The representation ring of the unitary groups and Markov processes of algebraic origin
aa r X i v : . [ m a t h . R T ] M a r THE REPRESENTATION RING OF THE UNITARY GROUPSAND MARKOV PROCESSES OF ALGEBRAIC ORIGIN
GRIGORI OLSHANSKI
To the memory of Andrei Zelevinsky
Abstract.
The paper consists of two parts. The first part introduces the rep-resentation ring for the family of compact unitary groups U (1), U (2), . . . . Thisnovel object is a commutative graded algebra R with infinite-dimensional homoge-neous components. It plays the role of the algebra of symmetric functions, whichserves as the representation ring for the family of finite symmetric groups. Thepurpose of the first part is to elaborate on the basic definitions and prepare theground for the construction of the second part of the paper.The second part deals with a family of Markov processes on the dual object tothe infinite-dimensional unitary group U ( ∞ ). These processes were defined in ajoint work with Alexei Borodin (J. Funct. Anal. 2012). The main result of thepresent paper consists in the derivation of an explicit expression for their infin-itesimal generators. It is shown that the generators are implemented by certainsecond order partial differential operators with countably many variables, initiallydefined as operators on R . Contents
1. Introduction 31.1. Preliminaries: the symmetric group case 31.2. The results 41.3. The representation ring for the unitary groups: the algebra R U ( ∞ )? 61.5. The Markov generators 71.6. Lifting of multivariate Jacobi differential operators to algebra R R R R { σ λ } related to the Schur rational functions 132.4. Example: bases related to Macdonald polynomials 142.5. Structure constants of multiplication 142.6. The isomorphism R → Rep( gl (2 ∞ )) 15 R with Sym 162.8. The subalgebras R and R U ( ∞ ) 203.1. Description of extreme characters: the Edrei-Voiculescu theorem 203.2. The quotient algebra b R = R/J n + , n − ) 253.4. Symmetries 263.5. The homomorphisms R → C (Ω) and R → C (Ω) 273.6. Analog of the Vershik-Kerov ring theorem 294. The operator D z,z ′ ,w,w ′ S N ∞ N D m,m + a, ,b → Jacobi 599.6. Completion of proof 6210. Appendix: uniform boundedness of multiplicities 63References 64 Introduction
Preliminaries: the symmetric group case.
The present paper deals withcertain combinatorial and probabilistic aspects of the representation theory of theinfinite-dimensional unitary group U ( ∞ ). A parallel theory also exists for the infinitesymmetric group S ( ∞ ). That theory is simpler and better developed, and it servedas a motivation for the present paper. So I start with a brief overview of somerelevant results which hold in the symmetric group case.In the modern interpretation, classical Frobenius’ construction [12] of irreduciblecharacters of the symmetric groups S ( N ) relies on the isomorphism of graded al-gebras Rep( S (1) , S (2) , . . . ) ≃ Sym, where Sym denotes the algebra of symmetricfunctions and Rep( S (1) , S (2) , . . . ) is our notation for the representation ring of thefamily { S ( N ) : N = 1 , , . . . } of the finite symmetric groups.The algebra Rep( S (1) , S (2) , . . . ) can be described as follows:Rep( S (1) , S (2) , . . . ) := ∞ M N =0 Rep SN (1.1)where Rep SN is the space of class functions on S ( N ), and the multiplicationRep SM ⊗ Rep SN → Rep SM + N is given by the operation of induction from S ( M ) × S ( N ) to S ( M + N ).(This definition should not be confused with that of the representation ring of anindividual group, see, e.g., Segal [39]).The algebra Rep( S (1) , S (2) , . . . ) has a distinguished basis formed by the irre-ducible characters of the symmetric groups. Under the isomorphism Rep( S (1) , S (2) , . . . ) → Sym, called the characteristic map , this basis is transformed into the distinguishedbasis in Sym formed by the Schur symmetric functions.These facts are well known, see e.g. Macdonald [25, Chapter I, Section 7].The infinite symmetric group S ( ∞ ) is defined as the union of the infinite chain S (1) ⊂ S (2) ⊂ · · · ⊂ S ( N − ⊂ S ( N ) ⊂ · · · (1.2)of finite symmetric groups. For S ( ∞ ), the conventional notion of irreducible charac-ters is not applicable. However, there exists a reasonable analog of normalized irre-ducible characters (that is, irreducible characters divided by dimension). These arethe so-called extreme characters whose definition, first suggested by Thoma [41], wasinspired by the Murray–von Neumann theory of factors. Thoma discovered that theextreme characters of S ( ∞ ) admit an explicit description: they are parameterizedby the points of the Thoma simplex Ω S , a convex subset in the infinite-dimensionalcube [0 , ∞ . Note that Ω S is compact in the product topology of [0 , ∞ .The dual object to the group S ( N ) is defined as the set [ S ( N ) of its irreduciblecharacters, and it can be identified with the set Y N of Young diagrams with N boxes. Likewise, we regard the set of extreme characters of the group S ( ∞ ) as (one GRIGORI OLSHANSKI of the possible versions of) the dual object \ S ( ∞ ) and identify it with the Thomasimplex Ω S .Vershik and Kerov [42], [43] initiated the asymptotic theory of characters (seealso Vershik’s foreword to [17]). They explained how the extreme characters of thegroup S ( ∞ ) arise from the normalized irreducible characters of the groups S ( N ) ina limit transition as N goes to infinity. In the asymptotic theory of characters, thealgebra Sym still plays an important role. In particular, the so-called ring theorem of Vershik and Kerov says that the extreme characters of S ( ∞ ) are in a one-to-onecorrespondence with those linear functionals on Sym that are multiplicative, takenonnegative values on the basis of Schur functions, and vanish on the principal ideal( e − ⊂ Sym, where e is the first elementary symmetric function (see Vershik-Kerov [18] and also Gnedin-Olshanski [13]).Now I proceed to probabilistic results. First, note that the embedding S ( N − ⊂ S ( N ) gives rise, by duality, to a canonical “link” [ S ( N ) \ S ( N − X Y between two spaces I mean a “generalized map” which assigns toevery point of X a probability distribution on Y ; in other words, a link is given bya Markov kernel (which in our case is simply a stochastic matrix). As explained inBorodin-Olshanski [6], the dual object \ S ( ∞ ) can be viewed as the projective limitof the chain d S (1) L99 d S (2) L99 · · ·
L99 \ S ( N − L99 [ S ( N ) L99 · · · (1.3)taken in an appropriate category with morphisms given by Markov kernels. Thus, S ( ∞ ) is an inductive limit group while its dual object \ S ( ∞ ) is obtained by takinga kind of projective limit.In [3], Borodin and I constructed a two-parameter family of continuous timeMarkov processes on the Thoma simplex. Our work was inspired by our previousstudy of the problem of harmonic analysis on S ( ∞ ) and substantially used thecanonical links from (1.3). We proved that the Markov processes in question havecontinuous sample trajectories and consequently are diffusion processes. The proofrelied on the computation of the infinitesimal generators of the processes: we showedthat the generators are given by certain second order differential operators initiallyacting on the the quotient algebra Sym / ( e − / ( e − ֒ → C (Ω S ) , (1.4)where C (Ω S ) denotes the Banach algebra of continuous functions on the compactspace Ω S .1.2. The results.
Let us turn to the compact unitary groups. They are organizedinto a chain similar to (1.2), U (1) ⊂ U (2) ⊂ · · · ⊂ U ( N − ⊂ U ( N ) ⊂ · · · , and we set U ( ∞ ) := S ∞ N =1 U ( N ). The extreme characters of the group U ( ∞ ) werefirst investigated by Voiculescu [45]. They are parameterized by the points of aninfinite-dimensional space Ω, which can be realized as a convex subset in the productof countably many copies of R + (see Subsection 3.1 below). Note that Ω is locallycompact. Like the dual object to S ( ∞ ), the space Ω = \ U ( ∞ ) can be identified withthe projective limit of the dual chain [ U (1) L99 [ U (2) L99 · · ·
L99 \ U ( N − L99 \ U ( N ) L99 · · · (1.5)Although the groups S ( ∞ ) and U ( ∞ ) are structurally very different, there is asurprising similarity in the description of their characters. An explanation of thisphenomenon is suggested in Borodin-Olshanski [6].Here is a brief description of what is done in the present paper.1. The attempt to extend the definition of the representation ring to the family ofthe unitary groups leads us to a novel object — a certain graded algebra R , whichplays the role of the algebra Sym.2. An analog of the embedding (1.4) is found. As explained below, it may beviewed as a kind of Fourier transform on U ( ∞ ).3. The main result is the computation of the infinitesimal generators for thefour-parameter family of Markov processes on Ω, previously constructed in Borodin-Olshanski [5]. It is shown that the generators in question are implemented by certainsecond order partial differential operators, initially defined on R .Now I will describe the results in more detail. As will be clear, for all the similar-ities between S ( ∞ ) and U ( ∞ ), the unitary group case turns out to be substantiallymore complicated.1.3. The representation ring for the unitary groups: the algebra R . Atfirst it was unclear to me if there is a good analog of the representation ring for thefamily { U ( N ) } . The difficulty here is that, in contrast to the case of finite symmetricgroups, induced characters have infinitely many irreducible constituents. Therefore,directly following the definition of Rep( S (1) , S (2) , . . . ) we see that products of basiselements are infinite sums; how to deal with them? The proposed solution is toenlarge the space and allow infinite sums. This leads to the following definition:The algebra R , the suggested analog of the algebra Sym, is the graded algebra offormal power series of bounded degree, in countably many variables each of whichhas degree 1. The variables are denoted by ϕ n , where n ranges over Z .Recall that Sym is the projective limit of polynomial algebras:Sym = lim ←− C [ e , . . . , e k ] , (1.6)where k → ∞ and e , e , . . . are the elementary symmetric functions.Likewise, R also can be represented as the projective limit of polynomial algebras: R = lim ←− C [ ϕ − l , . . . , ϕ k ] , (1.7) GRIGORI OLSHANSKI where k, l → + ∞ .A substantial difference is that deg e k = k , while deg ϕ n = 1 for all n ∈ Z . Becauseof this, the homogeneous components of Sym have finite dimension, while thoseof R are infinite-dimensional. Nevertheless, it turns out that the projective limitrealization (1.7) is a kind of finiteness property which can be efficiently exploited.As in the case of the algebra Sym, in R there exist various interesting bases, butthese are topological bases . Two bases are of particular importance for the purposeof this paper. They are denoted as { ϕ λ } and { σ λ } , where the subscript λ rangesover the set of highest weights of all unitary groups. The basis { ϕ λ } is formed by themonomials in letters ϕ n and is similar to the multiplicative basis in Sym generatedby the elementary symmetric functions. The basis { σ λ } is an analog of the Schurfunctions. The interplay between these two bases plays an important role in thederivation of the main result.By the Schur-Weyl duality, the representation ring for the family { S ( N ) } is iso-morphic to a certain representation ring of a single object — the Lie algebra gl ( ∞ ).Likewise, using the fermion version of the Howe duality one can identify the repre-sentation ring for the family { U ( N ) } with a certain representation ring for the Liealgebra gl (2 ∞ ) (for more detail, see Subsection 2.6 below).1.4. What is the Fourier transform on U ( ∞ ) ? Let us consider first a finitegroup G and let M inv ( G ) denote the space of complex measures on G , invariantwith respect to inner automorphisms. Next, let b G stand for the set of normalizedirreducible characters and Fun( b G ) denote the space of functions on b G . By integratinga character χ ∈ b G against a measure m ∈ M inv ( G ) we get a linear map F : M inv ( G ) → Fun( b G ) . Using the functional equation for normalized irreducible characters one sees that F turns the convolution product of measures into the pointwise product of functions.So F is a reasonable version of Fourier transform.More generally, the above definition of Fourier transform F works perfectly when G is a compact group. Then as M inv ( G ) one can still take the space of invariant com-plex measures on G or, if G is a Lie group, the larger space of invariant distributionsor else an appropriate subspace therein, depending on the situation.But what happens for G = S ( ∞ ) or G = U ( ∞ )? The dual object b G has been de-fined, and one knows that it is large enough in the sense that the extreme charactersof these groups separate the conjugacy classes. The problem is that the above def-inition of M inv ( G ) no longer works. For instance, the only invariant finite measureon S ( ∞ ) is the delta measure at the unit element.This difficulty can be resolved as follows. For a group G which is an inductivelimit of compact groups G ( N ) we define M inv ( G ) := lim −→ M inv ( G ( N )) , where the map M inv ( G ( N − → M inv ( G ( N )) is given by averaging over the actionof the group of inner automorphisms of G ( N ). In more detail, given a measure M ∈ M inv ( G ( N − M inv ( G ( N )) is defined as Z g ∈ G ( N ) M g dg, where M g denotes the transformation of M (which we transfer from G ( N − G ( N )) under the conjugation by an element g ∈ G ( N ), and dg denotes thenormalized Haar measure on G ( N ).In the case of G = S ( ∞ ) it is readily verified that M inv ( S ( ∞ )) can be identified,in a natural way, with the quotient algebra Sym / ( e − G = U ( ∞ ) the situation is more delicate. In the first approximation,the analog of Sym / ( e −
1) is the quotient algebra
R/J , where J is the followingprincipal ideal J := ( ϕ − , ϕ := X n ∈ Z ϕ n . (1.8)However, this algebra is too large and one has to narrow it in order for the Fouriertransform to be well defined. We discuss two variants of doing this, both of whichseem to be quite natural. Note that there are also many other possibilities: theydepend on the concrete choice of the spaces M inv ( U ( N )). I did not go too far in thisdirection, because for the main result it was sufficient to dispose of the simplest wayto relate the algebra R to the space \ U ( ∞ ) = Ω.Note that in a number of cases involving those of G = S ( ∞ ) and G = U ( ∞ ), theset of conjugacy classes of G can be endowed with a natural semigroup structure(see [29], [30], [31]). Then one may endow M inv ( G ) with a multiplication, which isan analog of convolution product and which turns into pointwise multiplication on b G under a suitable version of Fourier transform.1.5. The Markov generators.
