Japan. J. Math. Vol. 19, No. 1, 1993
Finite dimensional representations of the quantum group
GLq(n;C) and the zonal spherical functions
on Uq(n-1)•_Uq(n)
By Masatoshi NOUMI, Hirofumi YAMADA and Katsuhisa MIMACHI
(Received March 16, 1992)
•˜ 0. Introduction
Representation theory of quantum groups is now studied in connection with
various fields of mathematics. In this paper, we discuss representations of quantum
groups and their homogeneous spaces from the viewpoint of spherical functions. In this direction, the spherical functions on the quantum group SUq(2) are studied
by Vaksman-Soibelman [VS1], Koornwinder [K1, 2], Masuda et al. [MM1, 2] and
Noumi-Mimachi [NM1, 2].
This paper, a detailed version of the announcement [NYM], deals with the quantum general linear group GLq(n;C) and its "compact" real form Uq(n), the quantum unitary group; the objects of our study are their algebras of functions A(GLq(n;C)) and A(Uq(n)). The main points we will discuss are the following:
(1) An analogue of the Borel-Weil construction of finite dimensional irre ducible representations of GLq(n;C).
(2) Irreducible decomposition of A(GLq(n;C)) as a two-sided A(GLq(n;C)) - comodule and the orthogonal decomposition of A(Uq(n)).
(3) Invariant functional and the zonal spherical functions on the quantum homogeneous space Uq(n-1)•_Uq(n).
We will give here a brief description of the contents without precise definition of the terminology.
(1) For each dominant integral weight A, we realize the finite dimensional irreducible representation of GLq(n;C) of highest weight A as a space of relative invariants in A(GLq(n;C)) with respect to the Borel subgroup Bn , and also its basis consisting of standard monomials (Theorem 2.4 and 2.5). This gives a quantum analogue of the space of sections of the associated line bundle on the flag manifold.
(2) We give the irreducible decomposition of A(GLq(n;C)) as a two-sided A(GLq(n;C))-comodule (Theorem 2.11), by using the complete reducibility the orem of finite dimensional Uq(si(n))-modules, due to Rosso [R] and Lusztig [L]. 32 M. NOUMI, H. YAMADA and K. MIMACHI
This leads to the existence of an invariant functional on A(GLq(n;C)) and the
parametrization of all finite dimensional irreducible A(GLq(n;C))-comodules by means of the dominant integral weights (Theorem 2.12). The coordinate ring
A(Uq(n)) is defined as a Hopf *-algebra (A(GLq(n;C)),*). We introduce two Her
mitian forms on A(Uq(n)) by means of the invariant functional. The irreducible
decomposition of A(Uq(n)) is orthogonal with respect to these Hermitian forms
(Theorem 3.4). This statement on Uq(n) is an algebraic reformulation of results of Woronowicz [W].
(3) The quantum homogeneous space Uq(n-1)•_Uq(n) is a quantum (2n-1) - sphere. The algebra of functions on this quantum homogeneous space is defined as
the *-subalgebra of all left Uq(n-1)-invariants in A(Uq(n)) . We also give its irre
ducible decomposition as a right A(Uq(n))-comodule (Proposition 4.3). Furthermore we determine the zonal spherical functions on Uq(n-1)•_Uq(n) explicitly in terms
of the little q-Jacobi polynomials (Theorem 4.7). It is remarkable that the algebra
of functions on the quotient space Uq(n-1)•_Uq(n)/H by the diagonal subgroup
H of Uq(n) falls into a commutative algebra. On this commutative subalgebra, the
invariant functional on Uq(n-1)•_Uq(n) is expressed by an iterated Jackson integral
(Theorem 4.6). After we completed this work, we received preprints by Taft-Towber [TT],
Hashimoto-Hayashi [HH] and Vaksman-Soibelman [VS2]. The first two, [TT] and
[HH], also deal with the standard monomial bases for finite dimensional irreducible representations of GLq(n;C). Our arguments are different from theirs in the sense
that we give a characterization of relative B-n-invariants (Theorem 2.2). The last
one [VS2] is also devoted to a harmonic analysis on the odd dimensional quan
tum spheres Uq(n-1)•_Uq(n). In our formulation, the zonal spherical functions are
realized as elements of the algebra of functions.
Throughout this paper we always assume that the parameter q •¸ E C•_{0} is
not a root of unity. When the real form Uq(n) is concerned, we assume that q is a
real number with 0<|q|<1.
Contents
•˜ 1. Quantum group GLq(n;C).
1.1. Quantum semigroup Matq(n;C).
1.2. Quantum minor determinants.
1.3. Quantum group GLq(n;C).
1.4. Quantum universal enveloping algebra Uq(g.?(n;C)).
1.5. A monomial basis for A(Matq(m,n;C)).
•˜ 2. Finite dimensional representations of GLq(n;C).
2.1. Quantum G-spaces and relative invariants.
2.2. Representations of GLq(n;C). Quantum group GLq(n;C) and zonal spherical functions 33
2.3. Irreducibility of the representations A(B 1 GLq (n; C); zA).
2.4. Irreducible decomposition of A(GLq(n;C)).
•˜ 3. Quantum group Uq(n) and the invariant functional.
3.1. Quantum group Uq(n) and its unitary representations.
3.2. Invariant functional.•˜
4. Zonal spherical functions on Uq(n-1)•_Uq(n).
4.1. Quantum homogeneous space Uq(n-1)•_Uq(n).
4.2. Invariant functional on A(Uq(n-1)•_Uq(n)).
4.3. Zonal spherical functions on Uq(n-1)•_Uq(n).
•˜ 1. Quantum group GLq(n;C)
1.1. Quantum semigroup Matq(n;C)
For any positive integers m and n, we denote by Mat(m,n,C) the space of all
m•~n complex matrices and by Mat(n;C) the space of all n•~n square matrices. We introduce the quantum semigroups Matq (n;C) and their action on the quantum
spaces Matq(m,n;C). Hereafter we fix a nonzero complex number q.
By definition, the coordinate ring A(Matq(m,n;C)) is the C-algebra generated
by mn letters x2j (1 < i < m, 1 < j < n) with fundamental relations
(1.1.a)
(1.1.b)
(1.1.c)
When m=n, we write simply Matq(n;C)=Matq(n,n;C). One can directly verify that there exist unique C-algebra homomorphisms L : A(Matq(rn, n, C)) -~ A(Matq(m, r, C)) ® A(Matq(r, n, C)) and : A(Matq(n, C)) --~ C satisfying
(1.2)
It is clear that each A(Matq(n;C)) becomes a bialgebra with coproduct •¢ and
counit ƒÃ, and that each A(Matq(m,n;C)) has the natural structure of two-sided
(A(Matq(m;C)), A(Matq(n;C)))-comodule. We remark that the commutation relations (1.1) of the algebra A(Matq(n;C))
are related to the so-called the constant R-matrix of type An-1(see[RTF]). Define
the matrix R •¸ Mat(n;C) ® Mat(n;C) by
(1.3) 34 M. NOUMI,H. YAMADAand K. MIMACHI where Eij are the matrix units. It is well-knownthat this matrix R satisfies the Yang-Baxter equation (see[J1]). With the matrix notation
the commutation relations (1.1) are simply expressed in the form
(1.4) where X ® X = xzaxkQE2~ ® Ei As preliminaries to the representation theory of the quantum group GLq(n;C), we investigate some fundamental comodules over the bialgebra A(Matq(n;C)). In the following, we will sometimes write Mn for (Matq(n;C). Let S be the C-algebra generated by n letters x1, ..., xn with fundamental relations
Then one can directly show that there exists a unique C-algebra homomorphism RMn: S --* S ® A(Matq(n, C)) such that
(1.5)
Then the algebra S=C [x1,..., xn] becomes a right A(Matq (n;C))-comodule with this homomorphism RM. Note that the algebra S has a graduation S = ~r>o Sr, where Sr denotes the subspace of S of all homogeneous elements of degree r. It is clear that the subspaces Sr(r•¸N) are right A(Matq(n;C))-subcomodules of S.
We call the right comodule Sr the symmetric tensor representation of the quantum semigroup (Matq(n;C) of degree r. The comodule S1 = Cx1® ... Cxn is also called the vector representation.
Let E be the C-algebra generated by n letters y1,...,yn with (anti-) commuta tion relations
Then one can directly show that there exists a unique C-algebra homomorphism RMn : E --~E ® A(Matq(n; C)) satisfying
(1.6) and that E becomes a right A(Matq(n;C))-comodule with this coaction. Similarly to the case of S, the right A(Matq(n;C))-comodule E decomposes into a direct Quantumgroup GLq(n;C) and zonalspherical functions 35 sum E = ® o Er of subcomodules according to the homogeneous degree. The right A(Matq(n;C))-comodule Er is called the alternating tensor representation of (Matq(n;C) of degree r; the comoduleE1 is isomorphic to the vector representation S1. Note that the subspace Er of degree r has a C-basis {Yji• • • yjr;1 We will see later that the representations Sr(r•¸N) and Er (0 < r < n) are irreducible. REMARK. The "exterior algebra" E=C[y1,...,yn] also has a structure of left A(Matq(n;C))-comodule whose coaction is given by the C-algebra homomor phism LM : E -~ A(Matq(n; C)) ® E such that Then the subspaces Er (0 < r < n) are again left subcomodules of E. •¡ 1.2. Quantum minor determinants We now investigate the matrix elements of the right comodule Er (0 < r < n). For each subset J of {1, ... ,n}, we denote by yJ the element yj1...Yjr. of Er, where J={j1,...,jr} and j1<... A(Matq(n;C)) of Er with respect to the C-basis {yJ;#J=r} by the formula (1.7) We also use the notation l = if I = {i1 < (1.8) Here ~r is the permutation group of the set {1,...,r} and, for each a e Car, £(a) = £(a(1)..... a(r)) stands for the number of inversions involved in ƒÐ: This element l is called the quantum r-minor determinant of the matrix X = (x1)1<,< 2 jn with respect to rows I and columns J. Since ƒÄIJ(#I=#J=r) are the matrix elements of an A(Matq(n;C))-comodule, they satisfy the relations 36 M. NOUMI, H. YAMADA and K, MIMACHI (1.9) and If r=n and I=J={1,...,n}, then the quantum minor determinant ƒÄIJ is denoted by detq=detq(X) and is called the quantum determinant of X=(xij)i,j: (1.10) As for the quantum determinant detq, formula (1.9) implies (1.11) We remark that the quantum determinant detq has an alternative expression (1.12) This is a consequence of the following simple fact: If ƒÐ,r E„qCan and r = a-1, then xa(1) ,1... = x1 T(1) ... xn T(n) and £(a) = ~(T). REMARK. As for the left comodule structure of Er, one sees that the matrix elements with respect to the C-basis {yI;#I=r} are eventually the same as those of the right comodule Er. Namely one has for each I with #I=r. This follows from the fact that the quantum r-minor determinants ƒÄIJ are also written in the form By using the alternatingtensor representationE = ®r-® Er, we give below some fundamental formulas for the quantum r-minor determinants. For two subsets I and J of {1,...,n}, we define the symbol sgnq(I;J) by if if where Quantum group GLq(n;C) and zonal spherical functions 37 PROPOSITION1.1 (Laplace expansions). Let r, r1 and r2 be positive integers with 1 (1.13.a) (1.13.b) where the summation ranges over all partitions I1•¾I2=I of I such that #Iv= rv(v=1,2). PROOF. First note that yJ1yJ2=sgnq(J1;J2)yJ. Then formula (1.13.a) is obtained from (1.14) by comparing the coefficients of yI in (1.14). Formula (1.13.b) is proved similarly by considering the left comodule structure of E=C[y1,...,yn]. •¡ As special cases, we obtain COROLLARY. For each 1 (1.15.a) (1.15.b) (1.16.a) (1.16.b) where k = { 1, ... , k -1,k + 1, ... , n}. •¡ These formulas (1.15) and (1.16) will play an important role in the rest of this paper. 