Traces of intertwining operators and

Traces of intertwining operators and Macdonald's polynomials Alexander A. Kirillov, Jr. May 1995 Let : V ! V U be an intertwining operator between representations of a simple Lie algebra (quantum group, a ne Lie algebra). We de ne its generalized character to be the following function on the Cartan subalgebra with values in U : (h) = TrV ( eh). This is a generalization of usual characters. These generalized characters are a rich source of special functions, possessing many interesting properties; for example, they are common eigenfunctions of a family of commuting di erential (di erence) operators. We show that the special functions that can be obtained this way include Macdonald's polynomials of type A, and this technique allows to prove inner product and symmetry identities for these polynomials (though proved earlier by other methods). Generalized characters for a ne Lie algebras are closely related with so-called correlation functions on the torus in the Wess-Zumino-Witten (WZW) model of conformal eld theory. We derive di erential equations satis ed by these correlation functions (elliptic Knizhnik-Zamolodchikov, or Knizhnik-Zamolodchikov-Bernard equations) and study their monodromies. Typeset by AMS-TEX Traces of intertwining operators and Macdonald's polynomials A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Alexander A. Kirillov, Jr. Dissertation Director: Professor Igor B. Frenkel May 1995 c 1995 by Alexander A. Kirillov All rights reserved CONTENTS Acknowledgements 1 Introduction 2 Chapter I. Basic de nitions 1.1 Simple Lie algebras and their representations 5 1.2 Quantum groups 9 Chapter II. Generalized characters 2.1 Weighted traces of intertwiners and orthogonality theorem 13 2.2 Center of Ug and commuting di erential operators 15 2.3 Center of Uqg and di erence operators 18 2.4 Generalized characters as spherical functions 20 Chapter III. Jacobi polynomials 3.1 Jacobi polynomials and corresponding di erential operators 24 3.2 Jack polynomials as generalized characters 27 Chapter IV. Macdonald's polynomials as generalized characters 4.1 De nition of Macdonald's polynomials 30 4.2 Macdonald's polynomials of type A as generalized characters 32 4.3 The center of Uqsln and Macdonald's operators 35 Chapter V. Inner product and symmetry identities for Macdonald's polynomials 5.1 Inner product identities 38 5.2 Algebra of intertwiners 41 5.3 Symmetry identities 48 Chapter VI. Generalized characters for a ne Lie algebras 6.1 A ne Lie algebras 52 6.2 Group algebra of the weight lattice 54 6.3 Generalized characters for a ne Lie algebras 57 Chapter VII. A ne Jacobi polynomials 7.1 De nition of a ne Jacobi polynomials 61 7.2 A ne Jack polynomials as generalized characters 65 Chapter VIII. Modular properties of a ne Jacobi polynomials 8.1 Functional interpretation of C [ b P ] 68 8.2 Normalized characters and modular invariance 71 Chapter IX. Correlation functions on the torus and elliptic KZ equations 9.1 Intertwiners and currents 76 9.2 Correlation functions on the torus and di erential equations 80 9.3 Monodromies of elliptic KZ equations 87 Chapter X. Perspectives and open questions 94 Appendix: List of formulas related to elliptic functions 96 References 98 ACKNOWLEDGEMENTS I would like to express my deep gratitude to my advisor Igor Frenkel for his guidance and encouragement throughout my study at Yale. Most of what I know about representation theory, mathematical physics and special functions comes from him. Also, I want to thank my friend and co-author Pavel Etingof; most of the results in this dissertation are based on our joint work, and would never be written without him; the idea of systematic study of the weighted traces of intertwiners belongs to him. His enthusiasm and new ideas often helped to nd a new way when I was ready to give up. My understanding of the subjects discussed in this dissertation bene ted a lot from fruitful discussions with other people, and among them Ivan Cherednik, Jintai Ding, Howard Garland, Ian Grojnowski, David Kazhdan, Ian Macdonald, Masatoshi Noumi, Gregg Zuckerman and many others. Financial support during my rst three years of study was provided by Yale and during the nal year by Alfred P. Sloan graduate dissertation fellowship. Finally { and most important of all { I want to thank my wife Var for her love and patience { even when I started discussing mathematics with Pavel while skiing in New Hampshire... INTRODUCTION This dissertation is devoted to study of some structures appearing in representation theory of simple Lie algebras (quantum groups, a ne Lie algebras) and their relations with the theory of special functions and mathematical physics. This subject, of course, has a long history. A representation-theoretic approach to special functions was developed in the 40-s and 50-s in the works of I.M.Gelfand, M.A.Naimark, N.Ya.Vilenkin, and their collaborators (see [V], [VK]). The essence of this approach is the fact that most classical special functions can be obtained as suitable specializations of matrix elements or characters of representations of groups. In geometrical terms, interesting special functions appear as spherical functions on symmetric spaces associated with G, which was studied by Harish-Chandra and Helgason (see [HC], [W]). However, this approach does not cover all interesting cases. For example, it was shown in recent works on representations of (quantum) a ne Lie algebras that matrix elements of intertwining operators between certain representations of these algebras are interesting special functions { (q-)hypergeometric functions and their generalizations [TK, FR]. In this dissertation we present a more general scheme, which includes both the classical theory of characters and matrix elements and intertwining operators. This scheme was suggested in the joint paper of the author and Pavel Etingof [EK1] and developed in papers [EK2{ EK5, E1, K1, K2]. Brie y it can be described as \theory of characters for intertwining operators", which we call generalized characters. This approach allows to get in a natural way many interesting special functions (for example, Lam e functions, Macdonald's polynomials) and easily prove a number of their properties. Here are the main ideas of this approach. Let g be a simple nite-dimensional Lie algebra over C , and let G be the corresponding compact real Lie group. An important role in the representation theory of G play the characters of nite-dimensional representations; in particular, they form a distinguished basis in the space of functions on G invariant with respect to conjugation; in other words, they are zonal spherical functions on the symmetric space G G=Gdiag . We can generalize this considering equivariant functions, i.e. functions on G with values in a representation U which satisfy f(gxg 1) = gf(x): This is a particular example of so-called -spherical functions which were studied in papers of Harish-Chandra (see [HC, W]). It turns out that this space has equally natural description. Namely, let V be any nite-dimensional representation of G, and : V ! V U be an intertwining