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invariant elements and define a coalgebra structure on it [4], [24], [25], [26]. ..... algebra of polynomials on the quantum group SUq(n) is a Hopf ∗-algebra.
PACIFIC JOURNAL OF MATHEMATICS Vol. 188, No. 1, 1999

QUANTUM STIEFEL MANIFOLD AND DOUBLE COSETS OF QUANTUM UNITARY GROUP G.B. Podkolzin and L.I. Vainerman Dedicated to the memory of Professor Yuri L. Daletskii

We study the homogeneous space of the quantum group Uq (n) related to the subgroup Uq (n − m) (m < n), classify its irreducible representations and get a formula for its invariant integral. We also study the double cosets Uq (n − m)\Uq (n)/Uq (n − m) and the hypergroup structure associated with them.

1. Introduction. If a group G acts on a set S transitively on the right, then one can view S as G/K, where K is a subgroup of G. Thus a function on S can be considered as a function on G, invariant with respect to the right shifts by elements of K. It is especially interesting to consider bi-invariant functions on G since they can be identified with the functions on the set of G-orbits in S. If G is a locally compact group and K is its compact subgroup with Haar measures µG and µK respectively, then the set B ⊂ L1 (G, µG ) of all bi-invariant functions is an algebra with respect to the convolution and has a natural hypergroup structure related to generalized translation operators (see the survey [23] and the references given there): Z Rh f (g) = (1) f (gkh)dµK (k) = ∆(f )(g, h), g, h ∈ G. K

If the subalgebra B is commutative ( or, equivalently, the coproduct ∆ is cocommutative) then (G, K) forms a Gel’fand pair [6]. In many cases the characters of B are well known special functions of mathematical physics. This explains the importance of the notion of a Gel’fand pair. The case of a noncommutative subalgebra B has been studied in a number of papers. In particular, a pair (SO(n), SO(m)) was considered in [8], [22]. It is known that the homogeneous space SO(n)/SO(m) can be regarded as a Stiefel manifold S n,n−m . The infinitesimal object for the corresponding hypergroup structure was investigated in [15], [16]. A similar situation arises while considering functions on compact quantum groups [29] (see also [20]). Here, for a pair of compact quantum groups 179

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(H1 , H2 ) and a surjection π : H1 → H2 , we also consider an algebra H of biinvariant elements and define a coalgebra structure on it [4], [24], [25], [26]. In this situation we say that H is endowed with a hypergroup structure. If the coalgebra is cocommutative, we call (H1 , H2 ) a Gel’fand pair [12], [24]. If, in addition to, the algebra is commutative, we call such a Gel’fand pair strict [24], [25]. When algebra and coalgebra of bi-invariant elements are noncommutative, a corresponding hypergroup structure is more complicated. One of the simplest examples of such a situation is given by the pair H1 = Uq (n), H2 = Uq (n−m) (quantum unitary groups, m < n). For this pair we study a structure of a homogeneous space Sqn,m = Uq (n)/Uq (n − m) which is a quantum analogue of a Stiefel manifold. We give its description in terms of generators and relations of commutation, obtain a classification of its irreducible representations and a formula for an invariant integral on it. These results generalize the results obtained in [14], [17], [21], [28] for m = 1. In this case Sqn,1 are quantum spheres. We also investigate the double cosets Uq (n − m)\Uq (n)/Uq (n − m) and the corresponding hypergroup structure. In a special case m = 1 these and related questions were studied in [4], [7], [11], [12], [24], [25], [26]. We are grateful to Prof. Yu.A. Chapovsky, A.A. Kaljuzhnui, A.U. Klimyk and T.H. Koornwinder for many useful discussions. The second author is grateful to Prof. C. Anantharaman and J. Renault for their kind hospitality and support during his stay in Orleans, where this work was finished. 2. Preliminaries. All modules, comodules, algebras, coalgebras, Hopf algebras [1], linear maps, homomophisms, tensor products are considered over the field C of complex numbers. Algebras (coalgebras) are associative (coassociative) unital (counital). 2.1. Quantum group GLq (n, C) [5], [14], [18], [19]. Let a Hopf algebra H := A(GLq (n, C)) be generated by letters tij (i, j ∈ {1, . . . , n}), det−1 q satisfying the following relations of commutation (q ∈ C): (2)

tik tjk = qtjk tik ,

tki tkj = qtkj tki (i < j),

(3)

til tjk = tjk til (i < j, k < l),

(4)

tik tjl − qtil tjk = tjl tik − q −1 tjk til (i < j, k < l).

(5)

−1 det−1 q tij = tij detq .