38 M. NOUMI, H. YAMADA and K. MIMACHI We define the (i,j)-cofactor xij of the matrix X=(xij)i,j by (1.17) for and call the matrix X=(xji)i,j in Mat(n;A(Matq(n;C))) the cofactor matrix of X=(xij)i,j. Then formulas (1.15.a) and (1.15.b) imply the following identity in Mat(n;A(Matq(n;C))): where Idn denotes the identity matrix of size n. From this, one sees that X.detq= XXX=detq.X. This means that the quantum determinant detq commutes with all (1 < i, j < n), hence belongs to the center of A(Matq(n;C)). By using Proposition 1.1, one can prove the generalized Plucker relations for quantum minor determinants. PROPOSITION 1.2 (Generalized Plucker relations). Let w and rv(v =1, 2) be positive integers with 4, + rv < n. Let Iv and Jv be subsets of {1,...,n} such that #Iv = .w + rU and #Jv = .w. Let K be a subset of {1,...,n} with #K=r1+r2. If #(I1•¾I2) (1.18.a) (1.18.b) where the summation ranges over the set of all partitions K'•¾K"=K of K with #K'=r1 and #K"=r2. PROOF. By using Proposition 1.1, the left hand side of (1.18.a) is calculated as follows: Any pair of sets (I"1;I"2) with I"v•¼Iv and #I"v=rv must intersect since #(I1•¾I2)< r1+r2. This means that the above expression is equal to zero. Formula (1.18.b) is similarly proved. •¡ Quantum group GLq(n;C) and zonal spherical functions 39 Above formulas correspond to the so-called Pluckier relations when I1=I2, J2=ƒÓ and r1=1(or I1=I2,J1=ƒÓ and r2=1). Let (j1,...,jr-1) and (ko,...,kr) be two strictly increasing sequences in (1,...,n). Then one has (1.19.a) (1.19.b) In the case of q=1, both (1.19.a) and (1.19.b) give the defining equations of the Grassmann manifold consisting of all r-dimensional vector subspaces of an n dimensional vector space. 1.3. Quantum group GLq(n;C) We introduce the quantum group GLq(n;C) followingthe frameworkof [RTF]. Its coordinate ring A(GLq(n;C)) is defined by adjoining the inverse det-1qof the quantum determinant to the algebra A(Matq(n;C)); the relations to be added to (1.1) are (1.20.a) (1.20.b) Since the quantum determinant detq belongs to the center of A(Matq(n;C) ), the algebra A(GLq(n;C)) is also written as (1.21) Hence, we have (1.22) regarding A(Matq(n;C)) as a C-subalgebra of A(GLq(n;C)). The algebra A(GLq(n;C)) has as structure of bialgebra induced from that of A(Matq(n;C)) by the conditions (1.23)®(detq 1) = detq 1 ® detq 1 and E (detq 1) =1. Moreover one can show that there exists a unique C-algebra antihomomorphism S : A(GLq(n; C)) -~ A(GLq(n, C)) such that (1.24) 40 M. NOUMI, H. YAMADA and K. MIMACHI In fact, as to the cofactor matrix X=(ji)i,j defined by (1.17), formula (1.4) N implies that the matrix (Ida ® X) (X ® I d1) commutes with the R-matrix R of (1.3), which guarantees the existence of S. Note also that S(detq) = det-1q. With this antipode S, the bialgebra A(GLq(n;C)) becomes a Hopf algebra. We recall that the antipode satisfies (1.25)poS=To(S®S)o® and EoS=E, where T: A(GLq(n;C)) ® A(GLq(n;C)) -~ A(GLq(n; C)) ® A(GLq(n; C)) stands for the flip By a quantum subgroupof a quantum group G, we mean a quantum group G', endowed with a surjective homomorphism 7rG,: A(G) -* A(G') of Hopf algebras. We define the diagonal subgroup Hn of GLq(n;C) as follows. Its coordinate ring A(Hn) is the commutative ring (1.26) of Laurent polynomials in n variables (t1,...,tn) with canonical homomorphism 7tH : A(GLq(n; C)) -~ A(H) defined by rrHn(xjj) = b2~tZ(1< i, j (1.27.a) (1.27.b) with commutation relations induced from (1.1). Note that the coordinates z11,..., znn on the diagonal commute with each other. The canonical homomorphism will be denoted by 71B~: A(GLq(n; C)) -> A(B). n Right or left A(GLq(n;C))-comodules will be called representations of the quantum group GLq(n;C). The coproduct •¢:A(GLq(n;C))•¨A(GLq(n;C)) A(GLq(n;C)) endows the coordinate ring A(GLq(n;C)) with a natural structure of two-sided A(GLq(n; C))-comodule, which will be referred to as the regular repre sentation of GLq(n;C). Note also that, via the natural injective homomorphism of bialgebras A(Matq(n;C)) A(GLq(n;C)), any A(Matq(n;C))-comodule can be regarded as an A(GLq(n;C) )-comodule. In particular, the symmetric and the alter nating tensor representations described in Section 1.1 give rise to representations of the quantum group GLq(n;C). Quantum group GLq(n;C) and zonal spherical functions 41 1.4. Quantum universal envelopingalgebra Uq(g !(n; C) ) Herewe discussthe connectionbetween the quantumgroup GLq(n;C) and the quantumuniversal enveloping algebra Uq(gl?(n; C)) of Jimbo [J1]or Drinfeld [D].When we deal with Uq(g~(n; C)), wealways fix a squareroot q2 of q. Wedenote by Lnthe freeZ-module of rank n withcanonical basis {e1,• .. , Ln = Ze. Wefix a symmetricbilinear form (, ) : LnX Ln -~ Z on Lndefined by (e27Ej) = bjj. An elementof Ln willbe calledan integralweight. We also use the extension2 Ln of Ln and the inducedbilinear form ( , ) on 2Ln. FollowingJimbo [J2], we introduce the quantumuniversal enveloping algebra Uq(gt(n; C)). Thisis the C-algebradefined by the generatorsek, f k (1 < k < n) and qA(A E 2Ln) and the followingfundamental relations: (1.28.a) (1.28.b) (1.28.c) (1.28.d) (1.28.e) (1.28.f) (1.28.8) This algebra also has the structure of a Hopf algebra with the following coproduct •¢ , counit ƒÃ and antipode S: (1.29.a) (1.29.b) (1.29.c) 42 M. NOUMI, H. YAMADA and K. MIMACHI There exists a natural pairing between the two Hopf algebras A(GLq(n;C) ) and Uq (gI?(n; C)). Given two Hopf algebras A and U over C, we say that a C bilinear form (,):U•~A•¨C is a pairing of Hopf algebras if it satisfies the following conditions: For any a,b•¸U and ƒÓ ,ƒÕ•¸A, (1.30.a) (1.30.b) (1.30.c) Here we used the same notation (, ) for the induced bilinear form U®U x A®A -~ C. PROPOSITION1.3. There exists a unique pairing of Hopf algebras (1.31) satisfying the conditions (1.32.a) (1.32.b) (1.32.c) (1.32.d) Proposition 1.3 can be checked by direct calculations. In the rest of this paper, we regard the elements of Uq(ge(n,C)) as linear functionals on A(GLq(n;C)) through the pairing of Proposition 1.3. If V is a right A(GLq(n;C))-comodule (resp. left A(GLq(n;C))-comodule) with structure map ping RG: V -~ V ® A(GLq(n, C)) (resp. LG : V -> A(GLq(n; C)) ® V), then V has a left ( resp. right) module structure over Uq(gI?(n,C)) defined by (1.33) for all a E Uq(g~(n, C)) and v•¸V. In particular the algebra A(GLq(n;C)) becomes a bimodule over Uq(ge(n; C)). The action of Uq(g~(n; C)) on A(GLq(n;C)) is described as follows: If a•¸Uq (ge(n; C)) and 0(a) = > ai ® a?, then one has (1.34) a. (cps)= (a2 °~P)(a 2 ' b) and (cps)° a = a) (b ' a2), 2 Z Quantum group GLq(n;C) and zonal spherical functions 43 for any cp,L' E A(GLq(n, C)). The action of the generators qA,ek, f k are given by (1.35.a) (1.35.b) (1.35.c) 1.5. A monomial basis for A(Matq(m,n;C)) Here we give an explicit monomial basis for the algebra A(Matq(m,n;C))= C [xia ;1 define a monomial xA in A(Matq(m;n;C)) by where the factors are arranged in the lexicographic order of the indices (i,j). THEOREM 1.4. The set of monomials {xA;A•¸Mat(m,n;N)} forms a C basis for the vector space A(Matq(m,n,C)),i.e., (1.36) This theorem can be proved by using the so-called "Diamond Lemma" [B]. For the proof of Theorem 1.4, we will introduce a total order among the words (in the free associative C-algebra) on the letters x2~ (1 (aij)i,j•¸ Mat(m,n;N), we associate a sequence of non-negative integers (1.37) We compare matrices A in Mat(m,n,N) by the lexicographic order of these se quences (1.37) in N1+mn. With this total order, Mat(m,n,N) becomes a well ordered set. Next we introduce a total order _??_among the words on the letters i (1.38) be a word in the free associative C-algebra on the letters x2~ (1 To such a ƒÓ, we associate a matrix A=(aij)i,j in appearing in (1.38). We compare 44 M. NOUMI, H. YAMADA and K. MIMACHI the words ƒÓ first by the total order _??_ of matrices A=(aij)i,j defined as above, and then by the lexicographic order of sequences (i1,j1,i2,j2,...,id,jd) in N2d. For example, one has The 12 words with the same A as X11X12X21are ordered as follows: With this total order _??_,the commutation relations (1.1) give rise to the fol lowing reduction system: (1.39.a) (1.39.b) (1.39.c) Note that, in either case, the words appearing on the right hand side are lowerunder _??