E.G. KALNINS AND WILLARD MILLER, JR. The notation used for -series and -integrals in this paper follows that of Gasper and Rahman [3].. A generalizati

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1 A NOTE ON TENSOR PRODUCTS OF -ALGEBRA REPRESENTATIONS AND ORTHOGONAL POLYNOMIALS E.G. KALNINSy AND WILLARD MILLER, Jr.z Abstract. We work out examples of tensor products of distinct generalized s`) algebras with a factor from the positive discrete series of representations of one algebra and a factor from the negative discrete series of the other. We show that the euation for the common eigenfunctions of the Casimir operator and the Cartan subalgebra generator is just the three term recurrence relation corresponding to orthogonality for special cases of the Askey-Wilson polynomials, and this connection yields an almost immediate resolution of the tensor product representation into a direct integral of irreducible representations. An identity for the matrix elements of the \group representation operators" with respect to the tensor product and the reduced bases follows easily. Cases where the measures for the orthogonal polynomials are not uniue correspond to cases where the tensor products and their resolutions are also nonuniue. PACS..+b, 3.65.Fd. Introduction. Zhedanov and others have introduced a product of generalized s`) algebras that allows one to take tensor products of representations corresponding to two distinct algebras, [,]. Here we work out examples of tensor products with a factor from the positive discrete series of representations of one algebra and a factor from the negative discrete series of the other. We show that the euation for the common eigenfunctions of the Casimir operator and the Cartan subalgebra generator is just the three term recurrence relation corresponding to orthogonality for special cases of the Askey-Wilson polynomials, and this connection yields an almost immediate resolution of the tensor product representation into a direct integral of irreducible representations. Furthermore, an identity for the matrix elements of the \group representation operators" with respect to the tensor product and the reduced bases follows immediately. Cases where the measures for the orthogonal polynomials are not uniue correspond to cases where the tensor products and their resolutions are also nonuniue. 99 Mathematics Subject Classication. 33D55, 33D45, 7B37, 8R5. Key words and phrases. basic hypergeometric functions, -algebras, uantum groups, Askey-Wilson polynomials. ydepartment of Mathematics and Statistics, University of Waikato, Hamilton, New Zealand zschool of Mathematics and Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota Work supported in part by the National Science Foundation under grant DMS 94{533. Typeset by AMS-TEX

2 E.G. KALNINS AND WILLARD MILLER, JR. The notation used for -series and -integrals in this paper follows that of Gasper and Rahman [3].. A generalization of s`). We consider a generalization of s`), denoted by [v; u]. This is an algebra with generators H, E +, E which obey the commutation relations.) [H; E + ]=E + ; [H; E ]= E ; [E + ;E ]= u H v H Here, u and v are real numbers and <<. For uv 6= this algebra is isomorphic to one of the true s`) type algebras, for uv =; u +v > it is isomorphic to a special realization of the -oscillator algebra, and for u = v = it is isomorphic to the Euclidean Lie algebra m), [,4-6]. This algebra has an invariant element.) C = E + E + vh u H ; [C; A] =; 8A[v; u] As pointed out by Zhedanov and others [,], the family of algebras admits a multiplication [v; u] [ u; w] = [v; w], dened by.3) F + =E + )=E + H + H E + F =E )=E H + H E L =H) =HI+IH The operators F, L satisfy the commutation relations.). Using.3) we can easily dene the tensor product of a representation of [v; u] and the representation of [ u; w], thereby obtaining a representation of [v; w]. This construction yields a convenient generalization of the tensor product computations in, for example, [5,6]. We consider the family of algebraically irreducible representations ", of the algebra [ u; w], where <, w< and w u<, dened as follows. A convenient orthonormal basis for the representation space is fe n n =;;g where.4) n )w n+ + u ) E e n = n+ ) w n + u ) E + e n = He n = +n)e n e n e n+ WehaveE + =E ) and H = H. The invariant element C = w + +u )) I for this representation, where I is the identity operator. In analogy with a standard relationship between special functions and the representations of Lie groups we can compute the \matrix elements" of -analogs of the group operators e E + e E with respect to the fe n g basis, in the representation ". Of course there are many -analogs of the exponential mapping, none of which have all the properties