operator. De ne the corresponding generalized character by (g) = TrV ( g): This is a natural generalization of usual characters; however, it takes values not in C but in the representation U . It is easy to see that is equivariant; moreover, every equivariant function is a linear combination of generalized characters. Note that due to the equivariance, the generalized character is completely determined by its values on the elements of the form g = eh; h from Cartan subalgebra of g. For this reason, from now on we consider as a function on Cartan subalgebra. In the classical case many properties of representations could be expressed in terms of their characters. The same holds here: many properties of intertwining operators can be expressed in terms of corresponding generalized characters. For example, the natural inner product on the space of intertwining operators gives an inner product on generalized characters and an analogue of the orthogonality theorem for them. Similarly, the fact that commutes with the center of Ug yields di erential equations satis ed by the { each element of the center gives a di erential equation. Thus, the generalized characters are common eigenfunctions of a family of commuting di erential operators, obtained from the center of Ug. These generalized characters are a rich source of special functions. In general, these functions have not been studied before. However, in some special cases they coincide with well-known objects. For example, if U is a trivial module then this is nothing but the usual characters and thus we recover the classical theory. More generally, the generalized characters (considered as functions on the Cartan subalgebra) take values in the zero-weight subspace U [0]; thus, if we consider the case of Lie algebra sln and take U to be the space of homogeneous polynomials in n variables of degree (k 1)n then U [0] is one-dimensional and thus the generalized characters can be considered as complex-valued. In this case we show that the ratio = k 1, where is the Weyl denominator, is the Jack symmetric polynomial. The simplest way to describe Jack polynomials { or more generally, Jacobi polynomials, which are generalization of Jack polynomials to the case of arbitrary root systems introduced by Heckman and Opdam { is to say that they are eigenfunctions of (conjugated by certain function) Sutherland operator: Lk = 2k(k 1) X 2R+ 1 (2 sinh ( ;h) 2 )2 ; where is the Laplace operator on h. For special values of k this operator is just the radial part of the Laplace operator on certain symmetric spaces associated with the group G, and the eigenfunctions are zonal spherical functions. Heckman and Opdam have showed that for any k, these polynomials have a number of remarkable properties. In particular, they showed that the Sutherland operator can be included in a commutative family of di erential operators, isomorphic to the algebra S(h)W ' Z(Ug) of Weyl group invariant polynomials on h; also, these polynomials are orthogonal with respect to a certain inner product. All of these results are highly non-trivial and required a good deal of ingenuity. We will show that our representation-theoretic approach allows to obtain both these properties immediately as a corollary of the general results about the generalized characters. This new interpretation also suggests new generalizations, such as vector-valued analogue of these polynomials or a ne analogue (see below). In a similar way, the generalized characters can be de ned for the quantum group Uqg, corresponding to g. All the constructions above can be generalized to this case with some minor changes; most importantly, di erential operators should be replaced by di erence operators. We will show that for the case g = sln and U chosen as above we get the famous Macdonald's polynomials { a family of symmetric polynomials which was recently introduced by I. Macdonald and has been the object of intensive study since that time. Again, this approach allows us to reprove many properties of Macdonald's polynomials (inner product identity, symmetry identity) in a very simple way. In a similar way, we can de ne generalized characters for a ne Lie algebras. It turns out that if we take U to be tensor product of evaluation representations then these characters are precisely the correlation functions on the torus of the WessZumino-Witten (WZW) model of conformal led theory. We deduce di erential equations, describing dependence of these generalized characters on the modular parameter of the torus and the parameters of the evaluation representations; we call these equations elliptic Knizhnik-Zamolodchikov equations (this was rst done, in a di erent language, by Bernard [Be]) and study their monodromies. Again, in the case g = sln, U { evaluation representation corresponding to the representation U of sln described above we get an a ne analogue of the theory of Jacobi polynomials. This is closely related with the elliptic analogue of the Sutherland operator. We can generalize to this case some results from the theory of usual Jacobi polynomials; there are also arise new phenomena, such as modular properties of these polynomials (as functions of the modular parameter of the torus). CHAPTER I BASIC DEFINITIONS 1.1 Simple Lie algebras and their representations In this whole work, all the objects are always de ned over the ground eld C of complex numbers or its extensions. In this section we brie y list the main facts on the simple Lie algebras and their representations which we are going to use; all of these facts are quite standard and can be found in any textbook on Lie algebras (see, for example, [Hu, B]). Let g be a simple Lie algebra over C of rank r, h g { its Cartan subalgebra, Ug { its universal enveloping algebra. Then g has the root decomposition: g = h M 2R g ; where R h is the corresponding root system. We x a polarization of R: R = R+t R+, where R+ is the subset of positive roots. We denote by 1; : : : ; r 2 R+ the basis of simple roots. The polarization of roots gives rise to polarization of g : g = n h n+, where n = L 2R g , and corresponding polarization of Ug : Ug = Un Uh Un+. We x an invariant symmetric bilinear form ( ; ) on g by the condition that for the associated bilinear form on h we have ( ; ) = 2 for long roots; this form allows us to identify h ' h : 7! h . Abusing the language, we will also use the notation ( ; ) for the associated bilinear form on h . We denote by h ; i the canonical pairing h h ! C . As usual, for every 2 R we de ne the dual root _ = 2h ( ; ) 2 h and introduce the following notions: Q =LZ i { root lattice; Q+ =LZ+ i; Q_ =LZ _i h { coroot lattice; P = f 2 h jh ; _i 2 Zg { weight lattice; P+ = f 2 h jh ; _i i 2 Z+g { cone of integer dominant weights; !i { fundamental weights: h!i; _j i = ij ; P_ = fh 2 hjhh; i 2 Zg { coweight lattice; = 1 2P 2R+ =P!i; 2 R { the highest root: 2 Q+ for every 2 R; h_ = h ; _i+ 1 { dual Coxeter number for g; We have a (partial) order on P de ned as follows: if 2 Q+. We denote by C [P ] the group algebra of the abelian group P , i.e. the algebra over C spanned by formal exponents e ; 2 P