_than the one on the left hand side. To prove Theorem 1.4, it sufficesto check that all the ambiguities in this reduction system are resolvable (see [B, Theorem 1.2]). There are 24 types of configuration of three indices to be checked, among which 9 are trivially resolved. We will not give all the details of this routine but for an example which shows how the non-trivial ambiguities are resolved. The word "fca" in the configuration is reduced to a single standard form as follows: Quantum group GLq(n;C) and zonal spherical functions 45 Here underlines indicate the submonomials to be reduced. By Theorem 1.4, an element ƒÓ in A(Matq(m,n;C)) can be written uniquely in the form (1.40) This expression is referred to as the standard form of ƒÓ. If cA•‚0, we say that ƒÓ has the leading term cAxA. The following lemma is obtained as a corollary to the proof of Theorem 1.4. LEMMA 1.5. a) For any monomial cp = x1 1x22 • • ' x2d7d in A(Matq(m,n; C)), define a matrix A=(aij)i,j in Mat(m,n;N) by setting aij to be the number of xij appearing in the expression of ƒÓ. Then the leading term of ƒÓ is in the form q-QxA, where l is the number of pairs (ƒÊ,v) (1 < ,u < v < d) such that (it, > iU, jµ = w) or (iµ = > JU)) b) The product xAxB of two monomials has the leading term q-lxA+B with £ = Ik >>(a1kbkj j+ ak bkj ), where A=(aij)i,j and B= (bij)i,j• Recall that the quantum determinant detq of (1.10) belongs to the center of A(Matq(n;C)). We can apply Theorem 1.4 to prove the following statement due to [ RTF]. THEOREM 1.6 ([RTF]). The center of A(Matq(n;C)) is generated by detq and is isomorphic to the polynomial ring in one indeterminate. PROOF. First note that the quantum determinant detq has the leading term xIdn. Hence the leading term of the power (detq)m(m•¸N) is given by xm.Idn . Let ƒÓ be a nonzero element in the center of A(Matq(n;C)) with leading term cxA(c•‚0). Then one has xa~(p = Spxap for all 1 < c~, /3 < n. By Lemma 1.5.b), one has (1.41) modulo lower order terms under _??_, where lƒ¿ƒÀ _ ~k (1.42) where ra1s _ >Ik>c ak/3 + ~k>Q aak. Comparing the leading terms of (1.41) and (1.42), one has qlƒ¿ƒÀ-rƒ¿ƒÀ=1, hence lƒ¿ƒÀ=rƒ¿ƒÀ since q is not a root of unity. This means that (1.43) 46 M. NOUMI, H. YAMADA and K. MIMACHI for all 1 < a, j3 < n. Formula (1.43) for (ƒ¿,ƒÀ)=(1,1) implies ak1=a1k=0 for k>1. Eventually formula (1.43) for (ƒ¿,ƒÀ) with ƒ¿=ƒÀ implies that the matrix A is diagonal. Moreover formula (1.43) for (ƒ¿,ƒÀ) with ƒ¿•‚ƒÀ implies that the diagonal components are all equal, i.e., A= m.Idn for some m•¸N.SetƒÕ=ƒÓ-c(detq)m. Then ƒÕ also belongs to the center and its leading term becomes strictly lower than xA. By the induction with respect to the well-ordering of Mat(n;N), we conclude that ƒÓ is a polynomial in detq. It also follows from Theorem 1.4 that the elements (detq)m(m•¸N) are linearly independent over C, since they have different leading terms. This shows that the center of A(Matq(n;C)) is isomorphic to a polynomial ring in one indeterminate. •˜ 2. Finite dimensional representations of GLq(n;C) 2.1. Quantum G-spaces and relative invariants Before discussing representations of GLq(n;C), we introduce some notions on quantum G-spaces and relative invariants. Let G be a quantum group with coordinate ring A(G). Then a quantum space X is called a quantum left G-space if the coordinate ring A(X) of X has a structure of left A(G)-comodule LG : A(X) * A(G) ® A(X) such that LG is a C-algebra homomorphism. An element x of A(G) is called a linear character of G if For a given linear character ƒÔ of G, an element ƒÓ of A(X) is called a left relative G-invariant with character ƒÔ if LG (cp) = x ® co. We denote by (2.1) the subspace of all left relative G-invariants in A(X) with character ƒÔ.IfƒÔ=1, we simply denote by (2.2) the subalgebra of all left G-invariants in A(X). The notions of right G-spaces and right relative G-invariants are defined similarly. We use the notations (2.3) (2.4) if X is a right G-space and ƒÔ is a linear character of G. Quantum group GLq(n;C) and zonal spherical functions 47 Suppose that a quantum space X has the structure of a left G-space and a right G'-space over quantum groups G and G'. Then X is called a two-sided (G, G') space if the structure mappings LG : A(X) -* A(G) ® A(X) and RG, : A(X) --f A(X) ® A(G') satisfy the compatibility condition We remark that, if ƒÔ and ƒÔ' are linear characters of G and G' respectively, then A(G•_X;ƒÔ) is a right A(G')-subcomodule of A(X) and that A(X/G'; ƒÔ') is a left A(G)-subcomodule of A(X). The subalgebra of all (G,G')-invariants in A(X) is denoted by (2.5) 2.2. Representations of GLq(n;C) In what follows, we sometimes write Gm= GLq(m;C) for simplicity. The al gebra A(Matq(m,n;C)) has a natural structure of (GLq(m;C), GLq(n;C))-space with coactions LGm:A(Matq(m,n;C))•¨A(GLq(m;C)) ® A(Matq(m,n;C)) and RGn : A(Matq(m, n, C)) -~ A(Matq(m, n; C)) ® A(GLq(n; C)) induced from the "coproduct" •¢ on A(Mat q(m,n;C)). By considering the action of the Borel sub groups of GLq(m;C) we construct finite dimensional representaitons of GLq(n;C) on the spaces of the relative invariants with respect to them. Let A = A1r1 + ... + AmEm be an element of Lm. We define linear characters t'1 and zA of the diagonal subgroup Hm and the Borel subgroups B•}m of GLq(m;C) by (2.6) respectively. Note that to • tµ = to+µ and zA • zµ = zA+~ . As we have remarked, the subspaces of relative invariants A(Hm•_Matq(m,n;C); tA) and A(Bm\Matq(m, n; C); zA) are right A(GLq(n;C))-subcomodules of A(Matq(m,n;C)). Note that the sum of the subspaces A(Bm\Matq(m, n, C); zA)(A E Lm) form an Lm-graded sub algebra of A(Matq(m,n;C)) by the property zA • zµ = zA+µ. First we remark that the monomials xA(A•¸Mat(m,n;N)) in A(Matq(m,n; C)) are relative invariants with respect to the two-sided action of (Hm,Hn) on Matq(m,n;C). For each matrix A = (ajj)i respectively. Then the monomial xA has the relative invariance (2.7) LHm(XA) = ta~A)® xA and RHn (xA) = xA ® t~~A~, 48 M. NOUMI, H. YAMADA and K. MIMACHI where LHm=(nHm®id)oLGm and RHn=(2d®7rHn)oRGn • The relative invariancs with respect to the Borel subgroup B-m are expressed in terms of quantum minor determinants. In the following, we use the abbreviation J = 313 r (resp. c = 3i3) to refer to the quantum r-minor determinant 12 r (resp. ~1~2 fir) where J={j1<... (2.s) For each positive integer r with 1 < r < m, we define the fundamental weight Ar by •Èr=ƒÃ1+...+ƒÃr. Then, for any subset J of {1....,m} with #J=r, the quantum r-minor determinant ƒÄJ has the relative invariance (2.9) with respect to the left action LB _ (71Bm® id) o LGm of the Borel subgroup B-m.This follows from (2.8) and the fact that nBm(ci) = z' if I = {1,... , r} and ~Bm(j) = 0 otherwise. Since LB-mis a C-algebra homomorphism, it is easily seen that any product of quantum minor determinants with #J8 = ms (1 < s < ~) has the relative invari ance (2.10) LB-() = zA ® with A = Amy + ... + Am e This means that ƒÓ=J1J2 • • •J~ E A(Bm1 Matq(m, n; C); zA) for A _ Aml + + Ame • Let A be an integral weight in Lm in the form (2.11) This condition is equivalent to saying that A is written as (2.12) A_A1E1+...+AmEm with Ai>A2>...>Am>Q, Note that the sequence (ƒÉ1,...,ƒÉm) is obtained from (m1,...,ml) by transposing the corresponding Young diagram, and vice versa (2.13.a) Quantum group GLq(n;C) and zonal spherical functions 49 (2.13.b) £=A1 and ms=#{r; 1 For such a ƒÉ, let T=(Trs; 1 {1,...,n}. If T satisfies the conditions (2.14.a) Tr-1,s < Tr,s for 1 (2.14.b) Tr,s-1 Tr,s for 1 {1..... n}. For each semi-standard tableau T=(Trs)r,s in SSTabn(ƒÉ), we define the standard monomial ƒÄT indexed by T as the product of quantum minor determinants (2.15) where Js = {T1,5,... , Tms,s} for 1 < s < ~. Here we give a remark on the linear independence of standard monomials. Let A = A1e1 + ... + AmEm E Lm be an integral weight with Al > ... Am > 0. For a semi-standard tableau T=(Trs)r ,s in SSTabn(ƒÉ), define a matrix A=(aij)i ,j•¸Mat(m,n;N) by setting aij to be the number of j's appearing in the i-th row of the tableau T: (2.16) Note that ƒÉ is recovered as the row sum ƒ¿(A). Then by Lemma 1.5 one sees that the standard monomial ƒÄT indexed by T has the leading term xA. This correspondence T A gives a bijection between the semi-standard tableaux T in SSTabn(ƒÉ) and the matrices A=(aij)i,j in Mat(m,n;N) satisfying (2.17.a) if (2.17.b) ai_1,k > ai,k for all 1 Combining this fact with Theorem 1.4, we have PROPOSITION 2.1. Let A = A1r1 + • • • + AmEm be an integral weight such that Al _ > ... >_ Am >_ 0. Then the standard monomials ƒÄT indexed by the semi-standard tableaux T in SSTabn(ƒÉ) are linearly independent over C. •¡ THEOREM 2.2. Let A _ A1E1 + ... + AmEm be any integral weight in Lm. Then, for any nonzero element ƒÓ in A(Hm•_Matq(m,n;C);tƒÉ), the following three conditions are equivalent: 50 M. NOUMI, H. YAMADA and K. MIMACHI a) cp E A(Bm\Matq(m, n, C); zA), i.e., LB_() = zA ®cp® b) ƒÓ is annihilated by the right action o f the elements fk (1 Uq (gt(m; C)):cp.fk=0 for all 1 c) Al>...>"m>0 and ƒÓ is a linear combination of the standard mono mials ƒÄT indexed by the semi-standard tableaux T in SSTabn(ƒÉ). •¡ First we will show the implication a)•Ëb). The linear functionals fk: A(GLq(m;C)) -~ C (1 < k < m) are factored through the canonical homomor phism 7tBm : A(GLq(m, C)) -k A(B). Denoting the corresponding linear function als A(B-m)•¨C by the same symbols fk, one has fk (zƒÉ)=0 for all ƒÉ•¸Lm. Hence co. f k = 0 for 1 < k the implication c)•Ëa). Therefore we have to prove b)•Ëc). LEMMA 2.3. Let A _ A1r1 + ... + AmEm by any integral weight in Lm. Let cp be an nonzero element of A(Hm•_Matq(m,n;C); tƒÉ) and suppose that its leading term has the form cxA (A•¸Mat(m,n;N),c•‚0). If ƒÓ • fk=0 for all 1 < k then the matrix A satisfies condition (2.17). •¡ Lemma 2.3 implies b)•Ëc) . In fact, let ƒÓ be a nonzero element in A(Hm•_Matq(m,n;C);tƒÉ) with leading term cxA. If ƒÓ • fk = 0 (1 < k < m), the matrix A satisfies conditions (2.17) by Lemma 2.3. Since ƒÉ is the row sum of A, one has Ai > a1 ,2+az,i+1+' ..