3 -ALGEBRA REPRESENTATIONS 3 needed to ensure that there is a true \group" associated with the -algebra. Among the -analogs we shall limit ourselves to the two that are most important, [3, page 9] e z) = X k= z k ; ) k ; E z) = X k= kk )= z k ; ) k If z is a complex number, the rst series converges to =z; ) for jzj < and the second series converges to z; ) for all z. Among the 8 possibilities, [7], we consider the matrix elements E+;e ) E E + )e E )e n = X n T ) n n ; )e n.5) T ) n n ; ) = p w = ) n n n n)n 3n )=4 ; ) n n w u n+ n ; ; ; un n n+ " u w ; ) n ; ) n u w ; ) n ; ) n ) n n! = A second family of irreducible representations of [v; u] where <, v > and v + u >, is dened as follows. A convenient orthonormal basis for the representation space is fj m m =;;g where.6) E + j m = m )v m+ + u ) j m m+ )v m + u ) E j m = j m+ Hj m = m)j m WehaveE + =E ) and H = H. The invariant element C =v + u )) I for this representation, where I is the identity operator. For this representation we consider the matrix elements [7] E+;e ) E E + )e E )j n = X n S ) n n ; )j n.7) S ) n n ; ) p v = ) n = ; ) n n n ; n n n)n +n )=4 un ; ) v n ; ) ) n n v u n+ ; ; un n n+ u v = ; ) n ; ) n u v ; ) n ; ) n

4 4 E.G. KALNINS AND WILLARD MILLER, JR. A third family of irreducible representations <z;>, of the algebra [v; w], where z is a complex number of absolute value one and is real, is dened in terms of the orthonormal basis for the representation space ff k k =;;;g and operators p p E + f k =) w k)= v z +k+)= f k+.8) E f k =) Hf k = +k)f k pw k+)= + p v z +k)= f k WehaveE + =E ),H =H and the invariant element C = p wv z + =z)i for this representation. Here we consider the matrix elements [7] E+;e ) E E + )e E )f k = X k R ) k k ; )f k.9) R ) k k ; ) p = w = ) k k k k )k+k )=4 ) k k )= k p z k ++= w v ; k ++3= z k k + k += z p += v ;;) z w p w v p v w ;k k + ; ) ; ; vk! Now we form the tensor product representation.) [v; u]" [ u; w] of [vw]. In this case the invariant operator is C = F + F + vl w L To decompose this representation we compute the common eigenfunctions of L and C. Clearly, eigenfunctions of L with eigenvalue are just those linear combinations of the basis vectors J m = j m e n where n = + m; m =;;.For <, they are linear combinations of the basis vectors J m = j m e n where n = + m; n =;;. Taking the case and applying C to the ON set fj mg we nd )CJ m = p wv.) + p wv h m+ ) +m+ ) + u v +m ) h m ) +m ) + u v +m ) u w ++m ) u w ++m ) +[ w + v ++ v ++ m ) + u v +m ) + w m++ ) u w +m+ )]J m i J m+ i J m

5 -ALGEBRA REPRESENTATIONS 5 The operator C is self-adjoint. If we introduce the spectral transform of this operator so that C corresponds to multiplication by the transform variable x, then.) takes the form of a three term recurrence relation for orthogonal polynomials Jm x) of order m in x. Indeed, comparing.) with the three term recurrence relation for the continuous Askey-Wilson polynomials.) p m x) p m x; a; b; c; dj) =ab; ac; ad; ) m a m 4 3 m ; abcd m ; ae i ; ae i ab; ac; ad [3, page 73], we get a match with ; ; ; a = v w ) +++= ; b= v w ) += ; c= p u = ; d=; vw and C p vwx= ). Making the identication " m+ Jmx; ; +m+ ; u w t) = +m ; u v +m ; ) p p m x)t ) m v w ; +++ e i ; ) where t = e i and x = cos, we can verify that.) holds, as well as the orthogonality relations.3) <Jm ;J m >= m;m ;; <f;g>= x) = x ) = p w v + Z Z x)dx d fx; t)gx; t) e i ;e i ; e i ; e i ; e i ; e i ; e i ; e i ; ) e i ; p w v + e i ; For = continuation and a limiting procedure u p vw + e i ; p u vw + e i ; ), =;;, the expression for Jm x; t) can be obtained by analytic.4) p Jm x; t) = )n+ w p v z + ; ) p u w ; ) n + ; ) z v [ w + ] n v w + z; ) " n+ ; u 3 n ; w +n u ; v +n+ ; ) +n+ ; ) + ; p v w + z; p v w + z u w ; ; t ;