QUANTUM STIEFEL MANIFOLD

181

The coproduct ∆ : H → H ⊗H and counit ε : H → C act in the following way: X −1 ∆(tij ) = tik ⊗ tkj , ∆(det−1 q ) = det−1 q ⊗ detq , k

ε(tij ) = δij , ε(det−1 q ) = 1. The antipode S is a homomorphism S : H → H such that: ˆ

S(tij ) = (−q)i−j ξˆij det−1 q , S(det−1 q ) = detq ,

where

ξJI =

X

(−q)l(τ ) tiτ (1) j1 · · · tiτ (r) jr

τ ∈Pr

is the quantum minor determinant; kˆ = (1, · · · , k − 1, k + 1, · · · , n), detq = 1,··· ,n ξ1,··· ,n I = (i1 , · · · , ir ); J = (j1 , · · · , jr ), Pr is the permutation group of the set (1, · · · , r); l(τ ) = (τ (1), · · · , τ (r)) is the number of inversions in τ . The commutation relations (2), (3), (4) can be rewritten as (6)

R(T ⊗ T) = (T ⊗ T)R, P where T := (tij )i,j=1,...,n = i,j tij Eij ∈ Mat(n, A(GLq (n, C))), T ⊗ T = P i,j,k,l tij tkl Eij ⊗ Ekl , R ∈ Mat(n, C) ⊗ Mat(n, C) is a so-called constant R-matrix of type An−1 [9], [18]: X X X Eii ⊗ Eii + Eji ⊗ Eij + (q − q −1 ) Eii ⊗ Ejj , R=q i

i6=j

i 1 q 0 = 1, q λ q µ = q λ+µ λ, µ ∈ Ln . This algebra has also a structure of a Hopf algebra with the following coproduct ∆, counit ε, and antipode S: ∆(q λ ) = q λ ⊗ q λ , ε(q λ ) = 1, S(q λ ) = q λ ∆(ek ) = ek ⊗ q −(εk −εk+1 )/2 + q (εk −εk+1 )/2 ⊗ ek , ε(ek ) = 0, S(ek ) = −q −1 ek ∆(fk ) = fk ⊗ q −(εk −εk+1 )/2 + q (εk −εk+1 )/2 ⊗ fk , ε(fk ) = 0, S(fk ) = −qfk . Given two Hopf algebras A and H over C, we say that a C-bilinear form ha, φi : H × A → C is a pairing of Hopf algebras if it satisfies the following conditions: ha, φψi = h∆H (a), φ ⊗ ψi, ha, 1i = εH (a) hab, φi = ha ⊗ b, ∆A (φ)i, h1, φi = εA (φ), hSH (a), φi = ha, SA (φ)i, for any a, b ∈ H and φ, ψ ∈ A. The following proposition can be found in [5], [19]: Proposition 1. There exists a unique pairing of Hopf algebras Uq (gl(n, C)) and A(GLq (n, C)) (briefly GLq (n, C)), such that: mhλ,ε1 +···+εn i (m ∈ Z) hq λ , tij i = δij q hλ,εi i hq λ , detm q i=q

hek , tij i = δik δj,k+1 ; hfk , tij i = δi,k+1 δjk m hek , detm q i = hfk , detq i = 0 (m ∈ Z).

Let V be a right GLq (n, C)-comodule (resp. left GLq (n, C)-comodule) with a structure mapping RG : V → V ⊗ GLq (n, C) (resp. LG : V → GLq (n, C) ⊗ V ), then V has a left (resp. right) module structure over Uq (gl(n, C)) defined by a · v = (id ⊗ a) ◦ RG (v)

(resp. v · a = (a ⊗ id) ◦ LG (v))

∀a ∈ Uq (gl(n, C)) and v ∈ V. In particular, GLq (n, c) is a bimodule over Uq (gl(n, C)). The actions of the generators q λ , ek , fk are given by q λ tij = tij q hλ,εj i ; ek tij = ti,j−1 δj,k+1 ; fk tij = ti,j+1 δjk ;

tij q λ = tij q hλ,εi i ; tij ek = δik ti+1,j ; tij fk = δi,k+1 ti−1j .