+a2,n > A2+1• This shows Al > ... > Am > 0. By condition (2.17), one can find a semi-standard tableau T in SSTabn (ƒÉ) such that the standard monomial ƒÄT has the leading term xA. Set ƒÕ=ƒÕcƒÄT. Then ƒÕ also satisfies the equations ƒÕ.fk= 0 (1 < k the induction with respect to the well-ordering of Mat (m,n;N), we conclude that cp is a linear combination of standard monomials ƒÄT (T•¸SSTabn(ƒÉ)). This proves b)•Ëc). PROOF OF LEMMA 2.3. We will use the induction on m. Suppose that the element ƒÓ in A(Hm•_Matq(m,n;C);tƒÉ) has the leading term cxA(c•‚0) with A= (aij)i,j•¸ Mat(m,n;N). By Theorem 1.4, we have the unique expression Here ƒÓƒ¿1...ƒ¿n are elements of A(Matq(m-1,n;C)) = C[x2a, 2 < i < m, l < j < nJ and ƒÓƒ¿1...ƒ¿n;ƒÀ1...ƒÀn are elements of A(Matq(m-2,n;C)) = C[x1j; 3 < i < m,1 < j < n]. Note that the element ƒÓa11...a1n;a21...a2n has the leading term cxA', where ) A'=(aij)3 (a11,..., a1n;a21,...,a2n).In the sequel, we use the notation of multi-indices Quantum group GLq(n;C) and zonal spherical functions 51 Cl=(Ctrl,...,cn),3=(i3,...,/3n)ENn.Putx =xi ...xn and 0a=(pal...an, = 7'al "°an;I31"'I3 ' By (1.35) we have for Since cp. fk = 0 for 2 < k < m, the elements ~a E C[xzj; 2 < i < m, l < j < n] satisfy the equations ƒÓƒ¿.fk=0 for any ƒ¿=(ƒ¿1,...,ƒ¿n) and 2 < k < m. By the induction hypothesis, the power matrix of the leading term of ƒÓƒ¿ satisfies the con dition (2.17) for any ƒ¿=(ƒ¿1,...,ƒ¿n). In particular the matrix (ajj)2 satisfies (2.17). CLAIM 1. a21=0 and all _??_ a22. To prove Claim 1, we use the expression where The right action of f1 on ƒÓ reads where [n]q2=1+q2+...+q2(n-1) and the symbol C stands for some half-integers which we do not need to make explicit. Since the left hand side is equal to zero by the assumption, one has (2.18) Choosing ƒ¿1=a11+1, one obtains This means that ƒÕa11;ƒÀ1+1=0 if ƒÀ_??_0. On the other hand, one knows that ƒÕa11;a21•‚0. Therefore we have a21=0. From (2.18) one sees by induction that 52 M. NOUMI, H. YAMADA and K. MIMACHI for any l•¸N, where {]q2!~_ [1]q2 []q2.~If 0.fi 0, then all > £. Therefore, to show all > a22, we have only to check that +all;o2 f 122 + 0. Expand ƒÕa11;0 in the form where the summation in the second term ranges over all (ƒ¿2,...,ƒ¿n;ƒÀ2,...,ƒÀn) lower than (a12,...,a1n;a22,...,a2n) with respect to the total order _??_. Applying fa221 to ƒÓa11;0 from the right, we see that the leading term of ƒÓa11;0•Efa221 is equal to This says that ƒÓa11;0• fa221•‚0 and that all > a22. Claim 2 is a special case of Lemma 2.3. To prove this, we suppose that (a11,..., a1n;a21,...,a2n)=(a11,0,...,0,a1k,...,a1n;0,a22,...,a2n). If an index (ƒ¿,ƒÀ)=(a11,ƒ¿2,...,ƒ¿n,0,ƒÀ2,...,ƒÀn) with ƒ¿=all is lower than (a11,0,...,0,a1k,...,a1n;0,a21,...,a2n), then ƒ¿1=...=ƒ¿k-1=0 and ƒ¿k _??_ a1k. Hence ƒÓa11;0 has the form where the summation ranges over all (ƒ¿k,...,ƒ¿n;ƒÀ2,...,ƒÀn)_??_(a1k,...,a1n;a22, ...,a2n). Applying fl1 to ƒÓ11;0, one has (2.19) where the summation runs over all (ƒ¿k,...,ƒ¿n,ƒÀ2,..., ƒÀn)_??_(a1k,...,a1n,a22, ...,a2n) and (v2,...,vn) with v2+...+vn=l,vj_??_ƒÀj(2 < j < n). Here we used the notation of Gauss' multinomial coefficient If l= a22+...+a2k, then one sees that, in (2.19), the coefficient of Quantum group GLq(n;C) and zonal spherical functions 53 does not vanish. In fact, the only index in the summation (2.19) that has effect on this term is (ƒ¿k,...,ƒ¿n;ƒÀ2,...,ƒÀn;v2,...,vn)=(a1k,...,a1n;a22,...,a2n;a22, ...,a2k,0,...,0). This argument guarantees that ~'aii;o ' fl 0 for £ = a22 + + a2k; hence all > a22 + ... + a2k• CLAIM 3. all + ... + al ,k-1 ~ a22 + ... + a2k for 2 k n. The general case can be reduced to Claim 2 by using the left action of e1,...,en-l. Recall that the indices (a11,...,a1n;a21,...,a2n) are extracted from the power matrix A=(aij)i,j of the leading term of ƒÓ. For each 3 < k < n + 1, define an element ƒÓk of A(Matq(m,n;C)) by Then one sees directly that the leading term of ƒÓk is in the form ckxAk with ck•‚0, where Ak is the matrix obtained from A replacing the first row by (a11+ ...+a1 ,k-1,0,...,0,a1k,...,a1n). Since ƒÓk fr = 0 for all 1 < r < m, one has a11+...+a1,k-1_??_a21+...+a2k by Claim 2, as desired. •¡ As we remarked before, Lemma 2.3 implies b)•Ëc). This completes the proof of Theorem 2.2. We remark that the assumption for q not to be a root of unity is essential in the argument above. Theorem 2.2 can be restated as follows. THEOREM 2.4. Suppose that q is not a root o f unity. (1) The Lm-graded subalgebra of A(Matq(m,n;C)) consisting of all left relative B-m-invariants is generated by the quantum minor determinants (J C {1,... , m}, J ~). (2) Let A = A16l + ... + ~mEm be an integral weight in Lm. Then one has A(Bm1 Matq(m, n; C); z') 0 if and only if Al > ... > Am > 0. (3) I f Al > ... > Am > 0, then the standard monomials ƒÄT=ƒÄJ1...ƒÄJl indexed by the semi-standard tableaux T in SSTabn(ƒÉ) form a C-basis for the right A(GLq(n;C))-comodule A(Bm,Matq(m, n, C); z''). •¡ By using a similar argument, we see that, for an integral weight ƒÊ with ƒÊl / 2 >_ >_ µn >_ 0, the standard monomials ƒÄS =ƒÄI1...ƒÄIl indexed by the semi standard tableaux S in SSTabm(ƒÊ) form a C-basis for the left A(GLq(m;C)) - comodule A(Matq(m,n;C)/B+n;zƒÊ). By Theorem 2.2 for the case m=n, we can determine the relative B-n invariants in the coordinate ring of the quantum group GLq(n;C). In fact, if ƒÉ= 54 M. NOUMI, H. YAMADA and K. MIMACHI a1~1 + + is an integral weight in Ln, then one has (2.20) for any l•¸Z. From this fact, we have THEOREM 2.5. If )1 > > fin, the monomials (detq)-lƒÄT indexed by the semi-standard tableaux T in SSTabry, (a + £(61 + + ~n)) form a C-basis for the right A(GLq(n;C))-comodule A(Bn 1GLq(n; C); z') for any £ E Z with ~n > -~. I f a2 We will say that an integral weight ) = )1E1 + + Am~m in Lm is dominant if it satisfies the condition ) > >_ 2.3. Irreducibility of the representations A(Bn \GLq(n, C); z~`) To investigate representations of the quantum groups GLq(n;C), we recall the following theorem due to Rosso [R]. THEOREM 2.6. ([R]). If q is not a root of unity, any finite dimensional Uq(sl(n;C))-module is completely reducible. In fact, Rosso proved the complete reducibility of finite dimensional modules over quantum universal enveloping algebra Uq (g) for an arbitrary complex simple Lie algebra g. In the following, we will make use of his result for the case g= sl(n;C). First we give some remarks on Uq(g1(n;C))-modules, which easily follow from Theorem 2.6. We say that an element of a Uq(gl(n;C))-module M is a weight vector if it is a simultaneous eigenvector of the commuting operators q°/2(1 < k < n). A Uq(gl(n;C))-module is called a weight module if it has a C-basis consisting of weight vectors. PROPOSITION2.7. Let M be a finite dimensional Uq(gl(n;C))-module. (1) If M is irreducible, then it is a weight module. (2) If M is a weight module, then it is completelyreducible as a Uq(gl(n;C)) module. (3) If the Uq(gl(n;C))-module M is irreducible, then it is irreducible as a Uq(st(n;C))-module. Moreover, M is a highest weight module. PROOF. We consider the case of left Uq(gl(n;C))-module. (1) Note that M has at least one weight vector. Since the subspace of all weight vectors of M becomes a Uq(gl(n;C))-submodule, M is a weight module if it is irreducible. (2) If M is a weight module, there exists a C-basis {vi}i for the subspace of highest weight vectors. Let Ni be the Uq(sl(n;C))-submodule generated by vi for each i. Then we have an irreducible decomposition M = ®i NZ of M as a Quantum group GLq(n;C) and tonal spherical functions 55 Uq(sl(n;C))-module. Since each vi is a weight vector, Ni becomes a Uq(gl(n;C)) submodule of M and is irreducible as a Uq(gl(n;C))-module. Hence M is completely reducible as a Uq(gl(n;C))-module. Statement (3) is also clear by these argument. •¡ In Proposition 2.7.(2), one cannot dispense with the assumption that M is a weight module. We remark that any finite dimensional left or right Uq(gl(n;C)) - submodule of the coordinate ring A(GLq(n;C)) is a weight module since so is A(GLq(n;C)). For each dominant integral weight ƒÉ in Ln, we define the right A(GLq(n;C)) - comodule VR(ƒÉ) and the left A(GLq(n;C))-comodule VL(ƒÉ) by (2.21) V R (A) = A(Bn \GLq (n; C); z~) and V L (A) = A(GLq (n, C) /Bn ; z'), respectively. Recall that VR(ƒÉ) is a left Uq(gl(n;C))-module and VL(ƒÉ) is a right Uq(gl(n;C))-module. PROPOSITION 2.8. Let ƒÉ be a dominant integral weight in Ln. Then the A(GLq(n;C))-comodules VR(ƒÉ) and VL(ƒÉ) are irreducible as Uq(gl(n;C)) - modules, hence as A(GLq(n;C))-comodules. LEMMA 2.9. Let ƒÉ and ƒÊ be two dominant integral weights in Ln. Then we have (2.22.a) (2.22.b) if PROOF. Note first that VR(ƒÉ)•¼A(Hn•_GLq(n;C);tƒÉ) and VL(ƒÉ)•¼ A(GLq(n;C)/Hn;tƒÉ). Statement (2.22.a) is clear since the weight spaces of Ha-weight ƒÉ of VR(ƒÉ) and VL(ƒÉ) are spanned by the same standard monomial 1 nJ rn [i] 2r2 1~ rl where A = r1A 1 + • • • +r A (r1,• • °. r 1 E N, r E Z). To prove (2.22.b), one may assume that An > 0 and tcn > 0, multiplying VR(ƒÉ) and VL(ƒÊ) by a suitable power of the quantum determinant detq. Recall that the standard monomial ƒÄT indexed by a semi-standard tableau T has the leading term xA with the upper triangular matrix A determined by (2.16). On the other hand, the standard monomial ƒÄT has the leading term xtA with the transposed matrix of A. Then comparing the leading terms explicitly, one sees easily that the union {ƒÄT;T•¸SSTabn(ƒÉ)}•¾{ƒÄS;S•¸SSTabn(ƒÊ)} is linearly independent. This shows (2.22.b). •¡ PROOF OF PROPOSITION 2.8. We consider the case of the left A(GLq(n;C)) - comodule V L(ƒÉ) . It is enough to show that VL(ƒÉ) is an irreducible right 56 M. NOUMI, H. YAMADA and K. MIMACHI Uq(gl(n;C))-module. Since VL(ƒÉ) is a weight module, it can be decomposed into a direct sum of irreducible submodules by Proposition 2.7. To determine its ir reducible components, it suffices to find all highest weight vectors. Let v be a highest weight vector in VL(ƒÉ) . Since q is not a root of unity, v is also a weight vector with respect to Hn, namely LHn(v)= tµ ® v for some ƒÊ•¸Ln. Then by the equivalence a)•Ëb) in Theorem 2.2, we see that v is a relative B-n-invariant: V E VR(,2) = A(Bn \GLq(ri, C); z~`). By Lemma 2.9, we conclude that ƒÊ=ƒÉ and that v is a constant multiple of the highest weight vector of VR(ƒÉ). This means that VL(ƒÉ) is irreducible as a right Uq(gl(n;C))-module. •¡ We remark that, if ƒÉ and ƒÊ are two distinct dominant integral weights, the left Uq(gl(n;C))-modules VR(ƒÉ) and VR(ƒÊ) are not isomorphic since they have different highest weights. 