with relations e0 = 1; e + = e e . If an element f 2 C [P ] can be written in the form f =P a e ; a 6= 0 then we say that a e is the highest term of f and write f = a e + lower order terms. As usual, we denote by W the Weyl group of R and by l(w) length of an element w 2 W with respect to the generators si = s i . This group acts naturally on h , preserving R; thus, it acts on P and C [P ]. Of special interest for us will be the algebra C [P ]W of W -invariant elements in C [P ]. We will often use the fact that the orbitsums (1.1.1) m = X 2W e ; 2 P+ form a basis in C [P ]W , which follows from the fact that for every 2 P the orbit W contains precisely one element from P+. We choose a basis e 2 g ; f 2 g for 2 R+ such that (e ; f ) = 1; then [e ; f ] = h . In particular, if = i is a simple root then the elements ei = 2 ( i; i)e i ; fi = 2 ( i; i)f i ; hi = _i = 2 ( i; i)h i satisfy the usual relations of the Lie algebra sl2. Moreover, ei; fi; hi; i = 1 : : : r generate g. Let ai be any orthonormal basis in g. De ne the Casimir element C 2 Ug by C =P a2i . This element is central and does not depend on the choice of orthonormal basis. Therefore, it is easy to check that it can be written as follows: let xl; l = 1 : : : r be an orthonormal basis in h; then C =P 2R+ e f + f e +Px2l . We will also use the following useful identity: (1.1.2) (C) = C 1 + 1 C + 2 ; where (1.1.3) =X ai ai = X 2R+ e f + f e +Xxl xl is the g-invariant element in g g, and : Ug! Ug Ug is the comultiplication: (x) = x 1 + 1 x; x 2 g and (ab) = (a) (b). In general, there is no explicit construction of the whole center of Ug. However, its structure as a graded algebra (and more over, as a module over Ug) is known. Theorem 1.1.1. Z(Ug) ' (Sh)W : We will construct the isomorphism (Harish-Chandra isomorphism) later; now we only say that under this isomorphism the Casimir element C 7! p2 : p2( ) = ( ; ) ( ; ). Note that since it is known that (Sh)W is a free polynomial algebra, the same is true for Z(Ug). Also, there exists a unique involutive algebra automorphism (Chevalley involution) ! : g! g such that (1.1.4) !(ei) = fi; !(fi) = ei; !(h) = h; h 2 h: Let us now consider the representation theory of g. If V is a g-module then we denote an action of x 2 g in V by V (x). We will always consider modules with a weight decomposition: V = L 2h V [ ], where V [ ] are nite-dimensional spaces such that hv = h ; hiv if v 2 V [ ]; h 2 h. We will call v 2 V [ ] vectors of weight . For every 2 h we de ne the Verma module M by the following conditions: 1. M is spanned over Ug by a single vector v (highest weight vector) such that e v = 0, v has weight . 2. M is a free module over Un . These modules are simplest examples of modules from category O. By de nition, a g-module V is said to be from categoryO, if it has weight decomposition, is nitely generated and satis es the following condition: for every v 2 V , the space Un+v is nite-dimensional. Let I is the maximal ideal in M which does not contain v (it exists and is unique); denote by L = M =I the corresponding irreducible highest-weight module. For generic (that is, for all except a subvariety of codimension 1), M = L . Moreover, it is known that there is a unique bilinear form on M (Shapovalov form) such that (v ; v ) = 1 and (v1; xv2) = (!(x)v1; v2) for all x 2 g. This form is symmetric and its kernel is I ; thus, it descends to a non-degenerate form on L . It is known that L is nite-dimensional i 2 P+, and every nite-dimensional irreducible representation of g is obtained in this way. Moreover, if 2 P+ then the dual representation L is also nite-dimensional irreducible: L = L , where = w0( ); w0 being the longest element in the Weyl group. If c 2 Z(Ug) is a central element, then cjM = (c)( + ) IdM , where : Z(Ug)! (Sh)W is the Harish-Chandra isomorphism mentioned before. The same holds for any subquotient of M , in particular, for L . Note that for the Casimir element we have CjM = ( ; +2 ) IdM ; thus, in the adjoint representation C acts by multiplication by 2h_. For a nite-dimensional representation V of g let chV = P 2h dimV [ ]e 2 C [P ]. For an irreducible representation, the character is given by the Weyl formula: (1.1.5) chL = Pw2W ( 1)l(w)ew( + ) ; 2 P+; where (1.1.6) = Y 2R+(e =2 e =2) = e Y 2R+(1 e ) is the Weyl denominator. Obviously, chL = e + lower order terms. More generally, if V is a module from category O then we can de ne its character by the same formula chV =P 2h dimV [ ]e . However, in this case it lies not in C [P ] but in its completion C [P ], which is de ned as follows: (1.1.7) C [P ] = f 1 X n=0 ane n j lim( ; n) = +1g: In the following we will be especially interested in the case g = sln, i.e. the root system of type An 1. In this case the root system and related objects admit the following explicit realization: h = f( 1; : : : ; n)jP i = 0g C n ; R = f"i "jgi6=j ; where "i is the standard basis in C n : "i = (0; : : : ; 0; 1; 0; : : : ; 0) 2 Zn (1 in the i-th place); ( ; ) =P i i; R+ = f"i "jgi 0; then either deg u+ 6= 0 or deg u 6= 0. We can assume that u 2 U [ ]; 2 Q+; 6= 0. Since is an intertwiner, Tr( u u0u+e2 ih) = Tr( (u ) u0u+e2 ih). From the de nition of comultiplication one easily sees that (u ) = u q =2 +Puj vj for some uj ; vj 2 U0U such that sdeg (uju0u+) < sdeg u. Thus, Tr( ue2 ih) = q =2 Tr( u0u+e2 ihu ) +X vj Tr( u0u+e2 ihuj): Since commuting with e2 ih does not change sdeg uj , by the induction assumption we can write Tr( ue2 ih) =q =2 Tr( u0u+e2 ihu ) +D0 Tr( e2 ih) =q( ; )0=2e Tr( u0u+u e2 ih) +D0 Tr( e2 ih) =q( ; )0=2e Tr( (u+ [u0u+; u ])e2 ih) +D0 Tr( e2 ih) for some D0 2 DOq Uqg. Since sdeg of all terms in [u0u+; u ] is less than sdeg u u0u+, we can again apply induction assumption and get Tr( ue2 ih) = 1 1 q( ; )0=2e D00 Tr( e2 ih): This proves the existence part of the theorem. Uniqueness follows from the following lemma: Lemma 2.3.2. Let us x a nite-dimensional Uqg-module U . Suppose that D 2 DO Hom(U [0]; U [ ]) is such that for any Uqg-intertwiner : V ! V U; V { nite-dimensional Uqg-module we have D = 0. Then D = 0. Proof of the lemma. Let us assume that D 6= 0. Multiplying D by a suitable element from C q [P ] we can assume that D has polynomial coe cients: D = P 2P e D( ), D( ) being di erence operators with constant matrix-valued coe cients. Let us take the maximal (with respect to the standard order in P ) such that D( ) 6= 0. Then if we have a generalized character such that = e u + lower order terms then, taking the highest term of D , we see that D( )(e u) = 0. On the other hand, if we take such that h ; _i i 0 then for every u 2 U [0] there exists a non-zero intertwiner :L ! L U such that the corresponding generalized character has the form = e u + lower order terms. Thus D( )(e u) = 0 for all 0; u 2 U [0]. It is easy to show that it is only possible if D( ) = 0, which contradicts the assumption D( ) 6= 0. Proposition 2.3.3. Let us keep the notations of Theorem 2.3.1. Then c 7! Dc is an algebra homomorphism of Z(Uqg) to DOq Uqg[0]=I. Proof. The same as in the classical case (Theorem 2.2.3) Unlike the classical case, in the quantum case we have an explicit