The Markov processes on Ω constructed in Borodin-Olshanski [5] depend on four complex parameters z, z ′ , w, w ′ subject to certain con-straints (see Definition 6.1). Let us ignore for a moment the constraints, so that z, z ′ , w, w ′ are arbitrary complex numbers, and consider a formal second order partialdifferential operator D z,z ′ ,w,w ′ = X n ,n ∈ Z A n n ( . . . , ϕ − , ϕ , ϕ , . . . ) ∂ ∂ϕ n ∂ϕ n + X n ∈ Z B n ( . . . , ϕ − , ϕ , ϕ , . . . ; z, z ′ , w, w ′ ) ∂∂ϕ n , (1.9)where the variables ϕ n are indexed by integers n ∈ Z , the second order coefficients A n n are certain (complicated) quadratic expressions in the variables, and the first GRIGORI OLSHANSKI order coefficients B n are certain linear expressions which involve the parameters, seethe explicit formulas (4.1) and (4.2) below.The main result of the paper can be informally stated as follows. Theorem 1.1.
Assume that the quadruple ( z, z ′ , w, w ′ ) satisfies the necessary con-straints, so that the construction of [5] provides a Markov process X z,z ′ ,w,w ′ on Ω .Then the generator of X z,z ′ ,w,w ′ is implemented by the differential operator D z,z ′ ,w,w ′ . A rigorous version is given in Theorem 7.1.Note that the Markov generator in question is defined on a dense subspace of C (Ω), the Banach space of continuous functions on Ω vanishing at infinity. Torelate such an operator with an operator acting on R we use the Fourier transformdiscussed in the preceding subsection. Here we use the fact that D z,z ′ ,w,w ′ preservesthe principal ideal J ⊂ R (see (1.8) above) and so also acts on R/J .The operator D z,z ′ ,w,w ′ is well adapted to the basis { ϕ λ } in R while the Markovgenerators are initially defined by their action on another basis, { σ λ } . This is themain source of difficulty in the proof of the main theorem: transition from one basisto another one is achieved by rather long computations.The construction of the processes X z,z ′ ,w,w ′ in our work [5] is based on a limittransition along the chain (1.5): we find jump processes on the dual objects \ U ( N )which are consistent with the “links” \ U ( N ) \ U ( N − X z,z ′ ,w,w ′ and what can be explicitly computed. The computationof the Markov generators in the present paper is the first step in this direction.The fact that the Markov generators are implemented by differential operatorsmakes plausible the conjecture that the sample trajectories of the processes arecontinuous (the diffusion property). In the symmetric group case (see Borodin-Olshanski [3]) we give a simple proof of the diffusion property for the processeson the Thoma simplex Ω S using the realization of their generators as differentialoperators on Sym. However, the structure of the differential operator D z,z ′ ,w,w ′ issubstantially more complicated, because, in contrast to the symmetric group case,the coefficients A n n are given by infinite series. This is an obstacle to extendingthe approach of [3].It seems that the Markov generators cannot be written in terms of the naturalcoordinates on Ω, and the same holds in the models related to S ( ∞ ), studied in[3] and [7] (a possible explanation is that the coordinate functions do not enterthe domain of the generators, see in this connection the discussion in Petrov [36,Remark 5.4] concerning a simpler model). This is why one needs to use a moreinvolved construction using the algebra R (or, in the symmetric group case, thealgebra Sym). Lifting of multivariate Jacobi differential operators to algebra R . Let m = 1 , , , . . . . The Jacobi partial differential operator in m variables t , . . . , t m isgiven by D ( a,b ) m := m X i =1 t i (1 − t i ) ∂ ∂t i + " b + 1 − ( a + b + 2) t i + X j : j = i t i (1 − t i ) t i − t j ∂∂t i ! . (1.10)Here a and b are parameters. In the simplest case m = 1 this operator turns intothe familiar hypergeometric ordinary differential operator D ( a,b ) = t (1 − t ) d dt + [ b + 1 − ( a + b + 2) t ] ddt . The operator D ( a,b ) is attached to the Jacobi orthogonal polynomials with the weightfunction t b (1 − t ) a on the unit interval 0 ≤ t ≤
1, that is, the Jacobi polynomialsare just the polynomial eigenfunctions of D ( a,b ) .In the case of several variables, despite the singularities on the hyperplanes t i = t j ,the operator D ( a,b ) m is well defined on the space of symmetric polynomials in t , . . . , t m and is diagonalized in the basis of m -variate symmetric Jacobi polynomials. Thelatter polynomials are a particular case of the Heckman-Opdam orthogonal polyno-mials, which corresponds to the root system BC m and a special choice of the “Jackparameter” (see e.g. Heckman [14], Koornwinder [22]). The operator D ( a,b ) m is wellknown; it appeared (in a more general form involving the Jack parameter) in manyworks, see, e.g., Baker-Forrester [1].Given m , let us fix two nonnegative integers k and l such that k + l = m . Weassume that m + 1 variables ϕ − l , . . . , ϕ k are expressed through m variables t , . . . , t m via k X n = − l ϕ n u n = k Y i =1 ( t i + (1 − t i ) u ) · m Y i = k +1 (1 − t i + t i u − ) , where the left-hand side should be viewed as a generating series for ϕ − l , . . . , ϕ k withan auxiliary indeterminate u (then, by equating the coefficients of monomials u n inthe both sides, we can write ϕ n ’s as polynomials in t i ’s). Setting u = 1 one sees thatthe constraint P kn = − l ϕ n = 1 holds. Moreover, we may identify the algebra Sym m of symmetric polynomials in variables t , . . . , t m with b R ( k, − l ) := C [ ϕ − l , . . . , ϕ k ] (cid:14) k X n = − l ϕ n − ! , the quotient by the principal ideal generated by the element P kn = − l ϕ n − b R ( k, − l ) as the quotient R/J ( k, − l ),where J ( k, − l ) denotes the ideal of R generated by the elements ϕ k +1 , ϕ k +2 , . . . ; ϕ − l , ϕ − l − , . . . ; ϕ − l + · · · + ϕ k − . (1.11) Note that the ideal does not change if ϕ − l + · · · + ϕ k − ϕ −
1, where ϕ is defined in (1.8) above.From the proof of Theorem 1.1 one can extract the following fact: Theorem 1.2.
Let us assume that parameters z and w are nonnegative integers,which are not both . Let us denote them by k and l , respectively.In this special case the differential operator D z,z ′ ,w,w ′ preserves the ideal J ( k, l ) ⊂ R and so determines an operator on R/J ( k, − l ) = b R ( k, − l ) . The latter operatorcoincides with the ( k + l ) -variate Jacobi operator (1.10) with parameters a = z ′ − k , b = w ′ − l . This fact clarifies the nature of the differential operator D z,z ′ ,w,w ′ . Indeed, fromTheorem 1.2 one can see that the sophisticated expression for D z,z ′ ,w,w ′ appears asthe result of formal analytic extrapolation, with respect to parameters ( k, l, a, b ), ofthe Jacobi differential operators D ( a,b ) k + l rewritten in a new set of variables. Note thatas k and l increase, the ideals J ( k, − l ) decrease and their intersection ∩ ∞ k,l =1 J ( k, − l )coincides with the principal ideal J ⊂ R generated by the sole element ϕ −
1. Notealso that the extrapolation procedure is purely formal, because the integers k and l ,whose sum m = k + l initially represents the number of variables, finally turn intocomplex parameters.It is interesting to compare this picture with what is done in the work of Sergeevand Veselov [40] which deals with the same Jacobi differential operators (involvingthe additional “Jack parameter”). However, in [40] the operators are lifted to the al-gebra Sym, while our target space is the algebra R . The initial motivation of Sergeevand Veselov is also different: they used the lifting to Sym as a tool for constructingsuper versions of quantum integrable systems in finite dimensions, while our interestis in infinite-dimensional Markov dynamics. (See also the papers Desrosiers-Halln¨as[9], Olshanski [32], [33] — in all these works the target space is Sym.)1.7. Organization of the paper.
Section 2 introduces the algebra R and Section3 relates it to the dual object \ U ( ∞ ). Section 4 introduces the differential operator D z,z ′ ,w,w ′ . In Sections 5 and 6 we recall some general facts about Feller Markovprocesses, next describe the “method of intertwiners” [5], and then explain howit produces a special family of Markov processes on \ U ( ∞ ) out of continuous timeMarkov chains on the discrete sets \ U ( N ). In Section 7 we formulate the maintheorem and outline the plan of its proof. The proof itself occupies Sections 8 and9. The last Section 10 is an appendix, where we prove the uniform boundednessof multiplicities in certain induced representations of compact groups; this fact wasused in Section 3.1.8. Acknowledgement.
I am grateful to Igor Frenkel for an important commentwhich I used in Subsection 2.6, and to Vladimir L. Popov who confirmed that thestatement of Proposition 10.1 is true and communicated its proof to me. I am also grateful to the anonymous referee for valuable suggestions. This research waspartially supported by a grant from Simons Foundation (Simons-IUM Fellowship)and by the RFBR grant 13-01-12449.2. The algebra R Definition of algebra R . Throughout the paper { ϕ n } stands for a doublyinfinite collection of formal variables indexed by arbitrary integers n ∈ Z .We define R as the commutative complex unital algebra formed by arbitraryformal power series of bounded degree, in variables ϕ n , n ∈ Z . Here we assume thatdeg ϕ n = 1 for every n . The algebra R is graded: we write R = L ∞ N =0 R N , wherethe elements of the N th homogeneous component R N have the form ψ = X n ≥···≥ n N a n ,...,n N ϕ n . . . ϕ n N (2.1)with no restriction on the complex coefficients a n ,...,n N .Equivalently, R can be defined as a projective limit of polynomial algebras.Namely, for a pair of integers n + ≥ n − we set R ( n + , n − ) := C [ ϕ n − , ϕ n − +1 , . . . , ϕ n + − , ϕ n + ] . Then one can write R = lim ←− R ( n + , n − ) , n + → + ∞ , n − → −∞ , where the limit is taken in the category of graded algebras.We call the natural homomorphisms R → R ( n + , n − ) the truncation maps . Let I ( n + , n − ) denote the kernel of the truncation R → R ( n + , n − ). As n ± → ±∞ , theideals I ( n + , n − ) decrease and their intersection equals { } . We take these ideals asthe base of a topology in R , which we call the I -adic topology .Following Weyl [46] we define a signature of length N as an arbitrary vector λ = ( λ , . . . , λ N ) ∈ Z N with weakly decreasing coordinates: λ ≥ · · · ≥ λ N . Theset of all such vectors is denoted by S N . In particular, S = Z . By agreement, S consists of a single element denoted by ∅ .With a signature λ ∈ S N we associate a monomial of degree N , ϕ λ := ϕ λ . . . ϕ λ N , and we agree that ϕ ∅ = 1. With this notation, (2.1) can be rewritten as ψ = X λ ∈ S N a λ ϕ λ . Initially, ψ is a formal series, but, alternatively, the above sum can be interpretedas the limit, in the I -adic topology, of the truncated finite sums, ψ = lim n ± →±∞ X λ ∈ S N : n + ≥ λ , λ N ≥ n − a λ ϕ λ . Therefore, one can say that the monomials ϕ λ form a homogeneous topological basis of R .2.2. Bases in R . We are going to describe a general recipe for constructing varioustopological bases in R which are all consistent with the projective limit realization R = lim ←− R ( n + , n − ).Let us introduce a partial order on signatures: two signatures λ , µ may be com-parable only if they have the same length N , and then λ ≥ µ ⇔ λ − µ ∈ Z + ( ε − ε ) + · · · + Z + ( ε N − − ε N ) , where ε , . . . , ε N is the natural basis of the lattice Z N . In particular, λ ≥ µ implies P λ i = P µ i . We write λ > µ if λ ≥ µ and λ = µ . Note that the signaturesof length N are precisely the highest weights of the irreducible representations of U ( N ), and the introduced order is nothing else than the standard dominance partialorder on the set of weights of the reductive Lie algebra gl ( N, C ), the complexifiedLie algebra of U ( N ).We will be dealing with various symmetric Laurent polynomials in several vari-ables u , . . . , u N , N = 1 , , . . . . The simplest example is the family of monomialsums m λ . Here λ ∈ S N and, by definition, m λ = X ( n ,...,n N ) ∈ S ( N ) · λ u n . . . u n N N , where S ( N ) · λ denotes the orbit of λ under the action of the symmetric group S ( N );in other words, the summation is over all distinct vectors ( n , . . . , n N ) ∈ Z N that canbe obtained from ( λ , . . . , λ N ) by permutations of the coordinates. By agreement, m ∅ := 1 (the same agreement is tacitly adopted for other families of polynomialsthat will appear below).Assume we are given an arbitrary