2.4. Irreducible decomposition of A(GLq(n;C)) Let M be a finite dimensional right A(GLq(n;C))-comodule with structure mapping RGn : M --~ M ® A(GLq (n; C)). Then the dual space M•É=Homc(M,C) has a unique structure of left A(GLq(n;C))-comodule LGn : My -+ A(GLq(n; C))® My satisfying (LGn(u),v)=(u,RG(v)) for any u•¸M•ÉMy and v•¸M, where (,) stands for the pairing defined by the contraction My ® M -f C. Then we define a homomorphim ~M : MV ®M --~ A(GLq(n, C)) of two-sided A(GLq(n;C)) - comodule by (2.23) ® v) = (u, RGn (v)) E A(GLq (n, C)) for any u E M" and v E M. We denote the image of ƒ³M by W (M). If M=VR(ƒÉ) for a dominant integral weight A, we will simply write ƒ³ƒÉ and W(ƒÉ) instead of ƒ³VR(ƒÉ) and W(VR(ƒÉ)). The two-sided comodule W (M) defined above is nothing but the C-subspace of A(GLq(n;C)) spanned by the matrix elements of the right A(GLq(n;C))-comodule M. To be precise, let {vi}i•¸I be a C-basis for M. Let wij(i,j•¸I) be the matrix elements of M with respect to the basis {v}i•¸I: (2.24) RGn (vi) = vi ® wij for all j E I. iEI Let {ui}i•¸I be the basis for M•É dual to {vi}i•¸I:(ui,vj)=ƒÂij. Then by definition the left comodule structure of M•É is described by (2.25) LG (ui) = wij ® uj for all i E I. jEl Quantum group GLq(n;C) and zonal spherical functions 57 In other words, the dual comodule M•É has the same matrix elements wij as those of M. The homomorphism ƒ³M: MV®M -} A(GLq(n; C)) defined above is then given by ƒ³M (u2 ® vj) = w2~ for all i, j E I, so that the two-sided subcomodule W(M) of A(GLq(n;C)) is spanned by wij(i,j•¸I). Note that the comodule structure of W(M) is given by (2.26) (wz~~ = wzk ® wkj and E (wzJ) = big We remark that if M is a finite dimensional right A(GLq(n;C))-subcomodule of A(GLq(n;C)), then M is contained in W(M). In fact the coaction RGn in (2.24) is given by the coproduct •¢ of A(GLq(n;C)). Hence we have v~ _ > (v) zw2a E W(M) for all j•¸I by the property of the counit ƒÃ. PROPOSITION 2.10. Let ƒÉ be a dominant integral weight in Ln. Then the right A(GLq(n;C))-comodule VR(ƒÉ) and the left VL(ƒÉ) are dual to each other. Namely there exist an isomorphism of left comodules VL(ƒÉ) _??_ VR (ƒÉ)•É and an isomor phism of right A(GLq(n;C))-comodules VR(ƒÉ)_??_VL(ƒÉ)•É. PROOF. Note that VR(ƒÉ)•¼W(ƒÉ) as we remarked above. Recall that VL(ƒÉ)( has the same highest weight vector as that of VR(ƒÉ). Hence we have VL(ƒÉ)•¼W(ƒÉ) since W(ƒÉ) is a left A(GLq(n;C))-comodule and that VL(ƒÉ) is an irreducible left A(GLq(n;C))-comodule. Note that ƒ³ƒÉ:VR(ƒÉ)•É ® VR(A) -~ W(A) is surjective and that VR(ƒÉ)•É is an irreducible left A(GLq(n;C))-comodule. Hence, as a left A(GLq(n;C))-comodule, W(ƒÉ) is isomorphic to copy of finite number of VR(ƒÉ)•É This means that there exists an isomorphism VL(ƒÉ) -} V(A)v.R • THEOREM 2.11. (1) For any dominant integral weight ƒÉ in Ln, ƒ³ƒÉ: VR(A)v®VR(A) --~ W (A) is an isomorphism of two-sided A(GLq(n;C))-comodules. Namely the matrix elements of the right A(GLq(n;C))-comodule VR(ƒÉ) are linearly independent over C. (2) The coordinate ring A(GLq(n;C)) is decomposed into the direct sum of irreducible two-sided A(GLq(n;C))-comodules (2.27) where the summation runs over all dominant integral weights ƒÉ in Ln. (3) The coordinate ring A(Matq(n;C)) is decomposed into the direct sum of irreducible two-sided A(GLq(n;C))-comodules (2.28) where the summation runs over all dominant integral weights ƒÉ with (A, 6k) > 0 for all 1 < k PROOF. (1) Since VR(ƒÉ) is an irreducible left Uq(gl(n;C))-module, its dual space is an irreducible right Uq(gl(n;C))-module. By Schur's Lemma one can show that the tensor product VR(A)v ® V(A) is again an irreducible two-sided Uq(gl(n;C) )-module. This guarantees that ƒ³ƒÉ is injective since ƒ³ƒÉ is a nonzero homomorphism of two-sided Uq(gl(n;C))-modules. (2) Note that the two-sided Uq(gl(n;C))-modules W(ƒÉ) are irreducible and that W (ƒÉ) and W(ƒÊ) are not isomorphic unless ƒÉ=ƒÊ. Hence the sum > W(A) is actually a direct sum. Next we remark that the coordinate ring A(GLq(n;C)) is a union of finite dimensional left A(GLq(n;C))-comodules. In fact the subal gebra A(Matq(n;C))=C[xij;1 < i, j < n] is the union of finite dimensional left A(GLq(n;C))-subcomodules A(Matq(n;C))r(r•¸N), where A(Matq(n;C))r denotes the vector space of all elements in A(Matq(n;C)) with degree less than or equal to r. Since A(GLq(n;C))=•¾m•¸Z detmqA(Matq(n;C)),A(GLq(n;C)) is also a union of finite dimensional left A(GLq(n;C))-subcomodules. Hence the right Uq(gl(n;C))-module A(GLq(n;C)) can be written as a direct sum of finite di mensional irreducible Uq(gl(n;C))-submodules. Each irreducible component must have a highest weight vector v. However, we readily know by Theorem 2.5 that any element ƒÓ in A(GLq(n;C)) such that ƒÓ. fk= 0 (1 < k < n) belongs to the sum ®a VR(A). Since VR(ƒÉ)•¼W(ƒÉ), we see that all highest weight vectors in A(GLq(n;C)) are contained in the sum ®~ W (A). This implies that A(GLq(n, C)) = ®a W (A). Statement (3) is proved by a similar argument. One has only to note that A(Matq(n;C)) is a two-sided A(GLq(n;C))-subcomodule of A(GLq(n;C)) and that highest weight vectors in A(Matq(n;C)) are contained in ®~ VR(A) where the summation runs over all dominant integral weight ƒÉ with (A, ek) > 0 for all 1 A linear functional h : A(GLq(n;C))•¨C is said to be right GLq(n;C) -invariant (resp. left GLq(n;C)-invariant) if (2.29) for all ƒÓ•¸A(GLq(n;C)). This condition is equivalent to saying that h is a homo morphism of right (resp. left) A(GLq(n;C))-comodules to the trivial A(GLq(n;C)) - comodule C. THEOREM 2.12. (1) There exists a unique two-sided (GLq(n;C))-invariant linear functional h:A(GLq(n;C))•¨C with h(1)=1. (2) Any finite dimensional right (resp. left) A(GLq(n;C))-comodule is com pletely reducible. (3) Any finite dimensional irreducible right (resp. left) A(GLq(n;C))-comod ule is isomorphic to VR(ƒÉ)(resp.VL(ƒÉ)) for some dominant integral weight ƒÉ in Ln. Quantum group GLq(n;C) and zonal spherical functions 59 PROOF. (1) Let h be the projection A(GLq(n;C))•¨C=W(ƒÓ) in the decomposition (2.27). Then h is two-sided GLq(n;C)-invariant. By Theorem 2.11, the trivial representation appears with multiplicity one, in the irreducible decompo sition of A(GLq(n;C)) as a right (resp. left) A(GLq(n;C))-comodule. This implies that any homomorphism A(GLq(n;C))•¨C of right (resp. left) A(GLq(n;C)) - comodules must be a constant multiple of the projection h. (2) Let M be a finite dimensional right A(GLq(n;C))-comodule with struc ture mapping RGn : M M ® A(GLq(n, C)) and M' a subcomodule of M. Fix a C-linear mapping p:M•¨M' with P|M'=id. Define the averaging p : M -* M' of pby p = (id ® h) o (id ® m) o (RGn ® id) 0 (p® S) o RGn where S:A(GLq(n;C))•¨A(GLq(n;C)) is the antipode of the Hopf algebra A(GLq(n;C)) and m:A(GLq(n, C)) ® A(GLq(n; C)) --~ A(GLq(n; C)) is the mul tiplication. Then by the property (1.25) of the antipode S, one can check directly that p is also a projection onto M' and that p is a homomorphism of comodules. The kernel of p then gives an A(GLq(n;C))-subcomodule of M complementary to M'. (3) Let M be a finite dimensional irreducible right A(GLq(n;C))-comodule. For any nonzero linear functional f:M•¨C, define a homomorphism of right A(GLq(n;C) )-comodules F:M•¨A(GLq(n;C)) by F = (f ® id) o RGn . Then F is nonzero since ƒÃ„€ F=f. This implies that M is isomorphic to an irreducible right subcomodule of A(GLq(n;C)). By Theorem 2.11, each W(ƒÉ) is a direct sum of irre ducible subcomodules isomorphic to VR(ƒÉ). Hence any irreducible A(GLq(n;C)) - subcomodule of Aq(GL(n;C)) is isomorphic to some VR(ƒÉ). •˜ 3. Quantum group Uq(n) and the invariant functional 3.1. Quantum group Uq(n) and its unitary representations In the following sections, we assume that q is a real number with |q|•‚0,1. For a complex number c, we denote by c its complex conjugate. To discuss a "real form" of GLq(n;C), we introduce the notion of Hopf * - algebras. Let A be a Hopf algebra over C. We say that a conjugate linear mapping ƒÓ•¨ƒÓ*:A•¨A is a * -operation on A if the following conditions are satisfied: For any ƒÓ,ƒÓ•¸A, (1) (2) (3) The last condition is due to Woronowicz [W]. A Hopf algebra endowed with a *-operation will be called a Hopf *-algebra. 