construction of central elements (see Theorem 1.2.1). Applying the previous construction to those central elements, we get the following theorem: Theorem 2.3.4. Let V be a nite-dimensional representation of Uqg, cV { the corresponding central element in Uqg (see Theorem 1.2.1), and DcV { the corresponding di erence operator constructed in Theorem 2.3.1. Let : L ! L U be a Uqg-intertwiner, and { the corresponding generalized character. Then satis es the following di erence equation: (2.3.2) DcV = chV (q 2( + )) : Proof. The same as in classical case (see Corollary 2.2.2); the value of cV in L is taken from Theorem 1.2.1. Remark. As in the classical case, all the constructions of this section can be easily generalized to the case where V is an arbitrary module from category O. 2.4 Generalized characters as spherical functions In this section we show that generalized characters for g can be interpreted as spherical functions on the space G G=G. The constructions of this section are due to the author, I. Frenkel and P. Etingof (see [EFK]). As before, let g be a simple Lie algebra over C , and let G be the compact real simply connected Lie group corresponding to g. It is known that every nitedimensional complex representation of g can be lifted to G. Also, it is known that every nite-dimensional representation V of G is unitary: there exists a positive de nite hermitian form ( ; )V on V such that (gv1; gv2)V = (v1; v2)V . Let us x some nite-dimensional representation U of g. Let C1(G;U) be the space of all smooth functions on G with values in U . We can de ne a (hermitian) inner product in C1(G;U) by (2.4.1) (f1; f2) = ZG(f1(g); f2(g))U dg; where ( ; )U is the inner product in U and dg is the Haar measure on G. It is easy to see that the inner product de ned by (2.4.1) is positive-de nite. We denote by L2(G;U) the closure of the space C1(G;U) with respect to the norm kfk =p(f; f). Definition 2.4.1. A function f 2 C1(G;U) is called equivariant (notation: f 2 C1(G;U)G) if for every g; x 2 G we have (2.4.2) f(gxg 1) = gf(x): The same applies to f 2 L2(G;U). This de nition can be rewritten as follows. Let us consider the group G G, and let Gd G G be the diagonal subgroup. Consider the functions f : G G ! U such that for every k 2 Gd; x 2 G G we have (2.4.3) f(xk) = f(x); f(kx) = kf(x): This is a special case of what is called -spherical functions (see [W]) on the group G G with respect to the subgroup Gd. On the other hand, it is easy to see that f(x; y) 7! f(xy 1) establishes isomorphism between the space of spherical functions on G G=Gd dened by (2.4.3) and the space of equivariant functions in the sense of De nition 2.4.1. Here is yet another description of the same space. Let f 2 C1(G;U)G, and let u 2 U . De ne a complex-valued function on G by fu(g) = hu; f(g)i. De ne the action of G on scalar functions on G by (Tgf)(x) = f(g 1xg). Then it is easy to see that Tgfu = fgu, and thus fu satis es the following condition: (2.4.4) Under the action of G de ned above, fu spans a nite-dimensional subspace in C1(G), and as a representation of G, this space is isomorphic to U . Vice versa, it is easy to see that every scalar function onG satisfying the condition (2.4.4) can be obtained as fu(g) for some f 2 C1(G;U)G; u 2 U . Now, let V be a nite-dimensional representation of G and let : V ! V U be a G-intertwiner. De ne coresponding generalized character 2 C1(G;U) by (2.4.5) (g) = TrV ( g): Note that for g = e2 ih this coincides with the previously given de nition of the generalized character (see (2.1.2)). Lemma 2.4.2. (1) For every intertwiner : V ! V U , the generalized character de ned by (2.4.5) is equivariant: 2 C1(G;U)G. (2) Assume that V is irreducible. Then = 0 i = 0. Proof. (1) is quite trivial: TrV ( gxg 1) = Tr((g g) xg 1) = gTr( x): (2) Let us consider on the maximal torus, i.e. on the points of the form e2 ih; h 2 hR. As was explained in Section 2.1, we can as well consider it as an element of C [P ] U , and = 0 as a function on h i = 0 as an element of C [P ] U . Let v be a highest-weight vector in V . Then v = v u0 + lower order terms for some u0 2 U [0]. It is known that u0 = 0 i = 0. On the other hand, = e u0 + lower order terms, which proves the theorem. Let 2 P+. Recall that L is the irreducible highest-weight module over g with highest weight , which in this case is nite-dimensional and thus can be considered as a module over G. Let H = HomG(L ; L U) (note that this space is nite-dimensional). Due to Lemma 2.4.2, we have an injective map H ! C1(G;U)G : 7! : We will denote the image of H in C1(G;U)G also by H . Theorem 2.4.3. (1) For 6= , H and H are orthogonal with respect to the inner product in C1(G;U) (see (2.4.1)). (2) L2(G;U)G = M 2P+H : (direct sum should be understood in the sense of Hilbert spaces). We refer the reader to [EFK] for the proof of this theorem. Example. Let U = C be the trvial representation. Then Theorem 2.4.3 coincides with well-known Peter-Weyl theorem. Finally, since the conjugacy classes in G are the same asW -orbits in the maximal torus (2.4.6) T = exp(ihR) ' hR=Q_; hR =MR _i ; it is easy to see that every equivariant function on G is uniquely de ned by its values on T . More precisely, we have the following isomorphism: (2.4.7) C1(G;U)G ' C1(T; U [0])W ; i.e. functions on T with values in U [0] satisfying f(wt) = wf(t) for every element from the Weyl group. Thus, restriction of the inner product de ned by (2.4.1) to the space of equivariant functions can be rewritten in terms of integral over T . The answer is given by the following theorem: Theorem 2.4.4. If f1; f2 2 C1(G;U)G then (2.4.8) (f1; f2) = 1 jW j ZT (f1(t); f2(t))U  dt; where, as before, dt is the Haar measure on T , ( ; )U is the inner product in U , and is the Weyl denominator: (e2 ih) = Y 2R+ e ih ;hi e ih ;hi ; and  is the complex conjugate of . Remark. Again, in the case U = C this theorem is well known; it was rst proved by H. Weyl. Proof. It is very easy to check that the scalar function (g) = (f1(g); f2(g))U is conjugation invariant. Thus, the theorem is reduced to the following statement: for every central function on G, we have ZG (g) dg = 1 jW j ZT (t)  dt; which is well-known (essentially, this is the result of Weyl we referred to above). Note that this theorem along with the orthogonality statement of Theorem 2.4.3 gives a new proof of the hermitian version of the orthogonality theorem for generalized characters for g (Theorem 2.1.2); of course, these arguments wouldn't help in the q-case. Also, using the description of the space of equivariant functions as scalar-valued functions on G transforming under the conjugation as U (see (2.4.4)), it is easy to see that every conjugation-invariant scalar di erential operator D on G de nes an operator in C1(G;U)G, acting by a scalar in every H (see details in [EFK]). Since the space of such operators is isomorphic to (Ug)g ' Z(Ug), this gives a natural action of Z(Ug) by di erential operators on C1(G;U)G. Using the isomorphism (2.4.7), we can rewrite this action in terms of di erential operators on T with coefcients from End U [0] (this operation is often referred to as \taking