= 1,2,...,α,
−i i
z p(α) = p(α)z for k = α+1,α+2,...,n,
k −k
[e ,p(α)]= 0 for i= 1,2,...,n.
ii
The extremal projector p(α) depends on the index α that defines what elements are consi-
dered as “raising” and what elements are considered as “lowering”, i.e. in our case the ele-
ments z ,z ,...,z , z ,...,z are raising and the elements z ,z ,...,z ,z ,...,z
−1 −2 −α α+1 n 1 2 α −α−1 −n
are lowering. It should be stressed that the “raising” and “lowering” subsets generate disjoint
subalgebras in Z (gl(n+1),gl(n)). The operator p(α) can be constructed in an explicit form.
q
(nc)
Let us introduce on Z (gl(n+ 1),gl(n)) the following sesquilinear Shapovalov form [16].
q
(nc)
For any elements x,y ∈ Z (gl(n+1),gl(n)) we set
q
B(α)(x,y) = p(α)y∗xp(α). (4.4)
Therefore, the Shapovalov form also depends on the index α (α = 0,1,2,...,n). We fix α
(α = 0,1,2,...,n) and for each set of nonnegative integers {r} = (r ,r ,...,r ) introduce
1 2 n
(α) (nc)
a vector v in the space Z (gl(n+1),gl(n)) by the formula
{r} q
v(α) = zrα···zr1zrα+1 ···zrn . (4.5)
{r} α 1 −α−1 −n
Theorem 2. For each fixed α (α = 0,1,2,...,n) the vectors {v(α)} are pairwise orthogonal with
{r}
respect to the Shapovalov form (4.4)
B(α) v{(αr}),v{(αr′)} = δ{r},{r′}B(α) v{(αr}),v{(αr}) . (4.6)
(cid:0) (cid:1) (cid:0) (cid:1)
and
α n
[r ]![ϕ −α+r −1]! [r ]![ϕ +α+r ]!
B(α) v(α),v(α) = i i,n+1 i l n+1,l l
{r} {r} [ϕ −α−1]! [ϕ +α]!
i,n+1 n+1,l
i=1 l=α+1
(cid:0) (cid:1) Y Y
[ϕ +r −r ]![ϕ −1]! [ϕ −r +r ]![ϕ −1]!
ij i j ij kl k l kl
×
[ϕ +r ]![ϕ −r −1]! [ϕ −r −1]![ϕ +r ]!
ij i ij j kl k kl l
1≤i<j≤α α+1≤k<l≤n
Y Y
[ϕ +r −1]![ϕ +r −1]![ϕ ]
il i il l il (α)
× z , (4.7)
[ϕ +r +r ]![ϕ −1]! 0
il i l il
1≤i≤α<l≤n (cid:19)
Y
where z(α) ≡ p(α).
0
10 R.M. Asherova, Cˇ. Burd´ık, M. Havl´ıˇcek, Yu.F. Smirnov and V.N. Tolstoy
As a consequence of this theorem we obtain that the Shapovalov form is not degenerate on
(nc)
a subspace of Z (gl(n+1),gl(n)), generated by the vectors of form (4.5).
q
(c)
In the case of the compact Z-algebra Z (gl(n+1),gl(n)) the Shapovalov form B(x,y) is
q
defined by formula (4.4) where α = 0, p(0) is the standard extremal projector of the quantum
algebra U (gl(n+1)) and the involution is given by formulas (4.1). It is not difficult to see that
q
B(v{r},v{r′}) = δ{r},{r′}B(v{r},v{r}).
(0)
where v := v and
{r} {r}
n
B(v{r},v{r})= (−1)iP=1riB(0) v{(0r)},v{(0r)}
n (cid:0) (cid:1)
[r ]![ϕ −1]! [ϕ −r +r ]![ϕ −1]!
l l,n+1 kl k l kl (0)
= z .
 [ϕ −r −1]! [ϕ −r −1]![ϕ +r ]! 0
l,n+1 l kl k kl l
l=1 1≤k<l≤n
Y Y
 
5 Discrete series of representations for U (u(n,1))
q
As in the classical case [9] each Hermitian irreduciblerepresentation of thediscrete series for the
noncompact quantum algebra U (u(n,1)) is defined uniquely by some extremal vector |xwi, the
q
vector ofextremalweight4. Thisvector shouldbethehighestvector withrespecttothecompact
subalgebra U (u(n))⊕U (u(1)). Since the quantum algebra U (u(1)) is generated only by one
q q q
Cartan element qen+1,n+1, the vector |xwi should be annihilated by the raising generators eij
(1 ≤ i < j ≤ n) of the compact subalgebra U (u(n)). So the vector |xwi satisfies the relations
q
e |xwi = µ |xwi (i = 1,2,...,n+1),
ii i
e |xwi = 0 (1 ≤ i< j ≤ n),
ij
where the weight components µ (i = 1,2,...,n) are integers subjected to the condition µ ≥
i 1
µ ≥ ··· ≥ µ . Such weights can be compared with respect to standard lexicographic ordering,
2 n
namely, µ > µ′, where µ = (µ ,µ ,...,µ ) and µ′ = (µ′,µ′,...,µ′ ), if a first nonvanishing
1 2 n 1 2 n
component of the difference µ−µ′ is positive.
The component µ is also an integer. In the general case of finite-dimensional irreducible
n+1
representations of the compact quantum algebra U (u(n))⊕ U (u(1)), the weights µ=(µ ,µ ,
q q 1 2
...,µ ) and µ are not ordering. If we choose some ordering for these weights, for example,
n n+1
(µ ,...,µ ,µ ,µ ,...,µ ), then such n+1-component weights can be compared.
1 α n+1 α+1 n
(α)
The extremal vector |xwi has the minimal weight Λ := (λ ,λ ,...,λ )
n+1 1,n+1 2,n+1 n+1,n+1
where λ := µ (i = 1,2,...,α), λ := µ , λ := µ (l = α+1,...,n). The
i,n+1 i α+1,n+1 n+1 l+1,n+1 l
(α) (α)
vector |Λ i:= |xwi with the weight Λ satisfies the relations
n+1 n+1
(α)
z |Λ i = 0, for i =1,2,...,α,
−i n+1
(α)
z |Λ i= 0, for k = α+1,α+2,...,n.
k n+1
(α)
It is evident that any highest weight vector |Λ ;Λ ) with respect to the compact subalgebra
n+1 n
U (u(n)) has the form
q
|Λ(α) ;Λ ) = zrα···zr1zrα+1 ···zrn |Λ(α) i. (5.1)
n+1 n α 1 −α−1 −n n+1
4We assume that the vector|xwi is orthonormal, hxw|xwi=1.