60 M. NOUMI, H. YAMADA and K. MIMACHI We now define the "compact" real form Uq(n) of GLq(n;C) by introducing a *-operation on the Hopf algebra A(GLq(n;C)) supposing that q is real. Using the fact that the antipode S is a C-linear anti-homomorphism,one can show that there exists a unique conjugate linear anti-homomorphism cp H cp* : A(GLq(n; C)) -* A(GLq(n; C)) such that (3.1) j = S(x32) = (-q)detq 1 for 1 < i, j PROPOSITION 3.1. Endowed with this conjugate linear anti-homomorphism, the Hopf algebra A(GLq(n;C)) becomes a Hopf *-algebra. Let I and J be two subsets of {1...., n} with #I=#J=r.By (1.13), one sees directly that (ƒÄIJ)*=S(ƒÄJI). Then Proposition 1.1 implies that (3.2) where Ic denotes the complement of I in {1,...,n}. In particular one has detq*= detq-1. Using formula (3.2), one can check that (A(GLq(n;C)),*) is a Hopf * - algebra. In the sequel, we denote by A(Uq(n)) the Hopf *-algebra (A(GLq(n;C)),*) and regard it as the coordinate ring of the quantum unitary group Uq(n). We some times write Un=Uq(n) for simplicity. Let M be a finite dimensional right A(Uq(n))-comodule with coaction RU M --+ M ® A(Uq (n)) . Let ( , ) : M x M -~ C be a Hermitian form on M, conjugate linear in the first argument. We extend ( , ) to a Hermitian form on M ® A(Uq(n)) with values in A(Uq(n)) by (u ® co, v ® b) = (u, v)co b for u, v e M and ƒÓ,ƒÕ•¸A(Uq (n)). Then we say that the Hermitian form (, ) is Uq(n)-invariant if (3.3) (Ru(u),Ru(v)) = (u, v) .1 for all u,v EM. We also say that M is unitary with respect to ( , ) if the Hermitian form (, ) is positive definite and Uq(n)-invariant. Let W=(wij)i,j•¸I be the representation matrix of M with respect to a C-basis {ui}i•¸I and set J = ((u27 Then condition (3.3) is equivalent to W*JW=J, where W* is the adjoint matrix of W, i.e. W*=(w1),jei. ~ZIf the Hermitian form ( , ) is non-degenerate, it is also equivalent to W*=JS(W)J-1, where S(W)=(S(w1)),ei. 22j Suppose that M is irreducible and that M has a non-trivial Uq(n)-invariant Hermitian form ( , ). Then by Schur's Lemma one can show that any Uq(n)-invariant Hermitian form is a constant multiple of ( , ). The Uq(n)-invariance of Hermitian forms on a left A(Uq(n))-comodule is de fined similarly. We do not give all the details of this definition but remark that it is convenient to take Hermitian forms conjugate linear in the second argument. Quantum group GLq (n;C) and zonal spherical functions 61 Recall that the quantum r-minor determinants ƒÄIJ(#I=#J=r) are matrix elements of the alternating tensor representation Er = ®#J=r CyJ. The above remark (3.2) means that W*=S(W) for its representation matrix W= Hence we have LEMMA 3.2. For each 0 < r < n, define a Hermitian form ( , ) on the alternating tensor representation Er ®#J=r CyJ by (YI, yJ) = bi,J. Then this Hermitian form ( , is Uq(n)-invariant. Let M' and M" be two finite dimensional right A(Uq(n))-comodules and R'Un and R"Un the structure mappings of M' and M", respectively. Then the tensor product M' ® M" -> M' ® M" ® A(Uq(n)) defined by Ru = (id ® m) o (id ® T ® id) o (RU n ® R'), where m stands for the multiplication of A(Uq(n)) and „„:A(Uq(n)) ®M" -> M" A(Uq (n)) for the flip cp ® v H v ® cp. If M' and M" has Uq(n)-invariant Hermitian forms ( , )' and ( , )", respectively, then the Hermitian form ( , ) on M' ® M" defined by (u' ® u", v' ® v") _ (u', v')' (u", v")" is also Uq(n)-invariant. Consider any tensor product M = Emi ® ® Eme of alternating tensor representations. Then by the above remark, the canonical Hermitian form ( , ) on M induced from those in Lemma 3.2 is positive definite and Uq(n)-invariant. PROPOSITION 3.3. Let ƒÉ be a dominant integral weight in Ln. Then the right A(Uq(n))-comodule VR(ƒÉ) has a positive definite Uq(n)-invariant Hermitian form (, ). Furthermore, VR(ƒÉ) has a C-basis consisting of weight vectors, orthonormal under such a Hermitian form. PROOF. First assume that An > 0. Write the dominant integral weight ƒÉ in the form ƒÉ=•Èm1+...+•Èml. Then define a linear mapping F : Emi ®• • • ® Em i -> A(Uq(n)) by F(u1®... ®ut) _ (v,R(ui~®... ®u~)) E A(Uq(n)), where v1 = ymi ®• • •®y l ymt is the highest weight vector of Emi ®• • • ®Emi Then it is clear that F is a homomorphism of A(Uq(n))-comodules. Let J1,...,Jl be any sequence of subsets of {1, 2,...,n} such that #Js=ms for 1 < s Hermitian form. Since any one-dimensional A(Uq(n))-comodule C detkq(k•¸Z) has 62 M. NOUMI, H. YAMAHA and K. MIMACHI a Uq(n)-invariant Hermitian form such that (detkq,detkq)=1, Proposition is valid for any dominant integral weight ƒÉ. The last statement follows from the fact that the weight space decomposition V R(A) = V (A), V R(A)u= {v E VR(A); RHJv) = v ® tµ} is orthogonal under the Hermitian from. REMARK. In the previous section, we proved the irreducibility of VR(ƒÉ) by using Rosso's theorem on the complete reducibility of Uq(sl(n;C))-modules. In the case where q is real, however, we can dispense with that theorem since the complete reducibility of VR(ƒÉ) follows from the argument of Proposition 3.3. 3.2. Invariant functional In the case of Uq(n), we call the invariant linear functional h:A(Uq(n))•¨C of Theorem 2.12 the Haar measure of Uq(n). We introduce the Hermitian forms (, ) L and (, ) R on the coordinate ring A(Uq(n)) by using the Haar measure h as follows: (3.4) Ki)L=h(/)) ~P>~P*and (b)R=h(ob*), S~~ i<'z for all ƒÓ,ƒÕ•¸A(Uq(n)). Note that the Hermitian form ( , )R is defined to be conjugate linear in the second argument. It is easy to show that the Hermitian form ( , ) L is right Uq(n)-invariant in the sense (3.3) and that ( , ) R is left Uq(n) -invariant. THEOREM 3.4. The irreducible decomposition A(Uq(n)) = ®~ W1A) of A(Uq(n)) as a two-sided A(Uq(n))-comodule is orthogonal with respect to the Uq(n) invariant Hermitian forms ( , )i. and ( ,) R. PROOF. For each dominant integral weight ƒÉ, one can choose a C-basis {ui(ƒÉ)}i•¸I(ƒÉ) for the irreducible right A(Uq(n))-comodule VR(ƒÉ) so that the rep resentation matrix W~, = (wJ(A))1,JeI(A)2 of VR(ƒÉ) satisfies the condition W*ƒÉ= S(WƒÉ), which is a consequence of Proposition 3.3. Fixing two dominant integral weights ƒÉ and ƒÊ, let C be an arbitrary I(ƒÉ)•~I(ƒÊ) complex matrix. Then the av erage C=h(W*ƒÉCWƒÊ) satisfies WƒÉC=CWƒÊ. Hence C=0 unless ƒÉ=ƒÊ since the two comodules VR(ƒÉ) and VR(ƒÊ) are both irreducible and not isomorphic to each other. Assuming that ƒÉ•‚ƒÊ, let C be the matrix unit Eir(i•¸I(ƒÉ),r•¸I(ƒÊ)). Then one has Eir=0, hence (Wjj(ƒÉ),Wrs(ƒÊ))L=0 for all j•¸I(ƒÉ),s•¸I(ƒÊ). This means that the decomposition A(Uq(n)) = ®~ W(A) is orthogonal with respect to (, )L. Orthogonality with respect to ( , ) R is proved similarly by the averaging h(WƒÉCW*ƒÊ). Quantum group GLq(n;C) and zonal spherical functions 63 We now calculate the square length of matrix elements of irreducible A(Uq(n)) - comodules VR(ƒÉ). As a first step, we consider the vector representation VR(•È1). LEMMA 3.5. For 1 < i, j, r, s < n, one has (3.5.a) (3.5.b) PROOF. Note that, for any two-sided weight vector ƒÓ•¸A(Uq(n)) of weight(ƒÉ,ƒÊ), one has h(ƒÓ)=0 unless (ƒÉ,ƒÊ)= (0,0). The elements xij and x*i are two sided weight vectors of weight (ƒÃi,ƒÃj) and (-ƒÃi,-ƒÃj), respectively. Hence (xij,xrs)L =(Xij,Xrs)R=0 unles (i,j)=(r,s). By (1.15.a) and (1.15.b) we have (3.6.a) (3.6.b) On the other hand, the Uq(n)-invariance of h implies (3.7.a) (3.7.b) Note that x jxkj(1 < k < n) are linearly independentover C. In fact one can check directly that the leading terms of xkj have differentexponent matrices. Hence, by comparing(3.6.a) with (3.7.a), we see that there exist constants cLi such that (xij, xij)L = cL for all i,j. Similarlywe see, by (3.6.b) and (3.7.b), (x,..xa J)L (x xj)q-2(i-1), i.e., cL = ci q-2(' 1). Applyingh to 1=k x 1xk1, We have (3.8) Thus we have ci 1-q2n q2(n-1), hence cL = 1-q2q2(n-i). Formula (3.5.b) is q proved in a similar way. 64 M. NOUMI, H. YAMADA and K. MIMACHI The difference between the two Hermitian forms ( , )L and ( , )R can be measured by (a special value of) the modular automorphism in the Tomita-Takesaki theory. This point is already discussed in [W]. We give an explicit description of this modular property. We define a C-algebra homomorphism ƒÐ:A(Uq(n))•¨A(Uq(n)) by (3.9) a(pp) = q2P • co • q2P for each o e A(Uq(n)), where 2ƒÏ is the following integral weight in Ln: (3.10) This means that if ƒÓ•¸A(Uq(n)) is a two-sided weight vector of weight (ƒÉ,ƒÊ, then a(o) = q2(P \+µ)cp. We call this automorphism ƒÐ the modular automorphism of A(Uq(n)). In fact it has the following property. PROPOSITION 3.6. The automorphism a satisfies (3.11) h(cp~) = h(Q(~)cp) for any cp, % E A(Uq(n)). Or equivalently, one has (3.12) (coy~)L = (a(),co)R '~for any co E A(Ue(n))• PROOF. Define a sesquilinear form F:A(Uq(n))•~A(Uq(n))•¨C byfor Then one can directly check that F satisfies the following: for all ƒÓ,ƒÕ1,ƒÕ2•¸A(Uq(n)). Hence, in order to prove (3.12), it suffices to check that (3.12) holds for any ƒÕ in a generator system for A(Uq(n)). Consider the case of ƒÕ=Xij. Since ƒÕ•¸W(•È1), (3.12) holds trivially for ƒÓ•¸W(ƒÉ) with ƒÉ•‚A1 by Theorem 3.4. The remaining case ƒÓ•¸W(•È1) is readily seen by (3.5.b). Lastly consider the case of ƒÕ=detlq(l•¸Z). Formula (3.12) holds trivially for ƒÓ•¸W(ƒÉ) with ƒÉ•‚l•Èn. The case ƒÓ•¸W(l•Èn) is reduced to (det, qdetq)L = (det, qdetq)R =1. By means of the modular property (3.12), one can determine explicitly the square length of matrix elements of irreducible representations. Quantum group GLq(n;C) and zonal spherical functions 65 THEOREM 3.7. (Square length of matrix elements). Let ƒÉ be a dominant inte gral weight in Ln and {ui}j•¸I(ƒÉ)2a C-basis for VR(ƒÉ) consisting of weight vectros ui of weight A(i) E Ln : RHn (u2) = u2 ®ta(2) . Assuming that the basis {ui}i•¸I(ƒÉ) is or thonormal under a Uq(n) -invariant Hermitian form on VR (ƒÉ), let wij(i, j•¸I(ƒÉ)) be the matrix elements of VR(ƒÉ) with respect to {ui}i•¸I(ƒÉ). Then the basis {wij}i,j•¸I(ƒÉ) for the two-sided A(Uq(n))-comodule W (ƒÉ) is orthogonal with respect to ( , )L and (, )R. Moreover the square length of wij is determined by the formula (3.13.a) (3.13.b) where 2ƒÏ=(n+1-2k)ƒÃk. The symbol da stands for a q-analogue of the dimension of VR(ƒÉ) defined by (3.14) PROOF. Setting W=(wij)i,j•¸I(ƒÉ), consider the average Eir=h(W*EirW) of the matrix unit Eir(i,r•¸I(ƒÉ)). Then Eir must be a scalar matrix, say cLir• Id, by Schur's Lemma. This means (3.15) KWJ,W3)L 2r= S~sc r for all j, s E 1(A). Similar argument by the average h(WESjW*) shows (3.16) (WTS,WIJ)R = SZr.c.R for all i, r E 1(A), for some cRjs•¸C. By the assumption RHn (u2) = u2 ® one has 7rHn (w1) ~= 5t'().j 2This implies that w2j is a two-sided weight vector of weight (ƒÉ(i),ƒÉ(j)) by (2.16). Hence, by Proposition 3.6, we have (3.17) Namelyone has bjscr = b2rcRg2(p,a(r>+a(s)7 . This impliesthat c. = bircLand cR = 55cR for cL = c and cR+ cR. Thisshows in particularthat the matrix elementswij are orthogonalto eachother under( , )L and ( , )j. As for square lengths,we have proved (3.18.a) 66 M.NOUMI, H.YAMADA andK. MIMACHI and (3.18.b) cL= cR• g2(~'a(i)+A(j)) for all i, j E 1(A). The last equalityshows that there existsa constantc suchthat cL = c.g2(P,A(2)) and cR= c °q-2 (PA(3 )). SinceW*W=Id, onehas >iEI()Awij wi j = 1. Applying the Haarmeasure to this formula,we obtain ~i cL= c.