6 6 E.G. KALNINS AND WILLARD MILLER, JR. where z = e i and m = n +. Furthermore, it is straightforward, though tedious, to verify that in terms of the new variables z; t the action of the operators F ;L is.5) p F + = p t w )= T = F = t p p w +)= T = t + L = ; C = p wv z + z ) t p vz +)= T = t ; p v z )= T = t Comparing.8) and.5), we have proved the direct integral decomposition ;.6) [v; u]" [ u; w] = Z <z; > d where z = e i. The functions J mx; t) are the Clebsch-Gordan coecients for this decomposition; the orthogonality and completeness relations for the corresponding Askey-Wilson polynomials are the unitarity conditions for the C-G coecients. The decomposition.6) can be used to obtain an identity relating the matrix elements.5),.7) and.9). We can compute the matrix element T m n ;mn; ) =<E F + )e F )j m e n ;j m e n > in two diferent ways. On one hand we have the integral representation.7) T m n ;mn; ) =<E F + )e F )Jm s ) ;Js m >=<R ; )Km s ;Ks m >; where J s m x; t) =Ks m x)ts, and n = s + m; n = s + m. On the other hand we can use the fact that for linear operators X and Y such that YX =XY, the formal identities s s e X + Y )=e X)e Y); E X+Y)=E Y)E X) hold,[3, page 8; 5, 7], so that Thus, E F + )e F ) =E E + H )E H E + )e E H )e H E ) =E E + H )e E H )E H E + )e H E ).8) T m n ;mn; ) =T ) n n m )= ; m )= )S ) m m n )= ; n )= ) Note that the matrix elements in a tensor product basis actually factor. Euating.7) and.8), we have the desired identity.

7 -ALGEBRA REPRESENTATIONS 7 3. A non-uniue tensor product. The -oscillator algebra provides a particularly interesting illustration of the ideas presented in the last section. It is the associative algebra generated by the four elements H, E +, E, E that obey the commutation relations 3.) [H; E + ]=E + ; [H; E ]= E ; [E + ;E ]= H E; [E;E ]=[E;H]= It admits a class of algebraically irreducible representations "`; where `; are real numbers and `>, [,4,5]. These are dened on a vector space with basis fe n n =;;;g, such that 3.) E + e n = ` s s n e n+ ; E e n = n ` e n He n =+n)e n ; Ee n =` e n Since E is a constant for the representations "`;, they can be considered as representations of the algebras [;` ]. Similarly, the -oscillator algebra admits a class of algebraically irreducible representations `; where `; are real numbers and `>. These are dened on a vector space with basis fh m m =;;;g, such that s 3.3) r m E + h m = ` h m+; E h m = ` Hh m = +m)h m ; Eh m = ` h m m h m Note that the `; can be considered as representations of [ ` ; ]. Now we consider the tensor product representation 3.4) "`; [;` ] `; [ ` ; ] of the Euclidean Lie algebra []. In this case the invariant operator is C = F + F. To decompose this representation we compute the common eigenfunctions of L and C. Clearly, eigenfunctions of L with eigenvalue s are just those linear combinations of the basis vectors Hn s = e n h m where n + m = s; m =;;. For s<, they are linear combinations of the basis vectors Hn s = e n h m where now n + m = s; n =;;. Taking the case s< and applying C to the ON set fhng s we nd )CH s n = n+s )=` n ) s n ) H s n+ 3.5) + n+s+)=` s n ) n ) H s n [ +s n` n )+ n` s n )]H s n The operator C is symmetric. If we introduce a spectral transform for a self-adjoint extension of this operator so that C corresponds to multiplication by the transform variable

8 8 E.G. KALNINS AND WILLARD MILLER, JR. x, then 3.5) takes the form of a three term recurrence relation for orthogonal polynomials H s nx) of order n in x. Indeed, comparing 3.5) with the three term recurrence relation for the -Laguerre polynomials 3.6) L ) n x; ) ; ) n =+ ; ) n n + ; ; )x n++ ; [3, page 94], we get a match with C = ` x and the identication 3.8) Hnx; s t) = ) n x s= t s ; ) n n ) s s ; ) n ; ) s s s )= log L n s) x; ); where t = e i and x = cos, we can verify that 3.5) holds, as well as the orthogonality relations 3.9) <Hn s ;Hs n >= n;n s;s; <f;g>= x) = )x; ) Z Z x)dx d fx; t)gx; t) For s, the expression for Hnx; s t) can be obtained by analytic continuation and a limiting procedure 3.) Hnx; s t) = ) M x s= t s ; ) M M ) +s L s) +s ; ) M ; ) s s+s )= log M x; ); where M = that in terms of the new variables x; t the action of the operators F ;L is m =;;, and n = M + s. Furthermore, it is straightforward to verify 3.) F + = =`tx = ; F = =`t x = ; L Operators 3.) acting on the space of suare integrable functions of the periodic variable dene the unitary irreducible representation p x`) of the Euclidean Lie algebra [; ], [6]. Thus, we have derived a direct integral decomposition 3.) "`; [;` ] `; [ ` ; ] = Z p x`) x) dx The functions H s nx; t) are the Clebsch-Gordan coecients for this decomposition. However, as is well known [3, 8], the measure 3.7) for which the -Laguerre polynomials are orthogonal is not uniue. Indeed the symmetric operator C has no uniue self-adjoint extension the deciency indices are,)). Thus, there is a multiplicity of possible selfadjoint extensions for C and each such extension denes a dierent tensor product 3.4).