QUANTUM STIEFEL MANIFOLD

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2.3. Quantum G-spaces and relative invariants [14]. Let G be a quantum group with a coordinate ring A(G). Then a quantum space X is called a quantum left A(G)-space if the coordinate ring A(X) of X has a structure of a left A(G)-comodule LG : A(X) → A(G) ⊗ A(X) such that LG is a C-algebra homomorphism. An element χ of A(G) is called a linear character of G if ∆(χ) = χ ⊗ χ, ε(χ) = 1. For a given linear character χ of G, an element φ of A(X) is called a left relative invariant with character χ if LG (φ) = χ ⊗ φ. The subspace of all left relative G-invariants in A(X) with a character χ is denoted by (G\X; χ) = (φ ∈ A(X) : LG (φ) = χ ⊗ φ). The notions of right G-space and right relative G-invariants are defined similarly. If χ = 1, the subalgebras of all right- and left- invariants are called quantum homogeneous spaces and denoted by X\G and G/X respectively. In the similar way one can define a subalgebra of bi-invariants: G\X/G = {φ ∈ A(X) : RG (φ) = φ ⊗ 1, LG (φ) = 1 ⊗ φ}. 2.4. Corepresentations of GLq (n, C) [14]. In what follows, we use an abbrevation ξJ = ξj1 ...jr (resp. ξ J = ξ j1 ...jr ) j1 ...jr (resp. ξ1...r ) where J = to refer to a quantum r-minor determinant ξj1...r 1 ...jr (j1 < · · · < jr ). For each positive integer r with r ∈ {1, · · · n}, we define the fundamental weight Λr by Λr = ε1 +· · ·+εr . Let λ be an integral weight in Ln in the form λ = Λm1 + · · · + Λml , 0 ≤ ml ≤ · · · ≤ m1 ≤ n. This condition is equivalent to stating that λ is written as λ = λ1 ε1 + · · · + λn εn with 0 ≤ λn ≤ · · · ≤ λ1 . For such λ, let T = (Trs ; 1 ≤ r ≤ n, 1 ≤ s ≤ λr ) be a family of elements in (1, . . . , n). If T satisfies the conditions (7)

Tr−1,s < Tr,s for 1 ≤ s ≤ l, 2 ≤ r ≤ ms ,

(8)

Tr,s−1 ≤ Tr,s for 1 ≤ r ≤ n, 2 ≤ s ≤ λr ,

then T is called a semi-standard tableau of shape λ with labels in (1, . . . , n). We denote the totality of all semi-standard tableaus T = (Tr,s ) by SST abn (λ) and define the standard monomial ξT indexed by T as the product of quantum minor determinants ξT = ξJ1 . . . ξJl ∈ GLq (n, C), where Js = (T1,s , . . . , Tms ,s ) for s ∈ {1, . . . l}. Suppose now that q is not a root of unity.

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Proposition 2. If 0 ≤ λn ≤ · · · ≤ λ1 , then the standard monomials ξT = ξJ1 . . . ξJl indexed by the semi-standard tableaus T in SST abn (λ) form a C-basis for a right GLq (n, C)-comodule denoted by V R (λ). In a similar way one can construct a left irreducible GLq (n, C)-comodule V L (λ). We say that an integral weight λ in Ln is dominant if λn ≤ · · · ≤ λ1 . Proposition 3. (1) If λ is dominant, then the monomials (detq )−m ξT indexed by semi-standard tableaus T in SST abn (λ + m(ε1 + · · · + εn )) form a basis for the right GLq (n, C)-comodule V R (λ) for any m ∈ Z with λn ≥ −m. (2) Any finite-dimensional irreducible right (resp. left) GLq (n, C)- comodule is isomorphic to V R (λ) for some dominant integral weight λ in Ln . (3) Any finite-dimensional right and left GLq (n, C)-comodule is completly reducible. Proposition 4. The coordinate ring A(CLq (n, C)) is decomposed into the direct sum of irreducible two-sided GLq (n, C)-comodules: (9)

A(GLq (n, C)) = ⊕λ W (λ)

where the two-sided GLq (n, C)-comodule W (λ)is isomorphic to the tensor product of the left and right irreducible GLq (n, C)-comodules V L (λ) and V R (λ): W (λ) ∼ V L (λ) ⊗ V R (λ), and the summation runs over all dominant integral weights λ in Ln . 2.5. Invariant integral [14], [29], [21], [27]. Definition 1. A linear functional ν : GLq (n, C) → C is called right-invariant (resp. left-invariant) integral if (ν ⊗ id) ◦ ∆(φ) = ν(φ) · 1

(resp. (id ⊗ ν) ◦ ∆(φ) = 1 · ν(φ))

for all φ ∈ GLq (n, C). A bi-invariant integral is called a Haar integral. Proposition 5. There exists a unique Haar integral h with h(1) = 1 and it is the projection ν : ⊕λ W (λ) → W (0). 2.6. Quantum group Uq (n) and quantum homogeneous space Uq (n− 1)\Uq (n) [14], [28]. The definition of a Hopf *-algebra can be found in [20], [19]: Definition 2. A Hopf algebra H is a Hopf *-algebra if it is equipped with a conjugate linear mapping ∗ : H → H, such that: (1) 1∗ = 1; (φψ)∗ = ψ ∗ φ∗ . (2) ε(φ∗ ) = ε(φ); ∆ ◦ ∗ = (∗ ⊗ ∗) ◦ ∆; (∀φ, ψ ∈ H). (3) ∗ ◦ ∗ = id; ∗ ◦ S ◦ ∗ ◦ S = id.