> i g2(P'A(2))=1. DefinedA by (3.14);we have c = i-,a hence (3.19) This proves (3.13.a) and (3.13.b). COROLLARY. The Hermitian forms (, ) and (, )R on A(Uq(n)) are positive definite. REMARK. By the duality between VL(ƒÉ) and VR(ƒÉ) (Proposition 2.10), one can easily see that the statement of Theorem 3.7 is valid also for the left A(Uq(n)) - comodule VL(ƒÉ). The square length of matrix elements is given by the same formula (3.13). Note that the constant dƒÉ does not depend on the choice of the orthonormal basis {ui}i•¸I(ƒÉ) as far as ui are weight vectors. In the proof of Theorem 3.7, we used the equality W*W=Id to determine the constant c. If one uses WW*=Id instead, one obtains another formula for dƒÉ: (3.20) dA= > q-2(P,A(i)). jEI(A) Let us define the character of the irreducible A(Uq(n))-comodule VR(ƒÉ) by (3.21) 1'A = E A(Uq (n)) • iEI (A) Although ƒÔƒÉ is an element of the non-commutative algebra A(Uq(n)), its restriction to the diagonal subgroup Hn coincides with the usual Schur function S(t) = SA(tl,.. ° , tr,) ([M]). Since ~Hn (w1) i= tA(2), we have (3.22) 7rHn (XA) = tA(2) = S(t). ICI(A) In fact the weight multiplicities in VR(ƒÉ) are counted by the semi-standard tableaux in SSTabn(ƒÉ), exactly in the same way as the case q=1. Note that the q analogue dƒÉ of the dimension of VR(ƒÉ) is obtained from (3.22) by the functional q2ƒÏ: A(Hn)•¨C: (3.23) dA = g2P(SA(t)) = SA(gn-1, q3,... , q-n+1). Quantum group GLq(n;C) and zonal spherical functions 67 This procedure is called a principal specialization of the Schur function SƒÉ(t). Evaluation of (3.23) is also well-known as a q-analogue of the hook-length formula ([M]). Note that d~+mAn = dA for any m E Z, since q2 (P4) ) =1. With this remark, let A = AE 1 + + ~nen be a dominant integral weight (or a partition) with Al > > An > 0. We define the Young diagram Y(ƒÉ) by Y(A)={(i,j) EN x N; 1 For each p=(i,j), the content c(p) and the hook-length h(p) are defined by c(p)=n-i+j and h(p)=A2+~j-i-j+1. Here µj (1 < j <£ = A1) are determined by A = Aµ1 + + Aµ~ with n > µ1 > pe > 0. Then the "q-dimension" dƒÉ is expressed by the hook-length formula (3.24) •˜ 4. Zonal spherical functions on Uq(n-1)•_Uq(n) 4.1. Quantum homogenous space Uq(n-1)•_Uq(n) In what follows, we denote by G the quantum group Uq(n) and by K the quantum subgroup Uq(n-1) of G, supposing that q is a real number with 0 < q < 1. Let A(G) = C[xij (1 < i, j X = (x2 j )1 <2, j into G by the surjective homomorphism ƒÎK:A(G)•¨A(K) of Hopf *-algebras defined by nK (xzj) = yij (1 (4.1)71K(xkn) = 0, ltK(xnk) = 0 (1 < k < n), 7VK((detq X)-1) = (detqY).-1 In order to investigate the quotient space K•_G, we study in advance the irreducible decomposition of VR(ƒÉ) and VL(ƒÉ) as A(K)-comodules. LEMMA 4.1. Let A = Al s1 + • • • + Anen be a dominant integral weight in Ln. Regard the right (resp. left) A(G)-comodule VR(ƒÉ) (resp. VL(ƒÉ)) as a right (resp. left) A(K)-comodule by RK = (id ® 7rK) o RG (resp. LK = (itK ® id) o LG). Then it has the irreducible decomposition 68 M. NOUMI, H. YAMADA and K. MIMACHI (4.2) VR (A) = VR (A; ti) (resp.V L (A) = ® VL (A; ~)) , µ µ where the summation runs over all dominant integral weights ,u = p Ei + + Ian-lEn-1 2n Ln_i such that (4.3) Al ~ µl ~ A2 ~ /i2 > An-1 ~ hn-1 ~ An. Here VR(ƒÉ;ƒÊ) (resp. VL(ƒÉ;ƒÊ)) is an A(K)-subcomodule of VR(ƒÉ) (resp. VL(ƒÉ)) isomorphic to the irreducible A(K) -comodule o f highest weight p. In particular every irreducible component appears with multiplicity one. PROOF. Since ƒÎK(detq X)=detq Y, the A(G)-comodule C(detq X )l is iso morphic to C(detq Y)l as a A(K)-comodule for all l•¸Z. Hence, we may assume that An > 0. To decompose the right A(K)-comodule VR(ƒÉ) into irreducible compo nents, it suffices to find all elements v in VR(ƒÉ) such that ek.v = 0 for 1 < k < n-2. For each integral weight ƒÊ•¸Ln-l satisfying (4.3), define the standard monomial vƒÊ by bn _ an_ b a b1 al vN 1 n nsl n-1s1 2 ,n ' .. si2sin~1 n where ak = Ak hk, bk = tck Ak+1 for 1 < k < n. Then it is clear that vƒÊ is a weight vector of weight ,u + [>1 VR(ƒÉ;ƒÊ) have different highest weights, we have ®~ VR(A; t.c) C VR(A). Counting the dimensions, we have ®µ VR(A; µ) = VR(A). In fact, it is easily seen that there is a natural bijection SSTabn(A) U, SSTabn_i (µ), where the union is taken over all ƒÊ•¸Ln-1 satisfying (4.3). The irreducible decomposition of VL(ƒÉ) is similarly obtained. By Theorem 2.12, any finite dimensional irreducible left A(G)-comodule is isomorphic to VL(ƒÉ) for some dominant integral weight ƒÉ=ƒÉ1ƒÃ1+...+ƒÉnƒÃn in Ln. We say that the irreducible representation VL(ƒÉ) is of class 1 with respect to K if it contains a K-fixed vector, namely if Lk (v) =1 ® v for some nonzero vector v in VL(ƒÉ). PROPOSITION 4.2. Let ƒÉ be a dominant integral weight in Ln. Then the left A(G)-comodule VL(ƒÉ) is of class 1 with respect to K if and only if A = £E i men = £A1 + mAn_i mAn for some £, m E N. I f A = £e1 men (~, m E N), then the standard monomial (4,4)vp = [detqXj m [ci 11•n-l1 n_1~ m Xnl= [xnnJ* m xnl £ Quantum group GLq(n;C) and zonal spherical functions 69 is a K fixed vector of VL(ƒÉ) . Moreover any K-fixed vector in VL(ƒÉ) is a constant multiple of v0. PROOF. With the notations of Lemma 4.1, one sees easily that the weight ƒÊ =0 in Ln-1 satisfies condition (4.3) if and only if al = ... = An _i = 0, i.e., A _ £E1 men = £A1 + mnn_1 mAn for some l, m•¸N. Note that the algebra A(K•_G) of left K-invariants in A(G) is a *-subalgebra of A(G), since ƒÎK:A(G)•¨A(K) is a *-homomorphism. Recall that, in the case of q=1, this homogeneous space is a (2n-1)-dimensional sphere. PROPOSITION 4.3. The algebra A(K•_G) of left K-invariants in A(G) has the irreducible decomposition as a right A(G)-comodule (4.5) A(K\G) = V(I?,rn), Q,mEN where V(l,m) is the irreducible right A(G)-comodule containing [xnn]m Xnl as its highest weight vector o f weight lƒÃ1-mƒÃn. PROOF. By Theorem 2.12, the right A(G)-subcomodule A(K•_G) is decom posed into the direct sum of irreducible right A(G)-comodules with highest weight vectors. We have to determine the highest weight vectors in A(K•_G), namely, vec tors lying in ®~ V L (A) . K-fixed vectors in ®~ V L (A) are determined in Proposition 4.2. Hence we see that the subspace of all the highest weight vectors in A(K•_G) is spanned by (*)m1(,xxn~m E N). This implies the irreducible decomposition (4.5). We define the elements zk and wk (1 < k < n) in A(G) by (4.6) zk = xnk and wk = z _ (-q)kinl ...j~...n(detqX)-1, Then it is straightforward to see that zk and wk are left K-invariant. Hence the subalgebra C[zk, wk; 1 < k < n] of A(G) generated by zk and wk(1 < k < n) is contained in A(K•_G). As for the right action of G, one has (4.7) RG(z~) = zz ®xzj, RG(w3) = 2f1i®xz~, z z which means that the algebra C [zk, wk; 1 < k < n] is a right A(G)-subcomodule of A(K•_G). Actually we have PROPOSITION 4.4. (1) The algebra of all left K-invariants in A(G) is gen erated by the above zk and wk (1 < k < n) (4.8) A(K\G) = C[zk, wk; 1 < k (2) The above generators zk and wk (1 < k < n) satisfy the following rela tions: (4.9.a) zizj = gzjz2, gwzwj ` wjw2 (1 i < j n), (4.9.b) wjzi=gzzwj (1 (4.9.c) wkzk = zkwk+ (1 q2) zUwU (1 n (4.9.d) ~zkwk=l. k=1 The *-operation on this algebra is given by 4 = wk (1 < k < n). PROOF. (1) Recall that w;z zi is the highest weight vector of V(l,m)(l,m•¸ N). Since the right A(G)-subcomodule C[zk, wk; 1 < k < a] of A(K•_G) contains all wn zl (~, m E N), it must coincide with A(K•_G) by Proposition 4.3. (2) Relations (4.9.a) and (4.9.d) are clear. By (1.4), the matrix X* = satisfies the relation (X* + I dn)R(X + I dn) = (Ida ® X)R(Idn 0 X*) for the R-matrix R of (1.3). By making this explicit, one obtains relations (4.9.b) and (4.9.c). We remark that, if q=1, relation (4.9.d) gives the defining equation of the (2n-1)-sphere { z1~2 + ... + zn 2 = 1} in the complex afine n-space Cn with canonical coordinates (z1,... , zn). By (4.9.a)-(4.9.c), one can easily show that the C-algebra A(K•_G) is spanned by the monomials (4.10) zl 1... znnw~i...win (c,3j~kEN). We remark that relations (4.9.c) also implies (4.11) zkwk=wkzk(1 q2) q2(k-1-v)WUzvZ v By using (4.9.a), (4.9.b) and (4.11), one sees easily that the algebra A(K•_G) is also spanned by the monomials (4.12) wQ1...wanzl 1... znn (ak,k~E N). We also remark that zk and wk satisfy another quadratic relation n (4.13) :ii:g2(n-k) wk zk =1, k=1 Quantum group GLq(n;C) and tonal spherical functions 71 which follows either from Proposition 1.1 or from the above (4.9.c) and (4.9.d). 4.2. Invariant functional on Uq(n-1)•_Uq(n) Here we investigate G-invariant functionals on A(K•_G). Note that the restric tion of the Haar measure h of G to A(K•_G) gives rise to a right G-invariant linear functional h:A(K•_G)•¨C with h(1)=1. It is also clear by Proposition 4.3 that h is the only functional with this property. Let us denote by H the diagonal sub group of G=Uq(n); the coordinate ring A(H) is the ring of Laurent polynomials C[t1, t1',... , tn, to 1 ] with *-operation t? = t2 1(1 < i < n). Note that there exists a direct sum decomposition by the right action of H: (4.14) where A(K•_G)A = {,ocE A(K•_G); RH(cp) = cp ® to}. The subalgebra A(K•_G/H) =A(K•_G)0 of right H-invariants in A(K•_G) is particularly important since the G-invariant linear functional h : A(K•_G)•¨C is factored through the projection A(K•_G)•¨A(K•_G)0 in (4.14). In fact, if cp is a weight vector of weight ƒÉ, one has h(ƒÕ)=0 unless ƒÉ=0 by the G-invariance of h. Recall that zk=xnk and wk = (_q)k_n(detq x)-1 are weight vectors of weight ƒÃk and -ƒÃk, respectively. By this remark, it is clear that the subalgebra A(K•_G/H) is spanned by the monomials znnwl 1 w~n (ck e N). In what follows, we use the notation of q-shifted factorials (4.15)(a, q)m = (1 a)(1 aq) (1 aqm-1) for m E N, PROPOSITION 4.5. The G-invariant functional h:A(K•_G)•¨C is factored through the projection A(K•_G)•¨A(K•_G/H). On the subalgebra A(K•_G/H), h is determined by the formula (4.16) for all A1,... , An E N. PROOF. For each integral weight A = A1e1 + • • • + anon with A1,... , An > 0, we define the monomial zƒÉ and wƒÉ in A(K•_G) by (4.17) and Note that the monomials zƒÉ and wƒÉ are weight vectors of weight ƒÉ and -ƒÉ, respec tively. To prove (4.16), it suffices to show the recurrence formula (4.18) 72 M. NOUMI, H. YAMADA and K. MIMACHI for all 1 < k < n, where [m] q2 = 11 qq2 . First we remark that the G-invariance of h implies that h(a.