9 -ALGEBRA REPRESENTATIONS 9 For each of these cases the Hnx; s t) can be thought of as the Clebsch-Gordan coecients, with the proviso that the coecients satisfy orthogonality but not completeness relations. Awell-known example of alternate orthogonality relations for the -Laguerre polynomials is 3.3a) " ; ) n n Hnx; s t) = ) n x s= t ; c); ) s c) where s<, and 3.3b) H s nx; t) = ) M x s= t s " s ; ) n c s ; c s ); ; ) M M c s ) ; ) c) ; c); ) +s ; ) M c s ; c +s ); c s ) ; ) L n s) x; ); L s) n x; ); for s and M = m =;;, n = M +s. Here, c> The orthogonality relations hold with the inner product dened by <f;g> = x)= X k= )x; ) Z c k ) k d fc k ;t)gc k ;t) Thus, we have the direct sum decompositions 3.4) "`; [;` ] `; [ ` ; ] = X k= p c k`) These decompositions can be used just as in x to obtain identities relating the matrix elements of the operator E F + )e F ) with respect to the tensor product basis and the reduced basis. The identities express products of -Laguerre polynomials as integrals or sums over Hahn-Exton -Bessel functions, with expansion coecients that are themselves products of -Laguerre polynomials. 4. Final remarks. Closely related results have been derived in the following papers. In [] the Clebsch-Gordan coecients for the tensor product of representations in the positive discrete series is calculated for the case [v; u][ u; w]. The results are expressed in terms of -Hahn polynomials. In this case the eigenspace of L with eigenvalue is nite dimensional, so one is considering nite discrete Askey-Wilson polynomials. In [5] we considered the tensor product representation!)[; ]"`; [;` ] of the -oscillator algebra [` ] and worked out the related matrix element identity. The direct sum decomposition corresponded to the orthogonality relations for the polynomials H n x) = n ; = n n ; ` x ;; x! )! ) x ; ; x `

10 E.G. KALNINS AND WILLARD MILLER, JR. These polynomials are orthogonal with respect to a measure with support at the points x = m`; ) m! ; m =;;; In [6] the authors considered the tensor product representation!)[; ]! )[; ] of the Euclidean Lie algebra. In this case the three-term recurrence relation for the common eigenfunctions of L and C = F + F is unbounded above and below. Thus the solutions do not correspond to polynomials. Again in this case the tensor product decompositon is not uniue. One of the associated identities for the matrix elements generalizes Koelink's addition formula for Hahn-Exton -Bessel functions, [9]. In [] we considered tensor products of various discrete analogs of the Euclidean and oscillator algebras. References. Ya.I. Granovskii and A.S. Zhedanov 994), Hidden Symmetry of the Racah and Clebsch-Gordan Problems for the Quantum Algebra sl), J. Group Theoretical Methods in Phys. to appear).. Ya.I.Granovskii, A.S.Zhedanov and O.B.Grakhovskaya 99), Phys. Lett. 85, B G. Gasper and M. Rahman 99), Basic Hypergeometric Series, Cambridge University Press, Cambridge. 4. E.G. Kalnins, H.L. Manocha and W. Miller 99), Models of -algebra representations Tensor products of special unitary and oscillator algebras, J. Math. Phys. 33, 365{ E.G. Kalnins, S. Mukherjee and W. Miller 993), Models of -algebra representations Matrix elements of the -oscillator algebra, J. Math. Phys. 34, 5333{ E.G. Kalnins, S. Mukherjee and W. Miller 994), Models of -algebra representations The group of plane motions, SIAM J. Math. Anal. 5, E.G. Kalnins, S. Mukherjee and W. Miller 993), Models of -algebra representations Matrix elements of Usu), in Lie algebras, cohomology and new applications to uantum mechanics, a volume in the Contemporary Mathematics Series, American Mathematical Society to appear).. 8. D.S. Moak 98), The -analogue of the Laguerre polynomials, J. Math. Anal. Appl. 8, { H.T. Koelink 99), On uantum groups and -special functions, thesis University of Leiden.. E.G. Kalnins and W. Miller 994), -algebra representations of the Euclidean, pseudo-euclidean and oscillator algebras, and their tensor products, submitted for publication).