family { P λ } of homogeneous symmetric Lau-rent polynomials indexed by signatures and satisfying the following triangularitycondition : P λ = X µ : µ ≤ λ α ( λ, µ ) m µ , α ( λ, µ ) ∈ C , α ( λ, λ ) = 1 (2.2)(examples will be given shortly). In particular, the number of variables in P λ equalsthe length of λ .With every such a family { P λ } we associate a family { π λ } of homogeneous ele-ments of R in the following way. We form a generating series for ϕ n ’s:Φ( u ) := X n ∈ Z ϕ n u n ∈ R [[ u, u − ]] . (2.3)Then the elements π λ in question are obtained as the coefficients in the expansionΦ( u ) . . . Φ( u N ) = X λ ∈ S N π λ P λ ( u , . . . , u N ) , N = 1 , , . . . , (2.4) and we agree that π ∅ = 1 . If P λ = m λ for all λ , then the meaning of (2.4) is clear and we obtain π λ = ϕ λ .But in the general case one has to explain how to understand the sum in the right-hand side: the answer is that it converges coefficient-wise, in the I -adic topology of R .Here is an equivalent definition. The relation (2.4) is interpreted as an infinitesystem of linear equations, X λ : λ ≥ µ α ( λ, µ ) π λ = ϕ µ , ∀ µ. (2.5)The triangularity condition (2.2) gives a sense to the infinite sum in the left-handside of (2.5) and guarantees that the infinite matrix [ α ( λ, µ )] is invertible. Then weget π λ = X ν : ν ≥ λ β ( ν, λ ) ϕ ν (2.6)with some new coefficients β ( ν, λ ) such that β ( λ, λ ) = 1.It is evident that { π λ } is a topological basis in R . Moreover, { π λ } is consistentwith the ideals I ( n + , n − ) meaning that I ( n + , n − ) is (topologically) spanned by thebasis elements that are contained in it, that is, by the elements π λ , λ ∈ S N , suchthat λ violates at least one of the inequalities n + ≥ λ , λ N ≥ n − . The quotientalgebra R ( n + , n − ) is, on the contrary, spanned by the π λ ’s such that λ satisfies theboth inequalities.2.3. Example: the basis { σ λ } related to the Schur rational functions. Letus turn now to concrete examples. The most important example is obtained whenas { P λ } we take the rational Schur functions s λ . These are symmetric Laurentpolynomials given by the same ratio-of-determinants formula as the ordinary Schurpolynomials, only the index λ is an arbitrary signature, so that the integers λ i arenot necessarily nonnegative: s λ ( u , . . . , u N ) = det[ u λ j + N − ji ] V ( u , . . . , u N ) , where the determinant in the numerator is of order N and the denominator is theVandermonde, V ( u , . . . , u N ) = Y ≤ i Observe that theMacdonald polynomials in finitely many variables (as well their degeneration, theJack polynomials) have a natural Laurent version, because they satisfy the re-lation similar to (2.7), see Macdonald [25, chapter VI, (4.17)]. Moreover, theysatisfy the condition (2.2), see [25, chapter VI, (4.7)]. Therefore, one may take P λ ( u , . . . , u N ) = P λ ( u , . . . , u N ; q, t ) (the Laurent version of Macdonald polynomi-als with two parameters ( q, t )) or P λ ( u , . . . , u N ) = P ( α ) ( u , . . . , u N ) (the Laurentversion of Jack polynomials with parameter α ), and then we get a certain topologicalbasis in R . In particular, the case q = t gives the Schur polynomials and the basis { σ λ } , and the case ( q = 0 , t = 1) give the monomial sums m λ and the basis { ϕ λ } .2.5. Structure constants of multiplication. Let, as above, { P λ } be a familyof symmetric Laurent polynomials satisfying the triangularity condition (2.2) and { π λ } be the corresponding topological basis in R . Then any homogeneous element ψ ∈ R N can be uniquely represented in the form ψ = P λ ∈ S N a λ π λ with some complexcoefficients a λ . I am going to explain how to write the operation of multiplicationin this notation. Let M and N be two nonnegative integers and λ ∈ S M + N . Partitioning thevariables in P λ into two groups, of cardinality M and N , we get an expansion of theform P λ ( u , . . . , u M + N ) = X µ ∈ S M , ν ∈ S N c ( λ | µ, ν ) P µ ( u , . . . , u M ) P ν ( u M +1 , . . . , u M + N ) , (2.12)where c ( λ | µ, ν ) are certain coefficients. Indeed, the existence, finiteness, anduniqueness of this expansion is obvious in the case P λ = m λ , and the general caseis reduced to that case using the triangularity property and the fact that for anysignature λ , the set { µ : µ ≤ λ } is finite.Now it follows from (2.4) that the same quantities c ( λ | µ, ν ) are the structureconstants of multiplication in the basis { π λ } . That is, (cid:16)X a ′ µ π µ (cid:17) (cid:16)X a ′′ ν π ν (cid:17) = X a λ π λ , a λ := X µ,ν c ( λ | µ, ν ) a ′ µ a ′′ ν . (2.13)The latter sum makes sense because we know that the expansion (2.12) is finite.2.6. The isomorphism R → Rep( gl (2 ∞ )) . The remark below is based on a com-ment by Igor Frenkel.Let gl ( ∞ ) denote the Lie algebra of complex matrices of format ∞×∞ and finitelymany nonzero entries. It has a natural basis formed by the matrix units E ij withindices i, j ranging over { , , . . . } . The Schur-Weyl duality establishes a bijectivecorrespondence S λ ↔ V λ between the irreducible representations of various sym-metric groups and a certain class of irreducible highest weight gl ( ∞ )-modules. Here λ = ( λ , λ , . . . ) is an arbitrary partition, S λ is the corresponding irreducible S ( N )-module (where N = | λ | := P λ i ), and V λ is the irreducible polynomial gl ( ∞ )-modulewhose highest weight is ( λ , λ , . . . ) with respect to the Borel subalgebra spannedby the E ij with i ≤ j . Under the Schur-Weyl correspondence, the multiplicationin Rep( S (1) , S (2) , . . . ) turns into the the tensor product of gl ( ∞ )-modules. In thissense the algebra Rep( S (1) , S (2) , . . . ) = Sym can be identified with Rep( gl ( ∞ )),the representation ring of polynomial gl ( ∞ )-modules.A similar interpretation exists for the algebra R . Namely, we replace gl ( ∞ ) withits relative gl (2 ∞ ) — the latter Lie algebra has the basis { E ij } of matrix unitswith indices i, j ranging over Z . Instead of the Schur-Weyl duality we use a versionof the “fermion” Howe duality [15] between various unitary groups U ( N ) and theLie algebra gl (2 ∞ ). This duality establishes a different kind of correspondence ofrepresentations, T λ ↔ V λ , where λ ranges over the set of all signatures. Here, for λ ∈ S N , we denote by T λ the corresponding irreducible representation of U ( N ), while V λ now stands for the irreducible gl (2 ∞ )-module with highest weight b λ = ( b λ i ) i ∈ Z which is described as follows.Recall that every signature λ of length N can be represented as a pair ( λ + , λ − )of two partitions (=Young diagrams) such that ℓ ( λ + ) + ℓ ( λ − ) ≤ N , where ℓ ( · ) is the conventional notation for the number of nonzero parts of a partition. Namely, λ = ( λ +1 , . . . , λ + ℓ ( λ + ) , , . . . , , − λ − ℓ ( λ − ) , . . . , − λ − ) . In this notation, the weight correspondence λ → b λ looks as follows b λ i = ( λ + ) ′ i , i = 1 , , . . . ; b λ − ( i − = N − ( λ − ) ′ i , i = 1 , , . . . , where ( λ ± ) ′ denotes the conjugate to λ ± partition (=Young diagram).Note that the coordinates b λ i , i ∈ Z , weakly decrease; the fact that b λ ≥ b λ isequivalent to the inequality ℓ ( λ + ) + ℓ ( λ − ) ≤ N mentioned above.About this instance of Howe duality see also Olshanski [28, Section 2] and [31,Section 17].As in the case of the Schur-Weyl duality, the multiplication in R corresponds, onthe Lie algebra side, to the tensor product of modules, so that we get an isomorphism R → Rep( gl (2 ∞ )), where Rep( gl (2 ∞ )) is our notation for the representation ringfor a special class of gl (2 ∞ )-modules. This class is generated by the weight modulesthat are locally nilpotent with respect to the upper triangular subalgebra and suchthat, for every weight b µ = ( b µ i ) i ∈ Z , the coordinates b µ i are nonnegative integers whichstabilize to a nonnegative integer N as i → −∞ and to 0 as i → + ∞ . The irreduciblemodules V λ ∈ Rep( gl (2 ∞ ) correspond to the basis elements σ λ ∈ R .2.7. Comparison of R with Sym . The two algebras have both similarities anddifferences. The homogeneous components in Sym have finite dimension while thosein R are not. The latter fact seems to be the most evident difference between R andSym. On the other hand, both algebras are projective limits of polynomial algebras:Sym = lim ←− C [ e , . . . , e n ] , R = lim ←− C [ ϕ n − , . . . , ϕ n + ] . (2.14)These polynomial algebras can be viewed as truncations of the initial algebras.All familiar homogeneous bases in Sym are parameterized by partitions, and thosein R are parameterized by signatures, which are relatives of partitions. However,these two kinds of labels, partitions and signatures, are related to the grading in avery different way: the degree of a basis element in Sym is given by the sum of partsof the corresponding partition, while the degree in R corresponds to the length N of a signature λ .This is also seen from the comparison of the representation rings Rep( gl ( ∞ )) andRep( gl (2 ∞ )). As abstract Lie algebras, gl ( ∞ ) and gl (2 ∞ ) are isomorphic, but therespective classes of modules are different, and the degrees of the irreducible modulesare defined in a very different way.Truncation in Sym and R is also defined differently. Namely, a basis element inSym is not contained in the kernel of the truncation map Sym → C [ e , . . . , e n ] if andonly if the length of the corresponding partition does not exceed n , while truncationin R is controlled by the first and last coordinates of a signature λ . In the case when λ > λ N < 0, one has λ = ℓ (( λ + ) ′ ) and | λ N | = ℓ (( λ − ) ′ ). To define a homomorphism of the algebra Sym in a commutative algebra A (forinstance, an algebra of functions on a space) it suffices to specialize, in an arbitraryway, the images of the generators e , e , . . . . In the case of R , the situation is moredelicate. Although the elements ϕ n play the role similar to that of the e n ’s, to definea morphism R → A it does not suffice to specialize the image of the ϕ n ’s. The reasonis that these elements are not generators of R in the purely algebraic sense, but only topological generators. It may well happen that a given specialization of the ϕ n ’scan be extended only to a suitable subalgebra of R . Two examples of subalgebrasare examined below.2.8. The subalgebras R and R . For λ ∈ S N , letDim N λ := s λ (1 , . . . , . This is the dimension of the irreducible representation of U ( N ) with highest weight λ . As is well known (Weyl [46], Zhelobenko [48])Dim N λ = Y ≤ i We define R ⊂ R as the subspace of elements with finite norm.Obviously, R is graded, so that we may write R = P ∞ N =0 R N . Proposition 2.2. R is a normed algebra.Proof. We have to prove that for any elements ψ ′ , ψ ′′ ∈ R one has k ψ ′ ψ ′′ k ≤ k ψ ′ kk ψ ′′ k . (2.17)Indeed, assume first that ψ ′ and ψ ′′ are homogeneous of degree M and N , respec-tively, and write ψ ′ = P a ′ µ σ µ , ψ ′′ = P a ′′ ν σ ν . By (2.13) k ψ ′ ψ ′′ k = sup λ ∈ S M + N | P µ,ν c ( λ | µ, ν ) a ′ µ a ′′ ν | Dim λ . Note that in our case, when P λ = s λ , the structure constants describe the expansionof irreducible characters restricted from U ( M + N ) to U ( M ) × U ( N ). It follows thatthese constants are nonnegative integers. Next, by counting dimensions one gets X µ ∈ S M , ν ∈ S N c ( λ | µ, ν ) Dim M µ Dim N ν = Dim M + N λ. Therefore, for every λ ∈ S M + N , | P µ,ν c ( λ | µ, ν ) a ′ µ a ′′ ν | Dim M + N λ ≤ k ψ ′ kk ψ ′′ k P µ,ν c ( λ | µ, ν ) Dim M µ Dim N ν Dim M + N λ = k ψ ′ kk ψ ′′ k . This proves the desired inequality (2.17).Now the general case, when ψ ′ and ψ ′′ are not necessarily homogeneous, followsimmediately, by taking into account the definition of the norm for non-homogeneouselements, (2.16). (cid:3) Definition 2.3. For N = 1 , , . . . we define R N ⊂ R N as the subspace of thoseelements ψ = P λ ∈ S N a λ σ λ ∈ R N for which the ratio | a λ | / Dim N λ tends to 0 as λ goes to infinity. In other words, for every ε > S N outside of which | a λ | / Dim N λ ≤ ε .Next, we set R := ∞ M N =1 R N and observe that R is a norm-closed subspace of R .Let R fin denote the space of finite linear combinations of the basis elements σ λ ,where λ = ∅ . By the very definition of R , it coincides with the norm closure of R fin . Proposition 2.4. R is closed under multiplication and so is a subalgebra in R . Note that, according to our definition, R does not contain the unity element1 = σ ∅ . Proof. Step 1. For any fixed µ ∈ S M and ν ∈ S N , where M, N ≥ 1, there exists aconstant C ( µ, ν ) such