~o) = h(o)6(a) for any ƒÕ•¸A(G) and a E Uq(g?(n; C)), hence h(ek.co) = h(fk.cp) = 0(1 < k < n) for any ƒÕ•¸A(G). By fk_1.zJ = zk.bj,k_1 and fk_1.Wj _ -gwk_1.ba,k, one has (4.19) By the G-invariance of h, one has h(f k_1 . (w~+6kz~+6k-1)) = 0, hence (4.20) for any 2 < k (4.21) for any 2 < k < n. On the other hand, by equality (4.9) we have (4.22) Hence by (4.21) we compute (4.23) Thus we have the recurrence formula (4.18) by (4.21). In a suitable coordinate system of K•_G, the G-invariant functional h is ex pressed by the Jackson integral in q-analysis. If 0 < q~ < 1, the Jackson integral on the q-interval [0,c] is defined by Quantum group GLq(n;C) and zonal spherical functions 73 (4.24) We recall from [AA]a q-analogue of the beta integral: (4.25) for any ƒ¿,ƒÀ•¸N. For each 1 < k (4.26) THEOREM 4.6. The algebra A(K•_G/H) is a commutative C-algebra gener ated by the above (k(1 < k < n -1). Moreover A(K•_G/H) is isomorphic to the polynomial ring in n-1 indeterminates. For any polynomial p = F((1,... ,(n1) in (l, ... , (n-1, the value h(ƒÕ) is expressed by the following iteration of Jackson integrals: (4.27) PROOF. By (4.9) one can directly check that the elements zkwk(1 < k < n) are mutually commutative. Hence (k(1 < k < n-1) are also mutually commutative. We will show that any monomial wi' • • • wnn . znn zl 1(Ak E N) is written as a polynomial in ƒÄ1,...,ƒÄn-1. In the following we set ƒÄ0=0 and ƒÄn=1. It is straightforward to see that relations (4.9) imply (4.28.a) (4.28.b) (4.28.c) By means of these relations, one sees that (4.29) 74 M. NOUMI, H. YAMADA and K. MIMACHI In particular one sees that the algebra A(K•_G/H) is a commutative algebra gen erated by ƒÄ1,...,ƒÄ(n-1. In order to show that A(K•_G/H) is isomorphic to the polynomial ring in n-1 indeterminates, we have to prove that the monomials • • • 11(o E N) are linearly independent over C. Let Ak be the exponent matrix of the leading term of zkwk : Ak = F Since Al < AZ An, the leading term of (k detq has the exponent ma trix Ak for each 1 < k < n 1. Hence one sees that, if £ > Q'1 + + an-1, a_ I then the leading term of (11 • • • (n n -iidetq has the exponent matrix a1 Al + + q an-1An-1 + (~ a1 a, _i)Idn• Note that the n-th row of this matrix is given by (ai,... , Q'n_1, £ a1 cm-z). This shows that, for any l•¸N, the monomials ~11... (nnil with c1 + + cxn_1 < £ are linearly independent. To prove (4.27), it suffices to show the equality for all = wi wnn znn zi 1. In this case, the polynomial F(ƒÄ1,...,ƒÄn-1) is given by the right hand side of (4.29). Then by iteration of Jackson integrals of type (4.25), one can show easily that the right hand side of (4.27) takes the value of Proposition 4.5. 4.3. Zonal spherical functions on Uq(n-1)•_Uq(n) We call a right K-invariant element ƒÕ in A(K•_G) a zonal spherical function on K•_G if it belongs to some irreducible A(G)-subcomodule of A(K•_G). We will de termine all the zonal spherical functions on K•_G and show that they are expressed by the little q-Jacobi polynomials P+ (x; q). Recall that the algebra A(K•_G) has the irreducible decomposition (4.5) as a right A(G)-comodule. By Proposition 4.2, we see that each irreducible component V(l,m) contains the trivial representation of K with multiplicity one. We remark that there is a unique K-fixed vector m in V(l,m) with ƒÃ(ƒÕl,m)=1. In fact, such a K-fixed element appears as a matrix element of the irreducible representation VL(lƒÃ1-mƒÃn) of highest weight lƒÃ1-mƒÃn. To be precise, let v0 be a nonzero K fixed vector in VL(lƒÃ1-mƒÃn). Then VL(lƒÃ1-mƒÃn) is decomposed into the direct sum (4.30) of left A(K)-subcomodules. Hence one can find a unique element ƒÕ0 in A(G) such that (4.31) LG(vo) = cOO® vo mod A(G) ® V'. It is easy to see that ƒÕ0 is bi-K-invariant and ƒÃ(ƒÕ0)=1. Namely, ƒÕ0 is the unique K-fixed vector in V(l,m) such that ƒÃ(ƒÕ0)=1. In the sequel we denote this element coo in V(l,m) by ƒÕl,m. Normalize the G-invariant Hermitian form (, ) on VL(lƒÃ1 -mƒÃn) so that (v0,v0)=1. Since decomposition (4.30) is orthogonal under this Hermitian form, one has ƒÕl,m = (LG(vo), vo). Quantum group GLq(n;C) and zonal spherical functions 75 We will determine these zonal spherical functions explicitly in terms of the basic hypergeometric series (4.32) For each ƒ¿,ƒÀ•¸N, we define the little q-Jacobi polynomials Pea' (x; q) (k E N) by (4.33) THEOREM 4.7. For each l,m•¸N, the zonal spherical function ƒÕl,m in V(l,m) with ƒÃ(ƒÕl,m)=1 is expressed by the little q-Jacobi polynomial in Cn-1 = L-IIii n-1=z2wi = 1 znwn as follows: (4.34.a) if (4.34.b) if To compute this matrix element ƒÕl,m, we use the realization (4.35) For each k =1,...,n, setand The subalgebra ,A = C[xi,... , xn, yi, ... , yn] of A(G) generated by these xk and yk is a left A(G)-subcomodule.Among the commutation relations of the generators, we need the following: (4.36.a) (4.36.b) For each ƒÉ=ƒÉ1ƒÃ1+...+ƒÉnƒÃn with Ak•¸N, we set A = xn An ... xl Al and yA _ yl Al ... yn A (4.37) n . 76 M. NOUMI, H. YAMADA and K. MIMACHI Note that the monomial xƒÉyƒÊ belongs to VL(lƒÃl-mƒÃn) if |ƒÉ|=l and |ƒÊ|=m, where|ƒÉ|=ƒÉ1+...+ƒÉn. We choose the K-fixed vector v0 = x n yn in V L (~e 1 men ) and consider the direct sum decomposition (4.38) as a left A(K)-comodule. Note that (4.39) V L (~r1 mEn) e C V' for k =1, ... , n 2, since v0• ek=0. Before computing the zonal spherical function, we determine the projector po : VL(IE1 men) -> Cvo of the decomposition (4.30). LEMMA 4.8. Suppose that a monomial x~' yµ in V L (?e1 min) has left weight (Q m)en, z. e., ~~ = £, µ4 = m and A µ = (~ m)en. Then, the weights ƒÉ and ƒÊ are written as (4.40) = v + (I? r)en and µ = v + (m r)sn, where r is an integer with 0 < r min{i?, m} and v = V1E1 + + Un-16n-1 is an integral weight with vk E N and v) = v1 + + vn-1 = r. In terms of r and v, one can reduce xƒÉyv modulo V' as follows: (4.41) If A-(-m)6, ~ nthen po(xayµ) =0. Lemma 4.8 can be proved by an argument similar to the one we used in the proof of Proposition 4.6. We now compute the zonal spherical function m by using the K-fixed vector v0 = xn yn . Note that (4.42) where zi and wi are generators of A(K•_G). Hence we have (4.43) Quantum group GLq(n;C) and tonal spherical functions 77 In order to determine the K-fixed part, we apply the projection id ® Po : A(G) V L(F~1 rn~n) -* A(G) ® Cvo to this formula. Then by Lemma 4.8 we have (id ®po) o LG(xnyn ) (4.44) where ii = v1~1 + + Un-16n-1 = (1'i, , "n-1) with vk•¸N. Hence we obtain (4.45) n-1 r r LEMMA4.9. zvwv = zkwk = a -1' ~vI=r U q k=1 PROOF. For an element ƒÕ of A(G), denote by l(ƒÕ):A(G)•¨A(G) the operator of left multiplication ƒÕ•¨ƒÕƒÕ and by r(ƒÕ) the operator of right multipli cation ƒÕ•¨ƒÕƒÕ. Then one can directly see that the operators Pk = £(zk) o r(wk) (1 < k < n -1) satisfy the commutation relation P2 o P3 = q2P3 o Pi for i (4.46) On the other hand, one has (~-1 Pk) r (1) = (n-1 inductively since ƒÄn-1 commutes with zk(1 < k < n -1) by (4.28). Applying the above formula to 1, one obtains Lemma. By Lemma 4.9, we have (4.47) 2 78 M. NOUMI, H. YAMADA and K. MIMACHI Suppose that £ > m. Using zn-rwn-r = (en-1a q-2)m-r, we rewrite this formula into The last equality follows from a q-analogue of Pfaff's transformation formula (1.32) in [AW]. This proves (4.34.a). Formula (4.34.b) of the case of £ < m is proved by the same procedure. This completes the proof of Theorem 4.7. Recall that the zonal spherical function ƒÕl,m is a matrix element of VL(lƒÃl- mƒÃn) corresponding to the couple of weights ((l-m)ƒÃn,(l-m)ƒÃn). Hence the left comodule version of Theorem 3.7 implies that (4.48) where the "q-dimension" of VL(lƒÃ1-mƒÃn) is calculated by (3.24) as Let ƒÀ be a nonnegative integer and assume that £ m = £' m' = /3. Then by (4.29) one has Since we see by Theorem 4.6 that the left hand side of (4.48) represents the Jackson integral Quantum group GLq(n;C) and zonal spherical functions 79 Combining these formulas, we see that formula (4.48) gives a representation-theo retic interpretation of the orthogonality relation for the little q-Jacobi polynomials P2'((_1 q2)(f e N) due to Andrews and Askey [AA]. In fact, by replacing q and ƒÄ for q2 and ƒÄn-1 respectively, we have (4.49) for ƒ¿,ƒÀ•¸N and m,m'•¸N. References [AA] G.E. Andrews and R. Askey, Classical orthogonal polynomials, Lecture Notes in Math., 1171, Springer, 1985, 36-62. [AW] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that gen eralize Jacobi polynomials, Mem. Amer. Math. Soc., 319 (1985). [B] G.M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29 (1978), 178 -218. [D] V.G. 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Math. Phys., 111 (1987), 613-665. MASATOSHI NOUMI DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF TOKYO KOMABA, MEGURO-KU TOKYO 153, JAPAN HIROFUMI YAMADA DEPARTMENT OF MATHEMATICS TOKYO METROPOLITAN UNIVERSITY MINAMI-OHSAWA, HACHIOJI-SHI TOKYO 192-03, JAPAN KATSUHISA MIMACHI DEPARTMENT OF MATHEMATICS NAGOYA UNIVERSITY FURO-CHO, CHIKUSA-KU NAGOYA 464-01, JAPAN PRESENT ADDRESS DEPARTMENT OF MATHEMATICS KYUSHU UNIVERSITY HAKOZAKI, HIGASHI-KU FUKUOKA 812, JAPAN