the radial part of the Laplacian"). It can be easily shown that the di erential operators appearing in this way coincide with those constructed in Theorem 2.2.1. In particular, the operator DC given by (2.2.3) is exactly the radial part of the second order LaplaceBeltrami operator G. We refer the reader to [EFK] for more detailed information. Note that it would be quite di cult to calculate this radial part by geometrical arguments, whereas the algebraic approach makes it very simple. Again, we note that the radial part of the Laplace operator has been calculated in more general situation by Harish-Chandra (see [W, Chapter 9]), so (2.2.3) can be deduced from his results. CHAPTER III JACOBI POLYNOMIALS In this chapter we show that in some special cases the generalized characters for the Lie algebra sln coincide with so-called Jack polynomials, which were studied by Heckman and Opdam (see [HO, O1, H1, H2]). These results are due to the author and Pavel Etingof (see [EK2, EK4]). In fact, Heckman and Opdam de ned and studied analogues of these polynomials for arbitrary root systems; they called them Jacobi polynomials associated with a root system, the reason being that for g = sl2 these polynomials coincide with so-called ultraspherical (Gegenbauer) polynomials, which are a special case of Jacobi polynomials. In this chapter we only consider classical case, i.e. representations, intertwiners and generalized characters for g rather than for Uqg. 3.1 Jacobi polynomials and corresponding di erential operators In thie section we give de nition and main properties of Jacobi polynomials associated with an arbitrary reduced irreducible root system R, following papers of Heckman and Opdam. Recall (see Chapter 2) that C [P ] denotes the group algebra of the weight lattice, and C [P ](e 1) 1 is the ring obtained by adjoining to C [P ] the expressions of the form (e 1) 1; 2 R. Note that the elements of C [P ] may be considered as functions on h by the rule: e (h) = e2 ih ;hi. Under this identi cation elements of C [P ](e 1) 1 become functions on the real torus T = hR=Q_; hR = LR _i with singularities on the hypersurfaces e (h) = 1; 2 R. However, we will use the formal language as far as possible. Abusing the language, we will call elements of C [P ] polynomials, and elements of C [P ]W symmetric polynomials. Similarly, we will talk of divisibility of polynomials meaning divisibility in the ring C [P ]. Recall also (see Section 2.2) that we denote by DO the ring of di erential operators in h with coe cients from C [P ](e 1) 1; again, they can be treated formally as derivations of the ring C [P ](e 1) 1. In particular, for every 2 h we de ned @ so that @ e = ( ; )e and the Laplace operator h so that he = ( ; )e . Through this whole chapter, we x some positive integer k. Consider the following di erential operator (3.1.1) L = Lk = h k(k 1) X 2R+ ( ; ) (e =2 e =2)2 : This operator for the root system An was introduced by Sutherland ([Su]) and for an arbitrary root system by Olshanetsky and Perelomov ([OP]) as a Hamiltonian of an integrable quantum system. We will call L the Sutherland operator. As before, let be the Weyl denominator de ned by (1.1.6). De ne the following version of the Sutherland operator: (3.1.2) Mk = k(Lk k2( ; )) k: Proposition 3.1.1. ([HO]) (1) (3.1.3) Mk = h k X 2R+ 1 + e 1 e @ = h 2k X 2R+ 1 1 e @ + 2k@ : (2) Both Lk;Mk commute with the action of the Weyl group. (3) Mk preserves the algebra of symmetric polynomials C [P ]W C [P ](e 1) 1. Proof. We do not give the proof of (1) here, referring the reader to [HO]. Note that the proof involves some non-trivial statement about the root systems, which we will discuss later in the proof of the a ne analogue of this statement (Theorem 7.1.2). (2) is obvious, and (3) immediately follows from (1). Recall (see 1.1.1) that we denoted by m the basis of orbitsums in C [P ]W : m = P 2W e ; 2 P+. Lemma 3.1.2. (3.1.4) Mkm = ( ; + 2k )m + X < 2P+ c m : Proof. Explicit calculation. Now we can consider the eigenfunction problem for Mk. Let us consider the action of Mk in the nite-dimensional space spanned by m with . Then the eigenvalue ( ; +2k ) has multiplicity one in this space due to the following trivial but very useful fact: Lemma 3.1.3. Let ; 2 P+, < . Then ( + ; + ) < ( + ; + ). Thus, we can give the following de nition: Definition 3.1.4. Jacobi polynomials J ; 2 P+ are the elements of C [P ]W de ned by the following conditions: (1) J = m + P < c m . (2) MkJ = ( ; + 2k )J . Due to Lemma 3.1.3, these properties determine J uniquely. Note that this de nition is valid for any complex k, and components of J are rational functions of k. Let us introduce an inner product in C [P ]W . Let (3.1.5) hf; gi0 = 1 jW j [f g]0; where, as in Section 2.1, [ ]0 is the constant term of a polynomial, and the bar involution is de ned by e = e . More generally, let (3.1.6) hf; gik = hf k; g ki0: Note that for k = 1 this de nition coincides with previously given (2.1.3). Lemma 3.1.5. Mk is self-adjoint with respect to the inner product h ; ik. Proof. This is equivalent to Lk being self-adjoint with respect to the inner product h ; i0, which is obvious. Corollary 3.1.6. hJ ; J ik = 0 if < . In fact, one has a stronger result: Theorem 3.1.7. ([O1]) hJ ; J ik = 0 if 6= . We will prove this theorem for the root system An in the next section { see Theorem 3.2.4. In fact, the operator Mk can be included in a large commutative family of differential operators. De ne (3.1.7) D = fD 2 DOjD is W -invariant; [D;Mk] = 0g: Then we have the following theorem: Theorem 3.1.8. (1) Every operator D 2 D preserves the space C [P ]W , and J is a common eigenbasis for the action of D in C [P ]W : there exists a map : D ! (Sh)W such that (3.1.8) DJ = (D)( + k )J : (2) is an isomorphism D ' (Sh)W . We do not give the proof of this theorem here, referring the reader to the above mentioned papers of Heckman and Opdam (in fact, for the root system An this theorem was known before). However, we note here that part (1) is relatively easy, and so is injectivity of ; the di cult part is to prove surjectivity, or to construct 1. Again, we will reprove it by representation-theoretic methods for the root system An in the next section. 3.2 Jack polynomials as generalized characters In this section we show that one can get Jacobi polynomials for the root system An 1, in which case they are also known under the name \Jack polynomials", as generalized characters. This construction is due to the author and Pavel Etingof ([EK2, EK3]). In this section, we only consider the case g = sln. Recall that we have xed a positive integer k. De ne the representation U = Uk 1 (which we will later use to de ne the generalized characters) as the irreducible representation of sln with the highest weight (k 1)n!1; it can be described as the symmetric power of the fundamental representation: (3.2.1) U = S(k 1)nC n : In other words, U can be identi ed with the space of homogeneous polynomials in x1; : : : ; xn of degree (k 1)n. The action of sln is given by the following formulas: (3.2.2) ei 7! xi@i+1; fi 7! xi+1@i hi 7! xi@i xi+1@i+1; where @i = @ @xi . It is very important for us that all weight subspaces of U are one-dimensional. In particular, the same is true for U [0]; we x an element u0 = (x1 : : : xn)k 1 2 U [0], which allows us to identify (3.2.3) U [0] ' C : u0 7! 