QUANTUM STIEFEL MANIFOLD

185

Now we define the “compact real form” Uq (n) of GLq (n, C) by introducing an involution in the Hopf algebra GLq (n, C), if q is real, q ∈ / {−1, 0, 1}. The anti-homomorphism ∗ acts on A(GLq (n, C)) in the following way: t∗ ij = S(tji )

∀i, j ∈ {1, . . . , n}

∗ (det−1 q ) = detq .

One can check that T∗ T = TT∗ = 1, where T∗ = ((t∗ ji )ni,j=1 ), and that A(Uq (n)) = A((GLq (n, C)), ∗) is a Hopf ∗-algebra. Definition 3. The above Hopf ∗-algebra is the algebra of polynomials on the quantum unitary group Uq (n). Sometimes we denote it briefly Uq (n). The algebra of polynomials on the quantum group SUq (n) is a Hopf ∗-algebra specified by the condition detq = 1 with the same ∆, ε, S, ∗. For 1 ≤ m < n we define an epimorphism γm : A(Uq (n)) → A(Uq (n − m)) of Hopf ∗-algebras by: γm (tij ) = sij (1 ≤ i, j ≤ n − m); (10)

−1 γm (tkl ) = δkl 1 (k or l > n − m); γm (det−1 q ) = detq .

Proposition 6. The algebra Uq (n − 1)\Uq (n) is generated by tnk and t∗nk (1 ≤ k ≤ n), satisfying the following relations: tni tnj = qtnj tni , qt∗ni t∗nj = t∗nj t∗ni (1 ≤ i < j ≤ n), t∗nk tnk

t∗nj tni = qtni t∗nj , 1 ≤ i, j ≤ n X = tnk t∗nk + (1 − q 2 ) tnl t∗nl (1 ≤ k ≤ n) n X

l j}.

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The lines T−λn +1 , · · · , Tλ1 −λn do not contain the numbers 1, · · · , n − m by the construction. The comodule generated by the vectors vT is the Uq (n − m)-subcomodule of the weight µn−m = 0 in Uq (n − m)-comodule V R (λ). We denote it by V R (λ)0n−m . There is an isomorphism W (λ)/Uq (n−m) ∼ V L (λ)⊗V R (λ)0n−m and Uq (n)/Uq (n − m) ∼ ⊕λ V L (λ) ⊗ V R (λ)0n−m , where λ runs over the set of all weights such that µn−m = 0 ≺ λ. This isomorphism imposes a restriction on lower indices of the minors generating W (λ)/Uq (n − m). r Thus, the algebraic generators of Uq (n)/Uq (n − m) are the minors ξji11 ,···i ,···jr

r ∗ and (ξji11 ,···i ,···jr ) (1 ≤ i1 < · · · < ir ≤ n; n−m+1 ≤ j1 < · · · < jr ≤ n; 1 ≤ r ≤ m). These minors are polynomials in tij ; t∗ij (1 ≤ i ≤ n, n − m + 1 ≤ j ≤ n). The result of our considerations can be summarized as follows:

Theorem 1. 1) The quantum homogeneous space Sqn,m = Uq (n)/Uq (n − m) is the algebra generated by tij , t∗ij (1 ≤ i ≤ n, n − m + 1 ≤ j ≤ n) satisfying the following relations: tik tjk = qtjk tik , (17)

tki tkj = qtkj tki (i < j) til tjk = tjk til (i < j, k < l)

tik tjl − qtil tjk = tjl tik − q −1 tjk til (i < j, k < l)

t∗ik t∗jk = q −1 t∗jk t∗ik , t∗ki t∗kj = q −1 t∗kj t∗ki (i < j) t∗il t∗jk = t∗jk t∗il (i < j, k < l) (18)

t∗ik t∗jl − q −1 t∗il t∗jk = t∗jl t∗ik − qt∗jk t∗il (i < j, k < l) X X qtlp t∗lp + (q − q −1 ) tlm t∗lm = qt∗lp tlp + (q − q −1 ) t∗rp trp m>p

tij t∗is = qt∗is tij + (q − q −1 ) qtlp t∗jp + (q − q −1 ) (19)

(20)

X m>p

X

r