that c ( λ | µ, ν ) ≤ C ( µ, ν ) for all λ ∈ S M + N . This is a nontrivial claim whose proof is postponed to Section 10. Step 2. Let us fix µ and ν as above. We claim that σ µ σ ν ∈ R M + N . Indeed, by the definition of the multiplication in R , σ µ σ ν = X λ ∈ S M + N c ( λ | µ, ν ) σ λ . By the result of Step 1, the coefficients c ( λ | µ, ν ) are bounded from above. There-fore, to conclude that σ µ σ ν ∈ R M + N it remains to show that Dim N λ tends to infinityas λ goes to infinity along the subset X := { λ ∈ S M + N : c ( λ | µ, ν ) > } . Observe that λ ∈ X implies that the quantity λ + · · · + λ M + N remains fixed,because it is equal to ( µ + · · · + µ M ) + ( ν + · · · + ν N ).Therefore, as λ goes to infinity along X , the difference λ − λ M + N tends to + ∞ ,so that Dim N → ∞ , as it is seen from Weyl’s dimension formula (2.15). Step 3. Let us show that R is closed under multiplication. By the result of Step2, R fin R fin is contained in R . Since R fin ⊂ R is dense with respect to the normtopology, we conclude that R R ⊂ R . (cid:3) Remarks on comultiplication. By Frobenius’ reciprocity,Ind U ( M + N ) U ( M ) × U ( N ) s µ ⊗ s ν = X λ ∈ S M + N c ( λ | µ, ν ) s λ , where the left-hand side is the induced character. So, one could identify the formalsymbols σ λ with the irreducible characters s λ and say that the multiplication R M ⊗ R N → R M + N mimics the operation of induction from U ( M ) × U ( N ) to U ( M + N ).The reason to use the separate notation σ λ is that characters should be viewed as functions while elements of R behave as measures (or, more generally, distributions),which are dual objects with respect to functions.Of course, on a finite or compact group, one can use the normalized Haar measure m Haar to turn a function f into a measure, f m Haar . However, one should not forgetthat functions and measures have different functorial properties, so that when werestrict a character χ to a subgroup, we regard χ as a function, while if we induct χ from a subgroup, we tacitly treat χ as a measure. In the case of finite groups, theassignment f f m Haar is a linear isomorphism between the space of functions andthe space of measures. Because of this, Rep( S (1) , S (2) , . . . ) (the representation ringof the family of symmetric groups) possesses two dual operations, multiplication andcomultiplication making it a selfdual Hopf algebra (Zelevinsky [47]). For compactLie groups U ( N ), the situation is more delicate as the space of measures is muchlarger than the space of functions. This explains why the representation ring R , aswe have defined it, is not a Hopf algebra.Note that one can use the same structure constants c ( λ | µ, ν ) (in the basis { σ λ } )to construct a coalgebra R ◦ which is paired with R . Namely, a generic element of R ◦ is a possibly infinite sum of homogeneous elements which in turn are finite linearcombinations of symbols that we denote as χ λ ; the comultiplication in R ◦ is definedby setting, for λ ∈ S N , △ χ λ = X N ,N : N + N = N X µ ∈ S N ,ν ∈ S N c ( λ | µ, ν ) χ µ ⊗ χ ν . Then the pairing R × R ◦ → C is defined in a natural way, by proclaming { σ λ } and { χ λ } to be biorthogonal systems.Likewise, one can also define a suitable coalgebra R ◦ which is paired with thealgebra R . However, in contrast to the case of the representation ring for thesymmetric groups, I do not see any way to modify the definition of R so that itbecomes a selfdual Hopf algebra. Fortunately, for our purposes we do not need tohave both operations, multiplication and comultiplication, to be defined on the sameobject. 3. Characters of U ( ∞ )Here we study a relationship between the representation ring R and the dualobject Ω = \ U ( ∞ ). In the symmetric group case, there is a homomorphism of thealgebra Sym into the algebra of continuous functions on the dual object \ S ( ∞ ), andthe kernel of that homomorphism is the principal ideal of Sym generated by e − R and the dual object Ω.We exhibit three homomorphisms.First, R can be mapped into an algebra of functions defined on a certain subsetΩ ⊂ Ω (Ω is composed from some finite-dimensional “faces” of Ω). This map isfar from being the desired analog but it is useful for some technical purposes.Second, the subalgebra R can be mapped into C (Ω), the Banach algebra ofbounded continuous functions on Ω.Third, the above map sends the subalgebra R ⊂ R into the subalgebra C (Ω) ⊂ C (Ω) formed by continuous functions vanishing at infinity. The space C (Ω) is ofspecial interest for us because our main objects of study, the generators of Markovprocesses on Ω, are operators on the Banach space C (Ω).3.1. Description of extreme characters: the Edrei-Voiculescu theorem. For every N = 1 , , . . . , we identify U ( N ) with the subgroup of the group U ( N + 1)fixing the last basis vector in C N +1 . This makes it possible to define the inductivelimit group U ( ∞ ) = lim −→ U ( N ). In other words, elements of U ( ∞ ) are infinite unitarymatrices [ U ij ] ∞ i,j =1 such that U ij = δ ij when i or j is large enough.We endow U ( ∞ ) with the inductive limit topology, which plainly means thata function f : U ( ∞ ) → C is continuous if and only if for every N , the function f N := f (cid:12)(cid:12) U ( N ) is continuous on U ( N ).Notice that f is a class function (respectively, a positive definite function) if andonly if so is f N for every N . Definition 3.1. (i) By a character of U ( ∞ ) we mean a continuous class function f : U ( ∞ ) → C which is positive definite and normalized by f ( e ) = 1.(ii) Note that the set of all characters in the sense of (i) is a convex set. Itsextreme points are called extreme or indecomposable characters. The extreme characters of U ( ∞ ) are analogs of the normalized irreducible char-acters s λ ( u , . . . , u N )Dim N λ , λ ∈ S N . (3.1)To describe the extreme characters we need to introduce some notation.Let R + ⊂ R denote the set of nonnegative real numbers, R ∞ + denote the productof countably many copies of R + , and set R ∞ +2+ = R ∞ + × R ∞ + × R ∞ + × R ∞ + × R + × R + . Let Ω ⊂ R ∞ +2+ be the subset of sextuples ω = ( α + , β + ; α − , β − ; δ + , δ − )such that α ± = ( α ± ≥ α ± ≥ · · · ≥ ∈ R ∞ + , β ± = ( β ± ≥ β ± ≥ · · · ≥ ∈ R ∞ + , ∞ X i =1 ( α ± i + β ± i ) ≤ δ ± , β +1 + β − ≤ . We observe that Ω is a locally compact space in the topology inherited from theproduct topology of R ∞ +2+ .Instead of δ ± it is often convenient to use the quantities γ ± := δ ± − ∞ X i =1 ( α ± i + β ± i ) . Obviously, γ + and γ − are nonnegative. But, in contrast to δ + and δ − , they are not continuous functions of ω ∈ Ω.For u ∈ C ∗ and ω ∈ Ω setΦ( u ; ω ) = e γ + ( u − γ − ( u − − ∞ Y i =1 β + i ( u − − α + i ( u − 1) 1 + β − i ( u − − − α − i ( u − − . (3.2)For any fixed ω , this is a meromorphic function in variable u ∈ C ∗ with possiblepoles on (0 , ∪ (1 , + ∞ ). The poles do not accumulate to 1, so that the function isholomorphic in a neighborhood of the unit circle T := { u ∈ C : | u | = 1 } .Note that every conjugacy class of U ( ∞ ) contains a diagonal matrix with diagonalentries u , u , . . . ∈ T , where only finitely many of u n ’s are distinct from 1. Thesenumbers are defined uniquely, within a permutation. Thus every class function on U ( ∞ ) can be interpreted as a symmetric function Ψ( u , u , . . . ). Theorem 3.2 (Edrei-Voiculescu) . The extreme characters of the group U ( ∞ ) areprecisely the functions of the form Ψ ω ( u , u , . . . ) := ∞ Y k =1 Φ( u k ; ω ) , (3.3) where ω ranges over Ω . Note that the product actually terminates because Φ(1; ω ) = 1 and u k = 1 for k large enough. As compared with the normalized irreducible characters of thegroups U ( N ) given by (3.1), the extreme characters of U ( ∞ ) seem to be both moreelementary and more sophisticated objects. They are more elementary because theyare given by a product formula, but they are also more sophisticated as they dependon countably many continuous parameters.About various proofs and different facets of this fundamental theorem see Edrei[10], Voiculescu [45], Boyer [8], Vershik-Kerov [44], Okounkov-Olshanski [27], Borodin-Olshanski [4], Petrov [37]. Proposition 3.3. Given ω ∈ Ω , write the Laurent expansion of the function u Φ( u ; ω ) as Φ( u ; ω ) = X n ∈ Z b ϕ n ( ω ) u n . For n ∈ Z fixed, the coefficient b ϕ n ( ω ) is a continuous function on Ω vanishing atinfinity.Proof. See Borodin-Olshanski [4, Proposition 2.10]. (cid:3) Recall that we denoted by Φ( u ) the formal generating series assembling the vari-ables ϕ n , see (2.3) above. The fact that we employ now a similar notation is notoccasional. As explained below, the functions b ϕ n ( ω ) serve as the image of the gen-erators ϕ n ∈ R under the maps mentioned in the preamble to the section.3.2. The quotient algebra b R = R/J . Observe that Φ(1; ω ) ≡ 1, which implies X n ∈ Z b ϕ n ( ω ) = 1 , ω ∈ Ω . (3.4)This relation motivates the following definitions.Let us set ϕ := X n ∈ Z ϕ n . and let J := ( ϕ − ⊂ R be the principal ideal generated by the element ϕ − 1. Theideal J and the quotient algebra b R := R/J play an important role in our theory,similar to that of the ideal ( e − ⊂ Sym and the quotient ring Sym / ( e − 1) inVershik-Kerov’s theory [18], [19].The quotient ring b R is a filtered algebra: its filtration is inherited from the filtra-tion in R , which in turn is determined from the grading; the latter is not inheritedbecause the ideal J is not homogeneous.We will prove a few simple propositions concerning the algebra b R . Proposition 3.4. For every N = 0 , , , . . . , the intersection J ∩ R N is trivial. Proof. This is a formal consequence of the fact that R has no zero divisors (whichin turn follows from the isomorphism R = lim ←− R ( n + , n − )).Indeed, assume ψ ∈ J ∩ R N and show that ψ = 0. There exists ψ ′ ∈ R such that ψ = ( ϕ − ψ ′ . Since R has no zero divisors, the degree of ψ ′ cannot be larger than N − 1, so one can write ψ ′ = ψ + · · · + ψ N − , ψ i ∈ R i . Then ψ = N − X i =0 ( ϕ − ψ i = − ψ + ( ϕψ − ψ ) + · · · + ( ϕψ N − − ψ N − ) + ϕψ N − . Since ψ is homogeneous of degree N , we have ψ = ϕψ N − and − ψ = ( ϕψ − ψ ) = · · · = ( ϕψ N − − ψ N − ) = 0 . This implies ψ = · · · = ψ N − = 0 and finally ψ = 0. (cid:3) Let, as above, n + ≥ n − be a couple of integers. We denote by J ( n + , n − ) theideal in R generated by the ideals J and I ( n + , n − ). Under the homomorphism R → R ( n + , n − ), the image of J is the principal ideal generated by the element( ϕ n − + · · · + ϕ n + ) − 1. We set b R ( n + , n − ) := R/J ( n + , n − ) . (3.5)This algebra can be identified with the quotient C [ ϕ n − , . . . , ϕ n + ] (cid:14) ( ϕ n − + · · · + ϕ n + − n + − n − variables. Proposition 3.5. As n ± → ±∞ , the intersection of the kernels of the compositehomomorphisms R → R ( n + , n − ) → b R ( n + , n − ) coincides with J .Proof. This is a trivial consequence of the absence of zero divisors. Indeed, theideal J lies in the intersection of the kernels in question. Conversely, assume ψ ∈ R belongs to the intersections of the kernels and show that ψ ∈ J , that is, there exists ψ ′ ∈ R such that ψ = ( ϕ − ψ ′ .By the assumption, for every couple ( n + , n − ) there exists an element ψ ′ n + ,n − ∈ R ( n + , n − ) such that the image of ψ in R ( n + , n − ) is equal to( ϕ n − + · · · + ϕ n + − ψ ′ n + ,n − . Note that this element is unique and its degree is bounded from above by deg( ψ ) − ψ ′ = lim ←− ψ ′ n + ,n − . The elements ψ and( ϕ − ψ ′ have the same image under the map R → R ( n + , n − ), for every ( n + , n − ).Therefore, these elements are equal to each other. (cid:3) Corollary 3.6. The algebra b R can be identified with the projective limit of filteredalgebras b R ( n + , n − ) as n ± → ±∞ .Proof. Since R = lim ←− R ( n + , n − ), there is a natural homomorphism b R → lim ←− b R ( n + , n − ).Proposition 3.5 shows that it is injective. Let us check that it is also surjective. With-out loss of generality one can assume that n + > > n − . Then we use the relation ϕ n − + · · · + ϕ n + = 1 in b R ( n + , n − ) to eliminate ϕ and to lift b R ( n + , n − ) into R ( n + , n − )as the subalgebra R ′ ( n + , n − ) generated by ϕ n − , . . . , ϕ − , ϕ , . . . , ϕ n + . This makes itpossible to identify lim ←− b R ( n + , n − ) with lim ←− R ′ ( n + , n − ), where both limits are takenin the category of filtered algebras. Then the surjectivity in question becomes obvi-ous. (cid:3) We say that two signatures µ ∈ S N and λ ∈ S N +1 interlace if λ i ≥ µ i ≥ λ i +1 , i = 1 , . . . , N, (3.6)and then we write µ ≺ λ or, equivalently, λ ≻ µ . By agreement, any signature λ ∈ S is interlaced with the empty signature ∅ ∈ S . Proposition 3.7. For any µ ∈ S N , where N = 0 , , , . . . , one has ϕσ µ = X λ ∈ S N +1 : λ ≻ µ σ λ . (3.7) Proof. The classical Gelfand–Tsetlin branching rule says that for λ ∈ S N +1 , s λ ( u , . . . , u N +1 ) = X µ ∈ S N : µ ≺ λ s µ ( u , . . . , u N ) u | λ |−| µ | N +1 , where | λ | := P λ i , | µ | := P µ j . This gives us the structure constants c ( λ | µ, ν )(see Section 2) for the basis of rational Schur functions in the special case when λ ∈ S N +1 , µ ∈ S N , and ν = n ∈ S = Z . Namely, in this special case, c ( λ | µ, ν ) = ( , if µ ≺ λ and n = | λ | − | µ | , , otherwise.Combining this with the definition of the multiplication in R we get (3.7). (cid:3) The proposition shows that the ideal J coincides with the closed linear span ofthe elements of the form − σ µ + X λ : λ ≻ µ σ λ , where µ ranges over the set S ∪ S ∪ . . . of all signatures. This fact is used belowin the proof of Proposition 7.4. The simplices Ω( n + , n − ) . Let ( n + , n − ) be a couple of integers such that n + ≥ ≥ n − . We setΩ( n + , n − ) := (cid:8) ω = ( α ± , β ± , δ ± ) : α ± i = 0 for all i, β + i = 0 for i > n + ,β − j = 0 for j > | n − | , δ + = β +1 + · · · + β + n + , δ − = β − + · · · + β −| n − | (cid:9) ⊂ ΩThis a compact subset of Ω whose elements depend only on n + + | n − | independentparameters β +1 , . . . , β + n + , β − . . . , β −| n − | . The conditions on these parameters can bewritten in the form1 − β + | n + | ≥ · · · ≥ − β +1 ≥ β − ≥ · · · ≥ β − n − ≥ β +1 + β − ≤ n + , n − ) can be viewed as a simplex ofdimension n + + | n − | .If ω ∈ Ω( n + , n − ), then the function Φ( u ; ω ) drastically simplifies and takes theform Φ( u ; ω ) = n + Y i =1 (1 − β + i + β + i u ) · | n − | Y j =1 (1 − β − j + β − j u − ) . The function ϕ n ( ω ) vanishes identically on Ω( n + , n − ) unless n + ≥ n ≥ n − .Let C (Ω( n + , n − )) denote the algebra of continuous functions on the simplexΩ( n + , n − ). By Proposition 3.3, every function b ϕ n ( ω ) is continuous on Ω( n + , n − ).Recall that J ( n + , n − ) denotes the principal ideal in R ( n + , n − ) generated by theelement ( ϕ n − + · · · + ϕ n + ) − Proposition 3.8. The kernel of the homomorphism R ( n + , n − ) = C [ ϕ n − , . . . , ϕ n + ] → C (Ω( n + , n − )) assigning to ϕ n the function b ϕ n ( ω ) on Ω( n + , n − ) coincides with the ideal J ( n + , n − ) .Proof. Since b ϕ n ( ω ) vanishes on Ω( n + , n − ) unless n + ≥ n ≥ n − , the equality (3.4)shows that n + X n = n − b ϕ n (cid:12)(cid:12) C (Ω( n + ,n − )) = 1 . It remains to prove that this is the only relation.Let us examine the special case when n − = 0. To simplify the notation, set n + = m and ( t , . . . , t m ) := (1 − β + m , . . . , − β +1 )Let us write b ϕ n ( t , . . . , t m ) instead of b ϕ n ( ω ), where n = 0 , . . . , m . These are sym-metric polynomials in t , . . . , t m satisfying m Y i =1 ( t i + (1 − t i ) u ) = m X n =0 b ϕ n ( t , . . . , t m ) u n . For instance, for m = 2, b ϕ ( t , t ) = t t , b ϕ ( t , t ) = ( t + t ) − t t , b ϕ ( t , t ) = (1 − t )(1 − t ) . In the case under consideration, the claim of the proposition is equivalent to sayingthat the only algebraic relation between these m + 1 polynomials is that their sumequals 1. Let us prove the last assertion.Evidently, our polynomials lie in the linear span of the elementary symmetricpolynomials e n ( t , . . . , t m ), where n = 0 , . . . , m and e := 1. Therefore, it suffices tocheck that our polynomials are linearly independent.To do this, we evaluate them in the following m + 1 points of R m : x k := ( 1 , . . . , | {z } m − k , , . . . , | {z } k ) , k = 0 , . . . , m. At x k , the product Q ( t i + (1 − t i ) u ) equals u k . This implies that b ϕ n ( x k ) = δ nk ,which concludes the proof in our special case.Finally, the case n − < n − = 0 by usingthe twisting transformation τ defined in the next subsection. (cid:3) Proposition 3.8 shows that the quotient ring b R ( n + , n − ) = R ( n + , n − ) /J ( n + , n − )is embedded into the algebra C (Ω( n + , n − )) of continuous functions on the simplexΩ( n + , n − ) as the subalgebra of polynomial functions.Together with Proposition 3.5 this makes it possible to realize the quotient ring b R = R/J as an algebra of functions on the subsetΩ := [ n + ≥ n − Ω( n + , n − ) ⊂ Ω . (3.8)3.4. Symmetries. There exist natural transformations of characters of U ( ∞ ), whichpreserve the subset of extreme characters and thus induce transformations (or sym-metries ) Ω → Ω of the parameter space.One such transformation is the operation of conjugation mapping a character f ( U )to the conjugate character f ( U ) (here U ranges over U ( ∞ )). Conjugation inducesthe symmetry ω ω ∗ of Ω consisting in switching ( α + , β + , δ + ) ↔ ( α − , β − , δ − ).Another kind of transformation is the multiplication of f ( U ) by det( U ). In termsof the eigenvalues this amounts to multiplication by the product u u . . . . Thecorresponding symmetry of Ω leaves the parameters α ± intact and changes theremaining parameters in the following way:( β +1 , β +2 , . . . ) (1 − β − , β +1 , β +2 , . . . )( β − , β − , . . . ) ( β − , β − , . . . ) δ + δ + + (1 − β − ) δ − δ − − β − . Note that 1 − β − ≥ β +1 because of the condition β +1 + β − ≤ twisting symmetry of Ω and denote it as ω τ ( ω ). Obviously, τ is invertible.Under the symmetry ω ω ∗ , the subset Ω( n + , n − ) is mapped onto Ω( − n − , − n + ).If n − ≤ − 1, then the twisting symmetry τ maps Ω( n + , n − ) onto Ω( n + + 1 , n − + 1).Recall that so far we assumed n + ≥ ≥ n − . However, one can extend thedefinition of Ω( n + , n − ) so that the equality τ (Ω( n + , n − )) = Ω( n + + 1 , n − + 1) willbe valid for every couple n + ≥ n − , dropping the assumption that n + ≥ n − ≤ 0. For instance, if n − ≥ 1, then the first n − coordinates in β + are equal to 1and the actual parameters are β + n − +1 , . . . , β + n + .3.5. The homomorphisms R → C (Ω) and R → C (Ω) . Recall that the func-tions b ϕ n ( ω ) introduced in Proposition 3.3 belong to the Banach space C (Ω). At thismoment we only exploit the fact that they belong to C (Ω). Let us assign to everygenerator ϕ n ∈ R the function b ϕ n ( ω ). We are going to extend this correspondenceto a norm continuous homomorphism R → C (Ω).Let us start by assigning to every basis element σ λ a suitable function b σ ( ω ). Thiscan be done in two equivalent ways. First way . We use the determinantal formula (2.9) and set for λ ∈ S N and ω ∈ Ω b σ λ ( ω ) := det[ b ϕ λ i − i + j ( ω )] . (3.9) Second way . Restricting the extreme character Ψ ω defined in (3.3) to the sub-group U ( N ) ⊂ U ( ∞ ) gives us a normalized positive definite class function on U ( N ),which can be expanded into an absolutely and uniformly convergent series on theirreducible characters of U ( N ). Then the desired quantities b σ λ ( ω ) arise as the co-efficients of this expansion. Passing to matrix eigenvalues one can write this in theform Φ( u ; ω ) . . . Φ( u N ; ω ) = X λ ∈ S N b σ λ ( ω ) s λ ( u , . . . , u N ) . (3.10)From (3.9) it follows that the functions b σ λ ( ω ) belong to C (Ω) (even to C (Ω)),and from (3.10) we see that b σ λ ( ω ) ≥ u = · · · = u N = 1 we getthe identity X λ ∈ S N Dim N λ b σ λ ( ω ) = 1 . (3.11)Next, given an element ψ = P a λ σ λ ∈ R , we want to assign to it the function b ψ ( ω ) = P a λ b σ λ ( ω ) on Ω. Proposition 3.9. (i) For every element ψ = P a λ σ λ ∈ R , the series b ψ ( ω ) := P a λ b σ λ ( ω ) converges absolutely at every point ω ∈ Ω . Moreover, the resulting func-tion on Ω is bounded and its supremum norm does not exceed k ψ k . (ii) The map ψ b ψ ( · ) is an algebra homomorphism R → C (Ω) . (iii) The kernel of this homomorphism is the principal ideal J ⊂ R generated bythe element ϕ − . This ideal coincides with J ∩ R .Proof. Step 1. Let us check (i). We will assume first that ψ is homogeneous ofdegree N . Then we have (recall that b σ λ ( ω ) ≥ X λ | a λ | b σ λ ( ω ) = X λ | a λ | Dim N λ Dim N λ b σ λ ( ω ) ≤ k ψ k X λ Dim N λ b σ λ ( ω ) = k ψ k , (3.12)where the final equality follows from (3.11).The same holds for arbitrary (not necessarily homogeneous) elements, by the verydefinition of the norm in R . Step 2. Let us check that the map ψ b ψ ( · ) is consistent with multiplication.That is, for any two elements ψ ′ , ψ ′′ ∈ R and any ω ∈ Ω one has b ψ ′ ( ω ) c ψ ′′ ( ω ) = b ψ ( ω ) , ψ := ψ ′ ψ ′′ . Indeed, without loss of generality we may assume that ψ ′ and ψ ′′ are homogeneous,of degree M and N , respectively. Write ψ ′ = X µ ∈ S M a ′ µ σ µ , ψ ′′ = X ν ∈ S N a ′′ ν σ ν , ψ = X λ ∈ S M + N a λ σ λ . By virtue of (2.13), we have a λ = X µ,ν c ( λ | µ, ν ) a ′ µ a ′′ ν , where the structure constants correspond to the choice P λ = s λ .It readily follows that the desired statement is reduced to the following identity:for any fixed µ ∈ S M and ν ∈ S N one has b σ µ ( ω ) b σ ν ( ω ) = X λ ∈ S M + N c ( λ | µ, ν ) b σ λ ( ω ) , ω ∈ Ω . (3.13)This identity, in turn, follows from the second definition of the quantities b σ λ ( ω )(formula (3.10) above) and the identity s λ ( u , . . . , u M + N ) = X µ ∈ S M , ν ∈ S N c ( λ | µ, ν ) s µ ( u , . . . , u M ) s ν ( u M +1 , . . . , u M + N ) . Necessary interchanges of the order of summation are justified because all the seriesare absolutely convergent. Step 3 . Let us show that the functions b ψ ( ω ) are continuous on Ω. We may assumethat ψ is homogeneous of degree N . Then the corresponding function b ψ ( ω ) is givenby the series P λ ∈ S N a λ b σ λ ( ω ). We know that the functions b σ λ ( ω ) are continuous,but one cannot immediately conclude that b ψ is also continuous because the series isnot necessarily convergent in the norm topology of C (Ω). This difficulty is resolved in the following way. Since the space Ω is locally compact, it suffices to prove thatthe series for b ψ converges uniformly on compact subsets of Ω. Looking at (3.12)one sees that it suffices to do this for the series P λ Dim N λ b σ λ ( ω ). By (3.11), itconverges to the constant function 1 at every point ω ∈ Ω. Since all the summandsare nonnegative, the convergence is uniform on compact sets, as desired.Thus, we completed the proof of (ii). Step 4. Obviously, the element ϕ belongs to R , so that the principal ideal J ⊂ R generated by ϕ − J = J ∩ R . To do this wehave to check that if ψ ∈ R is such that ( ϕ − ψ ∈ R , then ψ ∈ R . This is provedby the same argument as in the proof of Proposition 3.4. Step 5. Finally, let us check that J coincides with the kernel of the homomor-phism ψ b ψ ( · ). We know that the function b ϕ ( ω ) is the constant function 1, so J is contained in the kernel.It remains to show that if, conversely, ψ ∈ R is such that b ψ ( ω ) ≡ ψ ∈ J . Here we apply the result stated at the very end of Subsection 3.3. It sufficesto use the fact that the function b ψ ( ω ) vanishes on Ω . Then that result says that ψ ∈ J . Because J ∩ R = J , we conclude that ψ ∈ J . (cid:3) Corollary 3.10. The homomorphism of Proposition 3.9 determines by restrictiona homomorphism R → C (Ω) .Proof. By the definition of the subalgebra R ⊂ R , the linear span of the basiselements σ λ is dense in R with respect to the norm topology. On the other hand,as it was pointed above, the functions b σ λ ( ω ) belong to C (Ω). Since C (Ω) is closedin C (Ω) and the homomorphism R → C (Ω) is norm continuous, this shows thatthe image of the whole subalgebra R is contained in C (Ω). (cid:3) Analog of the Vershik-Kerov ring theorem. Let R + ⊂ R denote theclosed (in the norm topology) convex cone spanned by the elements σ λ . For twoelements ψ , ψ ∈ R + , write ψ ≤ ψ if ψ − ψ ∈ R + .The following result is similar to the so-called ring theorem due to Vershik andKerov, see [18, Theorem 6] and [17, Introduction, Theorem 4]. Proposition 3.11. (i) The set of characters of U ( ∞ ) in the sense of Definition3.1 is in a natural one-to-one correspondence with linear functionals F : R → C satisfying the following properties : • F is norm-continuous and takes real nonnegative values on the cone R + . • If ψ ∈ R + is the least upper bound for a sequence ≤ ψ ≤ ψ ≤ . . . , then F ( ψ ) = lim n →∞ F ( ψ n ) . • F (1) = 1 and F ( ϕψ ) = F ( ψ ) for every ψ ∈ R . (ii) A character is extreme if and only if the corresponding functional F is multi-plicative, that is, F ( ψ ψ ) = F ( ψ ) F ( ψ ) for any ψ , ψ ∈ R . The proof is similar to that given in [18] (see also a more detailed version inGnedin-Olshanski [13, Section 8.7]).This result does not depend on the classification of the extreme characters andprovides one more proof of their multiplicativity.4. The operator D z,z ′ ,w,w ′ Definition 4.1. Fix an arbitrary quadruple ( z, z ′ , w, w ′ ) of complex parametersand introduce the following formal differential operator in countably many variables { ϕ n : n ∈ Z } D z,z ′ ,w,w ′ = X n ∈ Z A nn ∂ ∂ϕ n + 2 X n ,n ∈ Z n >n A n n ∂ ∂ϕ n ∂ϕ n + X n ∈ Z B n ∂∂ϕ n , where, for any indices n ≥ n , A n n = ∞ X p =0 ( n − n + 2 p + 1)( ϕ n + p +1 ϕ n − p + ϕ n + p ϕ n − p − ) − ( n − n ) ϕ n ϕ n − ∞ X p =1 ( n − n + 2 p ) ϕ n + p ϕ n − p (4.1)and, for any n ∈ Z , B n = ( n + w + 1)( n + w ′ + 1) ϕ n +1 + ( n − z − n − z ′ − ϕ n − − (cid:0) ( n − z )( n − z ′ ) + ( n + w )( n + w ′ ) (cid:1) ϕ n . (4.2)Note that only coefficients B n depend on the parameters ( z, z ′ , w, w ′ ). Proposition 4.2. The operator D z,z ′ ,w,w ′ is correctly defined on R . Note that not every formal differential operator in variables ϕ n can act on R . Hereis a very simple example: application of P n ∈ Z ∂∂ϕ n to the element ϕ = P n ∈ Z ϕ n givesthe meaningless expression P n ∈ Z 1. As is seen from the argument below, the validityof the proposition relies on the concrete form of the coefficients of D z,z ′ ,w,w ′ . Proof. (i) Obviously, when D z,z ′ ,w,w ′ is formally applied to a monomial in R , theresult is a well-defined element of R . We have to prove that, more generally, thesame holds when D z,z ′ ,w,w ′ is applied to any homogeneous element g ∈ R . In otherwords, the infinite sum arising in D z,z ′ ,w,w ′ g cannot contain infinitely many nonzeroterms proportional to one and the same monomial.