1: Also, we will use the fact that e f jU [0] = k(k 1) IdU [0] : Lemma 3.2.1. Let 2 P+. A non-zero sln-homomorphism :L ! L U exists i (k 1) 2 P+; if it exists, it is unique up to a factor. Proof. Standard exercise. For brevity, from now on we use the following notation: (3.2.4) k = + (k 1) : For every 2 P+ x an intertwiner (3.2.5) : L k ! L k U by the condition (v k) = v k u0 + lower order terms, where v k is the highest weight vector in L k . De ne by ' the corresponding generalized character: (3.2.6) ' = = TrL k ( e2 ih): It takes values in the space U [0]; since this space is one-dimensional, we can identify it with C using (3.2.3) and consider ' as scalar-valued function. Under this identi cation, ' has the following form: ' = e +(k 1) + lower order terms. Proposition 3.2.2. (3.2.7) '0 = k 1: The proof of this proposition will be given later in a more general case (see Proposition 4.2.2). Now we can prove the main theorem of this section: Theorem 3.2.3. Let ' ; 2 P+ be the generalized characters for sln de ned by formulas (3.2.6), (3.2.7). Then ' is divisible by '0, and the ratio ' ='0 is the Jack polynomial J (see De nition 3.1.4). Proof. Let us rst prove that ' is divisible by '0, and the ratio is a symmetric polynomial with highest term e . Consider the tensor product V = L L(k 1) . It decomposes as follows: V = L +(k 1) +P < N L +(k 1) . Consider the intertwiner = IdL 0 : V ! V U . On one hand, it follows from the de nition that = chL '0, where chL is the (usual) character of L . On the other hand, the decomposition of V implies that = ' + P < a ' , and thus ' ='0 = chL +P < a ' ='0. Since chL is a symmetric polynomial, it follows by induction in that ' ='0 is also a symmetric polynomial. Let us prove that ' ='0 satis es the equationMk(' ='0) = ( ; +2k )(' ='0). Since '0 = k 1 (see Proposition 3.2.2), this equation is equivalent to (3.2.8) Lk ' = ( + k ; + k ) ' : On the other hand, recall that we have proved in Section 2.2 that the generalized characters are eigenfunctions of a certain di erential operator DC which was obtained from the Casimir element C. Comparing the expressions (2.2.3) for DC and (3.1.1) for Lk, we see that (3.2.9) Lk = U (DC) 1; since e f u0 = k(k 1)u0. Therefore, (3.2.8) immediately follows from the di erential equation for generalized characters derived in Proposition 2.2.4. This immediately implies the orthogonality of Jack polynomials: Theorem 3.2.4. If J are Jack polynomials then (3.2.9) hJ ; J ik = 0 if 6= : Proof. It follows immediately from Theorem 3.2.3 and the orthogonality theorem for generalized characters (Theorem 2.1.2). Thus we have reproved by representation-theoretic arguments the orthogonality theorem 3.1.7 for Jacobi polynomials for the root system An 1. Note that the proof given by Heckman and Opdam uses transcendental (i.e., not algebraic) methods; the representation-theoretic approach outlined above gives probably the simplest proof of this fact. On the other hand, assuming (3.2.9) we could give an alternative proof of Theorem 3.2.3 { using the orthogonality theorem for generalized characters rather than the di erential equation satis ed by them. We will use this approach in the next chapter, where we discuss q-analogue of Jacobi polynomials { Macdonald's polynomials. Our construction also allows one to construct di erential operators commuting with Mk. Namely, it follows from Theorem 2.2.3 that the operators of the form D = (k 1) U (Dc) k 1, where c 2 Z(Ug), commute with each other (and thus, with Mk, which can be obtained from the Casimir element) and are W -invariant. This means that the map Z(Ug) ' (Sh)W ! D : c 7! (k 1) U (Dc) k 1 is the inverse map to the map de ned by (3.1.8). This proves (for the root system An 1) surjectivity of , i.e. the di cult part of Theorem 3.1.8 on the structure of D . CHAPTER IV MACDONALD'S POLYNOMIALS AND GENERALIZED CHARACTERS In this chapter we develop q-analogue of the constructions of Chapter 3. We dene the q-analogue of Jacobi polynomials, called Macdonald polynomials, and show that in some special cases the generalized characters for the quantum group Uqsln coincide with Macdonald's polynomials for the root system An 1. This construction is due to the author and Pavel Etingof [EK2]. In this chapter, the words \representation" etc. always stand for representation of the quantum group Uqg. 4.1 De nition of Macdonald's polynomials In this section we give the de nition of Macdonald's polynomials for reduced irreducible root system, following [M2]. We preserve the notations of Chapter 1. Consider the algebra of W -invariant polynomials C [P ]W . The main goal of this section is to construct a basis in C q;t [P ]W , where C q;t = C (q; t) is the eld of rational functions in two independent variables q; t. Unless otherwise stated, in this section \polynomial" will stand for an element of C q;t [P ], and divisibility will stand for divisibility in this ring; similarly, elements of C q;t [P ] will be called symmetric polynomials. Recall (see (1.1.1)) that we denoted by m the basis of orbitsums in C [P ]W : m = P 2W e ; 2 P+. Theorem 4.1.1. (Macdonald) There exists a unique family of polynomials P (q; t) 2 C q;t [P ]W ; 2 P+ such that(1) P = m +P < c m . (2) These polynomials are orthogonal with respect to the following inner product on C q;t [P ]: (4.1.1) hf; giq;t = 1 jW j [f g q;t]0; where, as before, the bar involution is de ned by e = e , [ ]0 is the constant term: [P c e ]0 = c0, and (4.1.2) q;t = Y 2R 1 Y m=0 1 q2me 1 q2mt2e : These polynomials are called Macdonald's polynomials (our notation slightly di ers from that of Macdonald: what we denote by P (q; t) in the notations of [M2] would be P (q2; t2)). Remark. In fact, for non simply-laced systems there this theorem can be generalized, allowing di erent variables t for roots of di erent lengths; see [M2] for details. It is often convenient to consider Macdonald's polynomials for t = qk; k 2 Z+ (see examples below). In this case, Macdonald's polynomials lie in C (q)[P ]; we will write h ; ik instead of h ; iq;qk, etc. (note that it agrees with De nition 3.1.6 for q = 1). However, most of the properties of Macdonald's polynomials obtained for t = qk can be generalized to the case when q; t are independent variables. Abusing the notations, we will sometimes say \t = qk, k is an independent variable" instead of saying that q; t are independent variables. Example. (1) For k = 0, we have P = m independently of q; for k = 1, P = chL { also independently of q. (2) In the limit q; t! 1 so that t = qk Macdonald's polynomials tend to Jacobi polynomials introduced in Section 3.1, which follows from Theorem 3.1.7. For the case g = sln (that is, when R is the root system of type An 1), one can slightly modify the above de nition and de ne Macdonald's polynomials for the gln as a basis in C [x1 ; : : : ; xn]Sn labeled by the partitions . We will keep the same notation P (x; q; t) for these polynomials. In this form they were introduced in [M1]. For the root system An 1 the polynomials P (x; q; t) can be de ned in a di erent way; namely, they can be de ned as the eigenfunctions of a certain family of commuting di erence operators. Recall that we have identi ed C [P ]W for the root system An 1 the space of with symmetric polynomials in x 1 1 ; : : : ; x 1 n of degree zero (see end of Section 1.1). Let us de ne the following operators acting in C [P ]W : (4.1.3) Mr = tr(r n) X I f1;::: ;ng jIj=r Yi2I j = 2I t2xi xj xi xj Tq2;I ; where (Tq2;if)(x1; : : : ; xn) = f(x1; : : : ; q2xi; : : : ; xn), Tq2;I = Qi2I Tq2;i and r = 1; : : : ; n. It is not too di cult (though it is not quite obvious) to show that these operators preserve the space of symmetric polynomials. Note that since the total degree is zero, Mnf = f for every f 2 C [P ]W , and any di erence operator acting in polynomials in x1; : : : ; xn is de ned by its action in C [P ]W uniquely up to a multiple of Mn. We could consider the case of gln rather then sln thus replacing C [P ]W by C [x1 ; : : : ; xn]Sn ; then all Mi act non-trivially. Let us, however, stick to the sln case. Theorem 4.1.2. (Macdonald) (1) [Mi;Mj] = 0. (2) Mr is self-adjoint with respect to the inner product h ; iq;t. (3) MrP (x; q; t) = cr P (x; q; t), and cr = X jIj=rYi2I q2 it2 i ; where i = n+1 2i 2 (see end of Section 1.1). This characterization of Macdonald's polynomials is analogous to the de nition of Jacobi polynomials as eigenfunctions of the commuting family D of di erential operators given in Chapter 3. In fact, one can show that any di erential operator D 2 D can be obtained as a certain linear combinations of coe cients of expansion of Macdonald's di erence operatorsMi in powers of (q 1). However, these expressions are rather messy; we will return to this relation later (see remark at the end of the next section). We also note (though it is not relevant for our purposes) that the same holds for an arbitrary root system: Macdonald's polynomials can be de ned as eigenfunctions of a certain family of commuting di erence operators (see [C2]); unfortunately, for root systems other than An it is very di cult to write these di erence operators explicitly. 4.2 Macdonald's polynomials of type A as generalized characters Through this section, we assume t = qk; k 2 N and show how one gets Macdonald's polynomials P (x; q; qk) for the root system An 1 as generalized characters. The construction is quite parallel to that of Section 3.2. In this section we only consider g = sln; unless otherwise speci ed, the words \representation", \intertwiner", \generalized character" stand for representation of Uqsln, etc. Also, all the objects are de ned over the eld C q = C (q1=2n). Let U be the nite-dimensional representation of Uqg with the highest weight (k 1)n!1; this is a q-analogue of the representation U = S(k 1)nC n considered in Section 3.2. As before, this representation can be realized explicitly in homogeneous polynomials of degree (k 1)n of n variables x1; : : : ; xn with the action of Uqsln given by (4.2.1) hi 7! xi @ @xi xi+1 @ @xi+1 ; ei 7! xiDi+1; fi 7! xi+1Di; (Dif)(x1; : : : ; xn) = f(x1; : : : ; qxi; : : : ; xn) f(x1; : : : ; q 1xi; : : : ; xn) (q q 1)xi : Since the multiplicities in tensor products for Uqg are the same as for g, we have the following proposition: Proposition 4.2.1. Let 2 P+. A non-zero Uqsln-homomorphism :L ! L U exists i (k 1) 2 P+; if it exists, it is unique up to a factor. As in Chapter 3, we denote k = + (k 1) and de ne the intertwiner and the corresponding generalized character ' : (4.2.2) : L k ! L k U; ' = : As before, we consider ' as a scalar-valued function, and choose the identi cation U [0] ' C so that ' = e +(k 1) + lower order terms. Proposition 4.2.2. (4.2.3) '0 = k 1 Y i=1 Y 2R+(e =2 q2ie =2): Proof. First, we prove the following statement: Lemma 1. ' is divisible by (1 q2je ) for any positive root and 1 j k 1. The proof is done in several steps. Let us introduce Fi = fiq dihi=2; then (Fi) = Fi q dihi + 1 Fi: Let F be a (non-commutative) polynomial in F1; : : : ; Fn 1 of weight ; 2 Q+. Let 'F = TrL ( Fe2 ih). Also, let us x a basis in U : U [ ] = C u ; 2 Q. Then Lemma 2. There exists a polynomial PF 2 C (q)[P ] such that (4.2.4) 'F = PF' Q 2Q+(1 q( ; )e )u : Proof is by induction in 2 Q+. For = 0 the statement is obvious. Now, let = Pmi i;Pmi = m and assume that the statement is proved for all 0 < . Take F = Fj1 : : : Fjm . Then F = (Fj1) : : : (Fjm) = Pi q iF ( i) ~ F ( i)q h i + F q h , where i 2 Q+; i < ; i 2 Z, and F ( ) has weight . Therefore, using the intertwining property of and the cyclic property of the trace, we get 'F = Tr( (F ) e2 ih) = q h Tr(F e2 ih) +A = q( ; )e 'F + A; where A =P q i ~ F ( i)q h i Tr(F ( i) e2 ih). Thus, 'F = 1 1 q( ; )e A: On the other hand, it follows from the induction assumption that A is an expression of the form (4.2.4) containing only the factors 1 q( ; )e with < in the denominator. This completes the proof of Lemma 2. Lemma 3. Let 2 R+, and let F be a (non-commutative) polynomial in Fi which in the limit q = 1 becomes a root element of sln. Then PFk 1 is a non-zero polynomial relatively prime to k 1 Q j=1(1 q2je ). It su ces to prove this lemma for q = 1. But for q = 1, (F ) = F 1+1 F , and therefore 'Fk 1 = Tr( F k 1 e2 ih) = F Tr( F k 2 e2 ih) + Tr( F k 2 e2 ihF ) = F 'Fk 2 + e 'Fk 1 ; so 'Fk 1 = (1 e ) 1F 'Fk 2 = : : : = (1 e )1 k' F k 1 u0: Since F k 1 u0 = c u(1 k) for some c 6= 0, we see that PFk 1 = c Q (k 1) (1 e ) (1 e )k 1 = c Y <(k 1) 6=s (1 e ) k 1 Y s=1(1 + e + : : :+ e(1 s) ): One can easily see that this polynomial is relatively prime to 1 e . Thus, we have proved Lemma 3. Now, let us return to the proof of Lemma 1. Let us write Tr( F k 1 e2 ih) = PFk 1 ' k 1 Q j=1(1 q2je ) : Since the left-hand side is a non-zero polynomial, and PFk 1 is relatively prime to k 1 Q j=1(1 q2je ), we see that ' must be divisible by k 1 Q j=1(1 q2je ). So, Lemma 1 is proved. Now it is easy to prove Proposition 4.2.2: Lemma 1 implies that we have the following identity: '0 = f k 1 Q j=1 Q 2R+(1 q2je ) for some polynomial f ; comparing the highest and the lowest terms on both sides we see that f = e(k 1) . This completes the proof of Proposition 4.2.2. Now we can prove the main theorem of this section: Theorem 4.2.3. If 2 P+ and ' ; '0 are the generalized characters for Uqsln de ned by (4.2.2), (4.2.3) then ' is divisible by '0, and the ratio ' ='0 is the Macdonald's polynomial P (q; qk) for the root system An 1. Proof. First, the same arguments as in the proof of Theorem 3.2.3 { no changes are needed { show that ' ='0 is a symmetric polynomial with highest term e . Therefore, to prove the theorem it su ces to prove that these ratios are orthogonal with respect to the inner product h ; ik. This immediately follows from the orthogonality theorem for generalized characters and formula for '0 (Proposition 4.2.2). Indeed, we know from the orthogonality theorem that [' ' ]0 = 0 if 6= . Therefore, [(' ='0)(' ='0)'0 '0 ]0 = 0. Due to Proposition 4.2.2, '0 '0 = Q 2 R k 1 Q i=0(1 q2ie ) = q;t, which proves the