(ii) Given a monomial ϕ λ = ϕ λ . . . ϕ λ N indexed by a signature λ , define its support supp ϕ λ as the lattice interval [ a, b ] := { a, . . . , b } ⊂ Z , where a = λ N =min( λ , . . . , λ N ) and b = λ = max( λ , . . . , λ N ). From (4.1) is is evident that for every monomial ϕ µ entering A n n ∂ ϕ λ ∂ϕ n ∂ϕ n , one has supp ϕ µ ⊇ [ a, b ].(iii) Likewise, from (4.1) it is clear that if a monomial ϕ µ enters B n ∂ϕ λ ∂ϕ n and [ a ′ , b ′ ] := supp ϕ µ , then one has | a ′ − a | ≤ | b ′ − b | ≤ g be a homogeneous element of R , and examinethe infinite sum D z,z ′ ,w,w ′ g resulting from application of D z,z ′ ,w,w ′ to g . Observethat there exist only finitely many monomials of a prescribed degree and with thesupport contained in a prescribed lattice interval. Therefore, (ii) and (iii) guaranteethat the undesired accumulation of infinitely many proportional terms in D z,z ′ ,w,w ′ g is excluded. (cid:3) Proposition 4.3. If z = n + and w = − n − , where n + ≥ n − are integers, then theoperator D z,z ′ ,w,w ′ preserves the ideal I ( n + , n − ) and hence correctly determines anoperator acting on the quotient ring R ( n + , n − ) = R/I ( n + , n − ) .Proof. The ideal I ( n + , n − ) consists of (possibly infinite) linear combinations ofmonomials whose support is not contained in the lattice interval [ n − , n + ]. Step(ii) of the argument above shows that the application of the second order terms in D z,z ′ ,w,w ′ enlarges the supports and so preserves the ideal I ( n + , n − ). Note that thisholds for any values of the parameters.Now let us examine the effect of the application of a first degree term B n ∂∂ϕ n .From (4.2) it is seen that the only danger may come from the quantities( n + w + 1)( n + w ′ + 1) ϕ n +1 (cid:12)(cid:12) n = n − − , ( n − z − n − z ′ − ϕ n − (cid:12)(cid:12) n = n + +1 . But these quantities vanish because, by our assumption, w = − n − and z = n + . (cid:3) Proposition 4.4. For any fixed integer m , the operator D z,z ′ ,w,w ′ is invariant underthe change of variables ϕ n ϕ n + m ( n ∈ Z ) combined with the shift of parameters z → z + m, z ′ → z ′ + m, w → w − m, w ′ → w − m. In connection with this proposition see also Remark 3.7 in [2]. Proof. Indeed, the indicated simultaneous shift of the variables and parameters doesnot change the coefficients A n n and B n . (cid:3) The next proposition is not so evident: Proposition 4.5. The operator D z,z ′ ,w,w ′ preserves the principal ideal J ⊂ R . Proof. We will prove that D z,z ′ ,w,w ′ commutes with the operator of multiplication by ϕ , which obviously implies that D z,z ′ ,w,w ′ preserves J .Take an arbitrary element F ∈ R and observe that D z,z ′ ,w,w ′ ( ϕF ) − ϕ D z,z ′ ,w,w ′ F equals 2 X n ∈ Z A nn + X n : n >n A n n + X n : n This method was proposed in Borodin-Olshanski [5]. The method allows one toconstruct Markov processes on dual objects to inductive limit groups like S ( ∞ ) or U ( ∞ ) by essentially algebraic tools. Here we describe its idea. For more details, see[5], Borodin-Olshanski [7], and the expository paper Olshanski [34].5.1. Generalities on Markov kernels and Feller processes. Let X and Y betwo measurable spaces. Recall that a Markov kernel with source space X and targetspace Y is a function P ( x, A ), where the first argument x ranges over X and thesecond argument is a measurable subset of Y ; next, one assumes that the followingtwo conditions hold (see e.g. Meyer [26]): • For A fixed, P ( · , A ) is a measurable function on X . • For x fixed, P ( x, · ) is a probability measure on Y (we will denote it by P ( x, dy )).When the second space Y is a discrete space, it is convenient to interpret thekernel as a function on X × Y by setting P ( x, y ) := P ( x, { y } ). In the case whenboth spaces are discrete, P ( x, y ) is a stochastic matrix of format X × Y .We regard a Markov kernel P as a surrogate of map between X and Y , denotedas P : X Y and called a link . Here the dashed arrow symbolizes the fact that alink is not an ordinary map: it assigns to a given point x ∈ X not a single point in Y but a probability distribution on Y .The superposition of two links P ′ : X Y and P ′′ : Y Z is the link P = P ′ P ′′ between X and Z defined by P ( x, dz ) = Z y ∈ Y P ′ ( x, dy ) P ′′ ( y, dz ) . If both X and Y are discrete, then the superposition becomes the matrix product.Every link P : X Y induces a contractive linear operator f P f from theBanach space of bounded measurable functions on Y to the similar function spaceon X : ( P f )( x ) = Z y ∈ Y P ( x, dy ) f ( y ) , x ∈ X. Assuming X and Y are locally compact spaces, we say that P : X Y is a Fellerlink if the above operator maps C ( Y ) into C ( X ). Note that the superpositionof Feller links is a Feller link, too. (We recall that C ( X ) consists of continuousfunctions on X vanishing at infinity. If X is a discrete space, then the continuityassumption is trivial and C ( X ) consists of arbitrary functions vanishing at infinity.)Now we recall a few basic notions from the theory of Markov processes (see Liggett[24], Ethier-Kurtz [11]).A Feller semigroup on a locally compact space X is a strongly continuous semi-group P ( t ), t ≥ 0, of contractive operators on C ( X ) given by Feller links P ( t ; x, dy ).A well-known abstract theorem says that a Feller semigroup gives rise to a Markov process on X with transition function P ( t ; x, dy ). The processes derived from Fellersemigroups are called Feller processes ; they form a particularly nice subclass ofgeneral Markov processes.A Feller semigroup P ( t ) is uniquely determined by its generator . This is a closeddissipative operator A on C ( X ) given by Af = lim t → +0 P ( t ) f − ft . The domain of A , denoted by dom A , is the (algebraic) subspace formed by thosefunctions f ∈ C ( X ) for which the above limit exists; dom A is always a densesubspace. Every subspace F ⊂ dom A for which the closure of A (cid:12)(cid:12) F equals A iscalled a core of A . One can say that a core is an “essential domain” for A . Veryoften, the full domain of a generator is difficult to describe explicitly, and then oneis satisfied by exhibiting a core F with the explicit action of the generator on F .5.2. Stochastic links between dual objects. Here we introduce concrete exam-ples of stochastic links we will dealing with.For a compact group G , we denote by b G the set of irreducible characters of G and call it the dual object to G . Given χ ∈ b G , we denote by e χ the correspondingnormalized character: e χ ( g ) = χ ( g ) χ ( e ) , g ∈ G. In the special case when G is commutative, e χ = χ and b G is a discrete group, butin the general case (when G is noncommutative), the dual object does not possessa group structure and we regard it simply as a discrete space.To every morphism ι : G → G of compact groups there corresponds a canonical “dual” link Λ : b G b G , defined as follows. For every irreducible character χ ∈ b G ,its superposition with ι is a finite linear combination of irreducible characters χ ′ ∈ b G with nonnegative integral coefficients. It follows that the superposition of e χ with ι is a convex linear combination of normalized irreducible characters of the group G ;the coefficients of the latter expansion are just the entries of the stochastic matrixΛ. That is, e χ ( ι ( g )) = X χ ′ ∈ b G Λ( χ, χ ′ ) e χ ′ ( g ) , g ∈ G , χ ∈ b G . If G → G and G → G are two morphisms of compact groups, then it is evidentthat the superposition of the canonical dual links b G b G and b G b G coincideswith the canonical link b G b G corresponding to the composition morphism G → G .Consider now the infinite chain of groups U (1) ⊂ U (2) ⊂ U (3) ⊂ . . . as defined in the beginning of Subsection 3.1. For every N < M , this chain definesan embedding U ( N ) ֒ → U ( M ), and we denote by Λ MN ( λ, µ ) : S M S N the corre-sponding dual link, which is a stochastic matrix of format S M × S N . In particular,for M = N + 1 this matrix takes the formΛ N +1 N ( λ, µ ) = Dim N µ Dim N +1 λ , if µ ≺ λ , otherwise , (5.1)where µ ≺ λ means that the two signatures interlace in the sense that λ i ≥ µ i ≥ λ i +1 , i = 1 , . . . , N, see Borodin-Olshanski [5, Section 1.1] for more details.Next, consider the embedding U ( N ) ֒ → U ( ∞ ) (the image of the former group inthe latter group consists of the infinite unitary matrices [ U ij ] such that U ij = δ ij unless both i and j are less or equal to N ). We define the dual object \ U ( ∞ ) as theset of extreme characters and identify it with Ω. Then the above definition of thedual link is still applicable with the extreme characters of U ( ∞ ) playing the roleof the (nonexisting) normalized irreducible characters. The resulting Markov kernelΩ S N has the formΛ ∞ N ( ω, λ ) = Dim N λ · b σ λ ( ω ) , ω ∈ Ω , λ ∈ S N , (5.2)where b σ λ ( ω ) is defined in Section 3. The derivation of this formula is simple: by (3.3),the restriction of the extreme character Ψ ω to the subgroup U ( N ) is given by thefunction Φ( u ; ω ) . . . Φ( u N ; ω ); the expansion of that function on the irreducible char-acters χ λ = s λ is given by (3.10), and we only need to introduce the factor Dim N λ to get the required expansion on the normalized characters e χ λ = s λ / Dim N λ . Proposition 5.1. The canonical links Λ MN : S M S N and Λ ∞ N : Ω S N areFeller links. For a proof, see Borodin-Olshanski [4, Corollary 2.11 and Proposition 2.12].5.3. The method of intertwiners. Let X and Y be locally compact spaces, P X ( t )and P Y ( t ) be Feller semigroups on X and Y , respectively, and Λ : X Y be a Fellerlink. We say that Λ intertwines the semigroups P X ( t ) and P Y ( t ) if the followingcommutation relation holds P X ( t )Λ = Λ P Y ( t ) , t ≥ . This relation can be understood as an equality of links or, equivalently, as an equalityof operators acting from C ( Y ) to C ( X ). Proposition 5.2. Assume we are given a family { P N ( t ) : N = 1 , , , . . . } of Fellersemigroups, where the N th semigroup acts on C ( S N ) . Further, assume that thesesemigroups are intertwined by the canonical links Λ N +1 N , so that P N +1 ( t )Λ N +1 N = Λ N +1 N P N ( t ) , N = 1 , , , . . . , t ≥ . Then there exists a unique Feller semigroup P ∞ ( t ) on C (Ω) characterized by theproperty P ∞ ( t )Λ ∞ N = Λ ∞ N P N ( t ) , N = 1 , , . . . , t ≥ . Proof. See Proposition 2.4 in Borodin-Olshanski [5]. The fact that the hypothesisof this proposition is satisfied in our concrete situation is established in Subsection3.3 of that paper. (cid:3) Proposition 5.3. We keep to the hypotheses of Proposition 5.2. Let A N and A ∞ denote the generators of the semigroups P N ( t ) and P ∞ ( t ) , respectively. (i) For every N = 1 , , . . . and every f ∈ dom( A N ) , the vector Λ ∞ N f belongs to dom( A ∞ ) and one has A ∞ Λ ∞ N f = Λ ∞ N A N f. (ii) Assume