orthogonality of f' ='0g with respect to the inner product h ; ik. Remark 4.2.4. Note that the proof only uses the following properties of U : (1) All weight subspaces are one dimensional. (2) ek u0 = 0; ek 1 u0 6= 0 for every 2 R+. (we also used Proposition 4.2.1, which ensures existence of intertwiners; however, it can be deduced from properties (1), (2) above.) Thus, the same theorem must be true if we replace U by any other representation satisfying these properties. However, one can check that for Uqsln we have only two possibilities: U = L(k 1)n!1 (which we used) or U = L(k 1)n!n 1 , which can be obtained from the previous one by an outer automorphism of g, i.e. by ip of the Dynkin diagram. For other Lie algebras, such representations do not exist at all (except some small number of exceptional cases, where such representations exist only for nite number of values of k), which explains why this theory can not be generalized to arbitrary root systems in a trivial way. Remark. This theorem can be generalized for the case of generic k, i.e., the case where q; t are independent variables; see details in [EK2]. 4.3 The center of Uqsln and Macdonald's operators In this section we show how one can get Macdonald's operators Mr introduced in Section 4.1 from the quantum group Uqsln. This construction is parallel to the one for q = 1 (see Section 3.2). As before, in this section we only consider g = sln and t = qk; k 2 N . Recall (see Section 2.3) that we have denoted by DOq the ring of di erence operators, i.e. operators of the form P 2 1 2P_ a T , where T e = qh ; ie , and a 2 C q [P ](qme 1) 1. We have also constructed for every element c of the center of Uqsln a di erence operator Dc 2 DOq such that for every intertwiner we have Tr( ce2 ih) = DcTr( e2 ih). These operators commute, and generalized characters are their common eigenfunctions. Let us show that applying this construction to the above case (i.e., g = sln and U is chosen as in the beginning of Section 4.2) we can get Macdonald's di erence operators Mr de ned by (4.1.3). Theorem 4.3.1. De ne the central elements cr 2 Z(Uqsln); r = 1 : : : ; n 1 by (4.3.1) cr = c n r q ; (cf. Theorem 1.2.1), where iq is the q-deformation of the representation of sln in the i-th exterior power iC n of the fundamental representation, and let Dcr 2 DOq be the corresponding di erence operator (see Theorem 2.3.1). Then (4.3.2) Mr = ' 1 0 Dcr '0; where Mr is Macdonald's operator introduced in (4.1.3), and '0 is the operator of multiplication by the function '0 de ned by (4.2.3). Proof. Let us rst prove that (4.3.3) ' 1 0 Dcr ('0P ) =XI q2Pi2I( +k )iP ; where the sum is taken over all subsets I f1; : : : ; ng of cardinality r. Indeed, we know from Theorem 2.3.4 that Dr' = ch n r(q 2( +k ))' : Since ch n r = en r r (1;:::;1)P e , where the sum is taken over all = ( 1; : : : ; n) such that i = 0 or 1, P i = n r, we get ch n r(q 2( +k )) =XI q2Pi2I( +k )i : Since P = ' ='0, we get (4.3.3). Comparing (4.3.3) with the formula (4.1.4) for eigenvalues of Macdonald's operators, we see thatMr and ' 1 0 Dcr '0 coincide on Macdonald's polynomials, and thus, on all symmetric polynomials. Repeating the uniqueness arguments outlined in the proof of Lemma 2.3.2, we see that it is only possible if they are equal. Remarks. (1) Recently a straightforward proof of Theorem 4.3.1 was found by Mimachi ([Mi]). (2) The central elements cr are closely related to those constructed in [FRT]. Essentially, the central elements constructed in [FRT] are traces of the powers of L-matrix, whereas cr are coe cients of the characteristic polynomial of L. This theorem allows one to see the relation between the di erential operators from D (see (3.1.7)) and Macdonald's di erence operators. This relation is nothing but the relation between the center of Uqsln (which, as we have seen, is spanned by the elements cV = (1 TrV )(R21R(1 q 2 ))) and the center of Usln, which does not have a nice explicit description, but can be described by means of HarishChandra isomorphism Z(Ug) ' (Sh)W . CHAPTER V INNER PRODUCT AND SYMMETRY IDENTITIES FOR MACDONALD'S POLYNOMIALS In this chapter we use the technique of generalized characters developed in the previous sections to prove so-called inner product (norm) and symmetry identities for Macdonald's polynomials of type An. These results are due to the author and Pavel Etingof ([EK5]). Inner product identities express the norm hP ; P i as a certain product over the positive roots. They have been conjectured by Macdonald for arbitrary root systems. He also gave a proof for An (unpublished); the proof for arbitrary root systems was given in a recent paper of Cherednik [C2]. Symmetry identity relates the values of P (q2( +k )) and P (q2( +k )). For An it was rst proved by Koornwinder (unpublished), so the rst published proof is in [EK5]. Again, recently Cherednik proved this identity for arbitrary root systems ([C3]). In this chapter we only consider root system of type An 1, i.e. g = sln. Our proofs use the realization of Macdonald's polynomials as generalized characters for Uqsln (see Chapter 4) and the technique of representing identities in the category of representations of a quantum group by ribbon graphs, developed by Reshetikhin and Turaev. We refer the reader to their papers [RT1, RT2] or to recent books of Turaev [T] and Kassel [Kas] for description of this technique; a very brief introduction can be found in the Appendix to [EK5]. 5.1 Inner product identities Let us x a positive integer k. Recall (see Chapter 4) that we have de ned Macdonald's polynomials P 2 C q [P ]; 2 P+ and an inner product h ; ik in C q [P ] such that hP ; P ik = 0 if 6= . The goal of this section is to calculate hP ; P ik; our proof is based on Theorem 4.2.3, which shows that P can be expressed in terms of generalized characters for Uqsln. Recall the notations U = Uk 1 = S(k 1)nC n ; u0 = uk 1 0 2 U [0]; k = + (k 1) ; : L k ! L k U; ' = = e k u0 + lower order terms introduced in Chapter 4; also, recall the Chevalley involution ! and Shapovalov form ( ; )V : V V ! ! C , discussed in Section 1.2. We assume that Shapovalov form in U is normalized so that (u0; u0)U = 1. Lemma 5.1.1. The inner product hP ; P ik can be calculated from (5.1.1) 1 = hP ; P ik1 ; where 1 is an invariant vector in L k L! k and : L k L! k ! L k L! k is the following operator: (5.1.2) L k L! k ! ! L k U U! L! k Id ( ; )U Id ! L k L! k : Proof. It follows from Theorem 4.2.3 that hP ; P ik = h' ; ' i1, where the inner product on generalized characters is introduced in Theorem 2.1.2. Now we can repeat the same arguments we used in the proof of Theorem 2.1.2. It will be convenient to rewrite this in a slightly di erent way as follows: Theorem 5.1.2. In the notations of previous Lemma, we have: (5.2.3) hP ; P ik = (hv k ; v ki)U ; where the intertwiner : L k ! L k U! is de ned by the condition (v k) = v k (u0)! + lower order terms. Proof. The proof is obvious if we use the technique of ribbon graphs. Namely, it follows from Lemma 5.1.1 that the inner product hP ; P ik = A can be de ned from the following identity of ribbon graphs: Φλ Φλ ω -1 φ ψ n(k-1) ω 1