additionally that for each N = 1 , , , . . . we are given a core F N ⊆ dom( A N ) for the operator A N . Then the linear span of the vectors of the form Λ ∞ N f ,where N = 1 , , . . . and f ∈ F N , is a core for A ∞ .Proof. Claim (i) directly follows from the definition of the generator. Claim (ii) isestablished in Borodin-Olshanski [7, Proposition 5.2]. (cid:3) The degenerate case. Let us fix a couple of integers n + ≥ n − and set S N ( n + , n − ) = { ν ∈ S N : n + ≥ ν ≥ · · · ≥ ν N ≥ n − } . (5.3)Note that this is a finite set.If µ ∈ S M ( n + , n − ) and N < M , then Λ MN ( µ, ν ) vanishes unless ν ∈ S N ( n + , n − ).So, Λ MN induces a link S M ( n + , n − ) S N ( n + , n − ). Likewise, if ω ∈ Ω( n + , n − ), thenΛ ∞ N ( ω, ν ) vanishes unless ν ∈ S N ( n + , n − ). So, Λ ∞ N induces a link Ω( n + , n − ) S N ( n + , n − ).When S N (with N = 1 , , , . . . ) and Ω are replaced by S N ( n + , n − ) and Ω( n + , n − ),respectively, all the results of the present section remain valid. The proofs areextended automatically, and we only point out some simplifications:In Proposition 5.1, the claim concerning the Feller property for the links Λ MN becomes redundant as the links S M ( n + , n − ) S N ( n + , n − ) are finite matrices.Next, because Ω( n + , n − ) is a compact space, the Feller property for the link Λ ∞ N :Ω( n + , n − ) S N ( n + , n − ) simply means that the functions of the form ω → Λ ∞ N ( ω, ν )are continuous on Ω( n + , n − ).In Proposition 5.2, one should replace C (Ω) by C (Ω( n + , n − )), the Banach spaceof all continuous functions on the compact space Ω( n + , n − ).In Proposition 5.3, because the sets S N ( n + , n − ) are finite, the generators A N arefinite-dimensional, so that dom( A N ) is the whole space of functions on S N ( n + , n − ). Markov processes on Ω and their generators This section contains some necessary material from Borodin-Olshanski [5], to-gether with a brief motivation. In that paper, we constructed a family { X z,z ′ ,w,w ′ } of continuous time Markov processes on the space Ω, indexed by the quadruple ofparameters ( z, z ′ , w, w ′ ) ranging over a certain subset of C . The infinitesimal gen-erator of X z,z ′ ,w,w ′ , denoted by A z,z ′ ,w,w ′ , is an unbounded operator on the Banachspace C (Ω). The results of [5] tell us how A z,z ′ ,w,w ′ acts on a subspace c F ⊂ C (Ω),the (algebraic) linear span of the functions b σ λ ( ω ), where λ ranges over the set of allsignatures except λ = ∅ . The explicit formulas for this action are the starting pointfor the computations in the remaining part of the paper. Note that c F serves as acore for the generator A z,z ′ ,w,w ′ , so that it is uniquely determined by its restrictionto c F .6.1. Special bilateral birth-death processes. Birth-death processes form a well-studied class of continuous time Markov chains. The state space of every birth-death process is the set Z + of nonnegative integers, and the process is determinedby specifying the quantities q ( n, n ± jump rates from state n ∈ Z + to theneighboring states n ± 1, with the understanding that q (0 , − 1) = 0, which preventsfrom leaving the subset Z + ⊂ Z . Under appropriate constraints on the jump ratesthe process is well defined (that is, does not explode, meaning that, with probability1, one cannot escape to infinity in finite time).The bilateral birth-death processes are defined in a similar way, only now thestate space is the whole lattice Z and the jump rates q ( n, n ± 1) are assumed to bestrictly positive for all n ∈ Z . Again, one needs some restrictions to be imposed onthese quantities in order that the process be non-exploding. Bilateral birth-deathprocesses are not so widely known as the ordinary ones. However, they were alsodiscussed in the literature.We are interested in bilateral birth-death processes whose jump rates q ( n, n ± n . We write them in the form q ( n, n − 1) = ( w + n )( w ′ + n ) , q ( n, n + 1) = ( z − n )( z ′ − n ) . (6.1)It is readily verified that these quantities are strictly positive for all n ∈ Z if andonly if each of pairs ( z, z ′ ) and ( w, w ′ ) belongs to the subset Z ⊂ C defined by Z := { ( ζ , ζ ′ ) ∈ ( C \ Z ) | ζ ′ = ¯ ζ }∪ { ( ζ , ζ ′ ) ∈ ( R \ Z ) | m < ζ , ζ ′ < m + 1 for some m ∈ Z } . (6.2)Note that if ( ζ , ζ ′ ) ∈ Z , then ζ + ζ ′ is real. Definition 6.1. We say that a quadruple ( z, z ′ , w, w ′ ) ∈ C is admissible if ( z, z ′ ) ∈ Z , ( w, w ′ ) ∈ Z , and z + z ′ + w + w ′ > − Proposition 6.2. Let ( z, z ′ , w, w ′ ) ∈ C be admissible. (i) There exists a non-exploding bilateral birth-process with the jump rates givenby (6.1) . (ii) This process is a Feller process. (iii) Its generator is implemented by the difference operator D z,z ′ ,w,w ′ on Z actingon functions f ( n ) , n ∈ Z by ( D z,z ′ ,w,w ′ f )( n ) = ( z − n )( z ′ − n )( f ( n + 1) − f ( n ))+ ( w + n )( w ′ + n )( f ( n − − f ( n )) , (6.3) and the domain of the generator consists of those functions f ∈ C ( Z ) for which D z,z ′ ,w,w ′ f ∈ C ( Z ) .Proof. Statements (i) and (ii) are the subject of Theorem 5.1 in Borodin-Olshanski[5], and (iii) is their formal consequence, as explained in [5, Proposition 4.6]. (cid:3) We refer to [5] for more details. Note that the property of non-explosion is thesame as regularity of the so-called Q-matrix (or the matrix of jump rates ), see [5,Section 4] and references therein. In our case, this matrix is simply the matrix ofthe difference operator D z,z ′ ,w,w ′ . This is a tridiagonal Z × Z matrix Q = [ q ( n, n ′ )]with the entries q ( n, n ± 1) given by (6.1), the diagonal entries q ( n, n ) = − q ( n, n + 1) − q ( n, n − , and all remaining entries equal to 0.6.2. Feller dynamics on S N . As explained in Borodin-Olshanski [5, Section 5.2],Proposition 6.2 admits an extension with Z replaced by S N , where N = 1 , , , . . . (recall that S = Z ). To state it we need first to define a matrix Q = [ q ( ν, µ )] offormat S N × S N . It depends on ( z, z ′ , w, w ′ ) and has the following form: • the entries q ( ν, µ ) equal 0 unless µ = ν or µ = ν ± ε i , where i = 1 , . . . , N and ε , . . . , ε N stands for the canonical basis of Z N ; • the (nonzero) off-diagonal entries are given by q ( ν, ν ± ε i ) = Dim N ( ν ± ε i )Dim N ν r ( ν, ν ± ε i ) , (6.4)where r ( ν, ν + ε i ) = ( z − ν i + i − z ′ − ν i + i − , i = 1 , . . . , N, (6.5)and r ( ν, ν − ε i ) = ( w + ν i − i + N )( w ′ + ν i − i + N ) , i = 1 , . . . , N ; (6.6) • the diagonal entries are given by q ( ν, ν ) = − X µ : µ = ν q ( ν, µ )= ( z + z ′ + w + w ′ ) N ( N − N − N ( N − − X µ : µ = ν r ( ν, µ ) . (6.7)When N = 1, this agrees with the definition of the preceding subsection. (Tocompare the above formulas with those from [5, Section 5.2], take into account ashift of parameters indicated in [5, (6.1) and (6.2)].)For ν = ( ν , . . . , ν N ) ∈ S N , we set ν ∗ = ( − ν N , . . . , − ν ) . The correspondence ν ν ∗ is an involutive bijection S N → S N . Proposition 6.3. One has q ( ν, µ ) = q ∗ ( ν ∗ , µ ∗ ) , where the matrix [ q ∗ ( · , · )] is obtained from the matrix [ q ∗ ( · , · )] by switching ( z, z ′ ) ↔ ( w, w ′ ) .Proof. This is readily checked. (cid:3) Proposition 6.4. Let ( z, z ′ , w, w ′ ) ∈ C be admissible in the sense of Definition6.1. (i) For every N = 1 , , , . . . , the S N × S N matrix Q = [ q ( ν, µ )] defined above isregular, so that there exists a non-exploding continuous time Markov process on S N with the jump rates given by the off-diagonal entries q ( ν, µ ) . (ii) This process is a Feller process. (iii) Its generator is implemented by the N -variate difference operator D z,z ′ ,w,w ′ | N on S N ⊂ Z N acting on functions f ( ν ) , ν ∈ S N by ( D z,z ′ ,w,w ′ | N f )( ν ) = X µ ∈ S N q ( ν, µ ) f ( µ ) = X µ ∈ S N \{ ν } q ( ν, µ )( f ( µ ) − f ( ν )) , (6.8) and the domain of the generator consists of those functions f ∈ C ( S N ) for which D z,z ′ ,w,w ′ | N f ∈ C ( S N ) .Proof. Statements (i) and (ii) are proved in Borodin-Olshanski [5, Theorem 5.4],and (iii) is their formal consequence, as explained in [5, Proposition 4.6]. (cid:3) Proposition 6.5. For any ( z, z ′ , w, w ′ ) ∈ C and any N = 0 , , , . . . the followingrelation holds D z,z ′ ,w,w ′ | N +1 Λ N +1 N = Λ N +1 N D z,z ′ ,w,w ′ | N ∀ N = 1 , , . . . . (6.9)(Recall that Λ N +1 N : S N +1 S N are the canonical links defined in (5.1).) Proof. In a slightly different notation, this is proved in [5, Proposition 6.2]. (cid:3) This result serves as the basis for the construction described in the next subsection.It is also used in Section 9 below.6.3. Feller dynamics on Ω . Throughout this subsection we assume, as before,that ( z, z ′ , w, w ′ ) is admissible (Definition 6.1). Proposition 6.6. For N = 1 , , . . . , we denote by P z,z ′ ,w,w ′ | N ( t ) the Feller semigroupon C ( S N ) afforded by Proposition 6.4. (i) These semigroups P z,z ′ ,w,w ′ | N ( t ) satisfy the hypothesis of Proposition 5.2, thatis, one has P z,z ′ ,w,w ′ | N +1 ( t )Λ N +1 N = Λ N +1 N P z,z ′ ,w,w ′ | N ( t ) , t ≥ , for every N = 1 , , , . . . . (ii) There exists a unique Feller semigroup P z,z ′ ,w,w ′ |∞ ( t ) on C (Ω) characterizedby the property P z,z ′ ,w,w ′ |∞ ( t )Λ ∞ N = Λ ∞ N P N ( t ) , N = 1 , , . . . , t ≥ . (Recall that Λ ∞ N : Ω S N are the links defined in (5.2).) Proof. Claim (i) is established in Borodin-Olshanski [5, theorem 6.1]. Claim (ii)follows from Claim (i) by virtue of Proposition 5.2. (cid:3) Definition 6.7. In what follows A z,z ′ ,w,w ′ | N denotes the generator of the semi-group P z,z ′ ,w,w ′ | N ( t ) on C ( S N ) and A z,z ′ ,w,w ′ denotes the generator of the semigroup P z,z ′ ,w,w ′ |∞ ( t ) on C (Ω).In the next proposition and its proof we use the quantities q ( ν, µ ) and r ( ν, µ ) thatwere defined in the preceding subsection. Note that they depend on the parameters z, z ′ , w, w ′ , and N . Proposition 6.8. Let N = 1 , , . . . . For every signature µ ∈ S N , the function b σ µ ∈ C (Ω) belongs to the domain of the generator A z,z ′ ,w,w ′ and A z,z ′ ,w,w ′ b σ µ = q ( µ, µ ) b σ µ + X ν ∈ S N : ν = µ r ( ν, µ ) b σ ν . (6.10) Proof. For µ ∈ S N , let µ denote the function on S N defined by µ ( ν ) = δ µν . By thedefinition of D z,z ′ ,w,w ′ | N , see (6.8), D z,z ′ ,w,w ′ | N µ = X ν ∈ S N q ( ν, µ ) ν . (6.11)For any λ ∈ S N we set e λ = (Dim N λ ) − λ . (6.12) Then, by (6.4), formula (6.11) can be rewritten as D z,z ′ ,w,w ′ | N e µ = q ( µ, µ ) e µ + X ν ∈ S N : ν = µ r ( ν, µ ) e ν . (6.13)Claim (iii) of Proposition 6.4 implies that all finitely supported functions on S N belong to the domain of A z,z ′ ,w,w ′ | N and for every such function f one has A z,z ′ ,w,w ′ | N f = D z,z ′ ,w,w ′ | N f . In particular, taking f = e µ we obtain from (6.13) A z,z ′ ,w,w ′ | N e µ = q ( µ, µ ) e µ + X ν ∈ S N : ν = µ r ( ν, µ ) e ν . (6.14)Next, by virtue of Proposition 6.6 one can apply Proposition 5.3, claim (i). Itimplies that for every λ ∈ S N , the function Λ ∞ N e λ on Ω belongs to the domain ofthe generator A z,z ′ ,w,w ′ and A z,z ′ ,w,w ′ Λ ∞ N e λ = Λ ∞ N A z,z ′ ,w,w ′ | N e λ (recall that the links Λ ∞ N : Ω S N are defined in (5.2)). Together with (6.14) thisgives A z,z ′ ,w,w ′ Λ ∞ N e µ = q ( µ, µ )Λ ∞ N e µ + X ν ∈ S N : ν = µ r ( ν, µ )Λ ∞ N e ν . (6.15)Finally, (5.2) shows that for any λ ∈ S N Λ ∞ N e λ = b σ λ . Substituting this into (6.15) gives the desired formula. (cid:3) Let c F ⊂ C (Ω) denote the linear span of the functions b σ λ , where λ range over S ⊔ S ⊔ S ⊔ . . . . As was shown in the proof of Proposition 6.8, c F coincides with thelinear span of the spaces Λ ∞ N C c ( S N ), where N = 1 , , , . . . and C c ( S N ) ⊂ C ( S N )stands for the subspace of finitely supported functions. By Proposition 6.8, c F is contained in the domain of the generator A z,z ′ ,w,w ′ . Moreover, this propositionexplains how the generator acts on c F . In particular, we see that c F is invariantunder the action of the generator. Theorem 6.9. The subspace c F ⊂ C (Ω) is a core for the generator A z,z ′ ,w,w ′ . This fact is not used in the arguments below, but it is a substantial complementto our main result, Theorem 7.1, which describes explicitly the operator A z,z ′ ,w,w ′ (cid:12)(cid:12) b F (the restriction of the generator to c F ). By virtue of Theorem 6.9, the latter operatoruniquely determines the generator, so Theorem 7.1 contains, in principle, a completeinformation about the generator. Proof. Theorem 6.9 is proved in [35]. Here we only indicate the idea of the proof.By Proposition 5.3, it suffices to show that C c ( S N ) is a core for A z,z ′ ,w,w ′ | N for every N . This, in turn, can be verified as in Borodin-Olshanski [7], by making use of a result due to Ethier and Kurtz (see its formulation in [7, Theorem 2.3 (iv)]. (Notetwo misprints in [7]: the claims of Corollary 6.6 (ii) and Corollary 8.7 (ii) concernthe subspace of finitely supported functions, so that instead of C ( · ) one shouldread C c ( · ).) (cid:3) The main theorem Formulation of the main theorem. In Section 4, we defined the differ-ential operator D z,z ′ ,w,w ′ which acts on R . It depends on an arbitrary quadruple( z, z ′ , w, w ′ ) ∈ C . We also showed that it preserves the ideal J ⊂ R and so deter-mines an operator on the quotient b R = R/J . Let us denote the latter operator by b D z,z ′ ,w,w ′ .Given ψ ∈ R , we will denote by b ψ ∈ b R the image of ψ under the canonical map R → b R . In particular, we may speak about the elements b σ λ ∈ b R . Note that inSection 3, we already used the same notation: namely, given ψ ∈ R , we denoted by b ψ ( ω ) the corresponding function on Ω (its definition is given just before Proposition3.9). Formally, the two definitions of b ψ look differently, but the new definition ismorally an extension of the old one, because, as shown in Proposition 3.9, the kernelof the homomorphism R ∋ ψ b ψ ( · ) coincides with J ∩ R . Theorem 7.1 (Main Theorem) . Let ( z, z ′ , w, w ′ ) ∈ C be an admissible quadruple ofparameters ( z, z ′ , w, w ′ ) ∈ C , see Definition 6.1, and recall that A z,z ′ ,w,w ′ denotes thegenerator of the semigroup P z,z ′ ,w,w ′ |∞ ( t ) , see Definition 6.7. We restrict A z,z ′ ,w,w ′ to the core c F ⊂ C (Ω) defined in the end of Section 6. Finally, let λ range over theset of all signatures, except λ = ∅ .Under the identification of the elements b σ λ ∈ b R with the functions b σ λ ( ω ) from thecore c F , the action of the generator A z,z ′ ,w,w ′ on those functions coincides with theaction of the operator b D z,z ′ ,w,w ′ on the corresponding elements b σ λ ∈ b R . Remark 7.2. As mentioned in the introduction, Theorem 7.1 gives a precise sense tothe informal statement (Theorem 1.1) that “the generator A z,z ′ ,w,w ′ is implementedby the differential operator D z,z ′ ,w,w ′ ”. It is tempting to regard Theorem 7.1 as theindication that X z,z ′ ,w,w ′ are diffusion processes, and it would be very interesting tofind out whether this is true. For instance, is it true that the operators A z,z ′ ,w,w ′ arediffusion generators as defined in Ledoux [23, Section 1.1].Theorem 7.1 will be proved in a slightly stronger form (Theorem 7.5 below).We are going to define a linear operator R → R that mimics the action of thegenerator A z,z ′ ,w,w ′ on c F . In the next proposition we use the I -adic topology in R ,introduced in Subsection 2.1. Proposition 7.3. For every quadruple ( z, z ′ , w, w ′ ) ∈ C there exists a unique linearoperator A z,z ′ ,w,w ′ : R → R , continuous in the I -adic topology, annihilating the unity element ∈ R , and such that for every N = 1 , , . . . and every µ ∈ S N , A z,z ′ ,w,w ′ σ µ = q z,z ′ ,w,w ′ | N ( µ, µ ) σ µ + X ν ∈ S N : ν = µ r z,z ′ ,w,w ′ | N ( ν, µ ) σ ν , (7.1) where q z,z ′ ,w,w ′ | N ( µ, µ ) and r z,z ′ ,w,w ′ | N ( ν, µ ) is a more detailed notation for the quan-tities q ( µ, µ ) and r ( ν, µ ) defined in the beginning of Subsection 6.2. It is worth emphasizing that here we drop the admissibility condition on the pa-rameters imposed in Section 6: the operator A z,z ′ ,w,w ′ is considered for any complexvalues of ( z, z ′ , w, w ′ ). This is possible because the formulas defining the quantities q ( µ, µ ) and r ( ν, µ ) make sense for arbitrary ( z, z ′ , w, w ′ ) ∈ C . Proof. Together with the condition A z,z ′ ,w,w ′ A z,z ′ ,w,w ′ on the linear span of the basis elements σ µ including σ ∅ = 1. The continuity of thisoperator immediately follows from the fact that A z,z ′ ,w,w ′ σ µ is a linear combinationof σ µ and “neighboring” basis vectors of the form σ µ ± ε i . The explicit form of thecoefficients is not important here. (cid:3) The next claim will be used in Section 9. Proposition 7.4. For any ( z, z ′ , w, w ′ ) ∈ C , the operator A z,z ′ ,w,w ′ preserves theideal J ⊂ R .Proof. It suffices to prove that A z,z ′ ,w,w ′ commutes with the operator of multiplica-tion by ϕ . We are going to show that the latter claim is merely a rephrasing of thecommutation relation (6.9).Indeed, for every N = 0 , , , . . . we define a linear isomorphism I N between thespace R N and the space Fun( S N ) of functions on the discrete set S N by setting I N : X µ ∈ S N a µ σ µ X µ ∈ S N a µ e µ , where a µ are arbitrary complex coefficients. By the very definition of A z,z ′ ,w,w ′ , wehave A z,z ′ ,w,w ′ (cid:12)(cid:12) R N = I − N D z,z ′ ,w,w ′ | N I N . On the other hand, Proposition 3.7 says that for every µ ∈ S N , ϕσ µ = X λ : λ ≻ µ σ λ . Comparing this with the definition of the canonical link Λ N +1 N (see (5.1)) and thedefinition of e µ (see (6.12)) we conclude that the operator R N → R N +1 given bymultiplication by ϕ coincides with the operator I − N +1 Λ N +1 N I N .Therefore, the commutation relation (6.9) just means that A z,z ′ ,w,w ′ and multipli-cation by ϕ commute. (cid:3) Theorem 7.5. Let ( z, z ′ , w, w ′ ) be an arbitrary quadruple of complex parameters.The operator A z,z ′ ,w,w ′ : R → R from Proposition 7.3 coincides with the differentialoperator D z,z ′ ,w,w ′ introduced in Definition 4.1. The theorem says that for every signature µ , the element ψ := D z,z ′ ,w,w ′ σ µ is a finite linear combination of basis elements σ ν (which is not evident!) and the cor-responding function b ψ coincides with A z,z ′ ,w,w ′ b σ µ . Obviously, this implies Theorem7.1.The rest of the paper is devoted to the proof of Theorem 7.5. The main essenceof difficulty is the fact that A z,z ′ ,w,w ′ is defined by its action on the elements of thebasis { σ µ } , whereas the action of D z,z ′ ,w,w ′ is directly seen in another basis, { ϕ ν } .The transition coefficients between the two bases seem to be too complicated toallow a direct verification of the theorem.In Subsection 7.3 we outline the plan of the proof, but first we need to recall anecessary formalism.7.2. Abstract differential operators. Let A be a commutative unital algebraand D : A → A be a linear operator. For x ∈ A , let M x : A → A denotethe operator of multiplication by x . Let us say that D has order ≤ k (where k = 0 , , , . . . ) if its ( k + 1)-fold commutator with operators of multiplication byarbitrary elements of the algebra vanishes:[ M x , [ M x , , . . . [ M x k +1 , D ] . . . ]] = 0 , x , . . . , x k +1 ∈ A . Let x , x , . . . be an arbitrary collection of elements of A . If D : A → A hasorder ≤ k , then its action on all monomials of any degree, formed from { x i } , isuniquely determined provided one knows the action on the monomials of degree ≤ k , including the monomial of degree 0, which is 1.We give a proof for k = 2 because we need this case only. Proposition 7.6. Let, as above, A be a commutative unital algebra and D : A → A be a linear operator of order ≤ . For any elements x , . . . , x n ∈ A , where n ≥ ,one has ( below the indices range over , . . . , n ) D ( x . . . x n )= X i Assume first that D has order ≤ 0. This means [ D , M x ] = 0 for any x ∈ A .Then D x = D M x M x D x D . (7.3)Next, assume D has order ≤ 1. This means that [ D , M x ] has order ≤ 0. Then,using (7.3), we have for any x, y ∈ AD ( xy ) = D M x y = x D y +[ D , M x ] y = x D y + y [ D , M x ]1 = x D y + y D x − xy D . (7.4) Finally, assume D has order ≤ 2. We are going to show that for any x, y, z ∈ AD ( xyz ) = x D ( yz ) + y D ( xz ) + z D ( xy ) − xy D z − xz D y − yz D x + xyz D . (7.5)Once this is established, the desired formula (7.2) is verified by induction on n .Namely, (7.5) is the base of the induction ( n = 3), and in order to pass from n to n + 1 one applies (7.5) with x = x . . . x n − , y = x n , z = x n +1 .It remains to prove (7.5), which is achieved using the same trick. We have D ( xyz ) = D M x ( yz ) = x D ( yz ) + [ D , M x ]( yz ) . As [ D , M x ] has order ≤ 1, we may apply (7.4), which gives[ D , M x ]( yz ) = y [ D , M x ] z + z [ D , M x ] y − yz [ D , M x ]1 . Next, y [ D , M x ] z = y D ( xz ) − xy D z, z [ D , M x ] y = z D ( xy ) − xz D y and − yz [ D , M x ]1 = − yz D x + xyz D . Putting all the pieces together we get (7.5). (cid:3) Plan of proof. The proof of Theorem 7.5 is reduced to the following twoclaims. Claim 7.7. The operators D z,z ′ ,w,w ′ and A z,z ′ ,w,w ′ coincide on the monomials ofdegree ≤ . Claim 7.8. The operator A z,z ′ ,w,w ′ : R → R has order ≤ in the abstract sense.Derivation of the theorem from these claims. Since both operators are continuousin the I -adic topology of R , it suffices to prove that they coincide on the monomials ϕ ν = ϕ ν . . . ϕ ν N .Since D z,z ′ ,w,w ′ is a second order differential operator, it has order ≤ A z,z ′ ,w,w ′ , by virtue of Claim 7.8.Thus, both operators have order ≤ N ≤ 2, and this holds by virtue of Claim 7.7. (cid:3) Claims 7.7 and 7.8 are proved in Section 8 and 9, respectively.The structure of the proof reflects the way of how the differential operator D z,z ′ ,w,w ′ has been found. Namely, assuming that A z,z ′ ,w,w ′ is a second order differential oper-ator we may write down it explicitly by computing its action on the monomials ofdegree ≤ 2, and this what we actually do in the proof of Claim 7.7.The proof is indirect, but it seems to me that a direct verification of the equality D z,z ′ ,w,w ′ = A z,z ′ ,w,w ′ , without recourse to Claim 7.8, is a difficult task. Proof of Claim 7.7 Beginning of proof. The differential operator D z,z ′ ,w,w ′ does not contain termsof order 0, so it annihilates the constants. The same holds for the operator A z,z ′ ,w,w ′ ,by the very definition.Let us verify that D z,z ′ ,w,w ′ ϕ n = A z,z ′ ,w,w ′ ϕ n , n ∈ Z . By the definition of D z,z ′ ,w,w ′ , the left-hand side equals B n = ( n + w + 1)( n + w ′ + 1) ϕ n +1 + ( n − z − n − z ′ − ϕ n − − (cid:0) ( n − z )( n − z ′ ) + ( n + w )( n + w ′ ) (cid:1) ϕ n . To compute the right-hand side we observe that ϕ n = σ ( n ) and then use thedefinition of A z,z ′ ,w,w ′ (see (7.1)). It says that A z,z ′ ,w,w ′ ϕ n = q ( n, n ) ϕ n + r ( n + 1 , n ) ϕ n +1 + r ( n − , n ) ϕ n − . Here the quantities r ( n ± , n ) and q ( n, n ) are given by formulas (6.5), (6.6), and(6.7), where we take N = 1, so that n and n ± r ( n, n + 1) = ( z − n )( z ′ − n ) , r ( n, n − 1) = ( w + n )( w ′ + n ) , which implies r ( n − , n ) = ( z − n + 1)( z ′ − n + 1) , r ( n + 1 , n ) = ( w + n + 1)( w ′ + n + 1) . Next, q ( n, n ) = − r ( n, n + 1) − r ( n, n − 1) = − ( z + n )( z ′ + n ) − ( w + n )( w ′ + n ) . This gives the same quantity B n , as desired.A more difficult task is to check that the two operators coincide on quadraticmonomials. That is, D z,z ′ ,w,w ′ ϕ κ = A z,z ′ ,w,w ′ ϕ κ , κ = ( k , k ) ∈ Z , k ≥ k . (8.1)The rest of the section is devoted to the proof of this equality.Below we use the notation: δ := ε − ε = (1 , − ∈ Z . Step 1. By (2.11), ϕ κ = ∞ X p =0 σ κ + pδ . So far we used the notation r ( ν, µ ) for ν = µ ± ε i only, but now it will be convenientto write r ( µ, µ ) instead of q ( µ, µ ). With this agreement we have A z,z ′ ,w,w ′ ϕ κ = ∞ X p =0 X ε r ( κ + pδ + ε, κ + pδ ) σ κ + pδ + ε , where ε ranges over {± ε , ± ε , } .Next, by (2.10), σ κ + pδ + ε = ϕ κ + pδ + ε − ϕ κ +( p +1) δ + ε . Consequently, A z,z ′ ,w,w ′ ϕ κ = X ε r ( κ + ε, κ ) ϕ κ + ε + ∞ X p =1 X ε (cid:2) r ( κ + pδ + ε, κ + pδ ) − r ( κ + ( p − δ + ε, κ + ( p − δ ) (cid:3) ϕ κ + pδ + ε . (8.2)The right-hand side is a linear combination of elements ϕ l l such that the differ-ence ( l + l ) − ( k + k ) takes only three possible values: ± A z,z ′ ,w,w ′ ϕ κ as the sum of three components, A z,z ′ ,w,w ′ ϕ κ = ( A z,z ′ ,w,w ′ ϕ κ ) + ( A z,z ′ ,w,w ′ ϕ κ ) − + ( A z,z ′ ,w,w ′ ϕ κ ) . On the other hand, it follows from (4.1) and (4.2) that D z,z ′ ,w,w ′ ϕ κ has the sameproperty, so we write D z,z ′ ,w,w ′ ϕ κ = ( D z,z ′ ,w,w ′ ϕ κ ) + ( D z,z ′ ,w,w ′ ϕ κ ) − + ( D z,z ′ ,w,w ′ ϕ κ ) . Thus we are led to check three equalities,( A z,z ′ ,w,w ′ ϕ κ ) = ( D z,z ′ ,w,w ′ ϕ κ ) , ( A z,z ′ ,w,w ′ ϕ κ ) − = ( D z,z ′ ,w,w ′ ϕ κ ) − , ( A z,z ′ ,w,w ′ ϕ κ ) = ( D z,z ′ ,w,w ′ ϕ κ ) . (8.3)The first two equalities are equivalent because of the symmetry consisting inswitching( z, z ′ ) ↔ ( w, w ′ ) , ( k , k ) ↔ ( − k , − k ) , ( l , l ) ↔ ( − l , − l ) . Therefore, it suffices to check the first and third equalities in (8.3).8.3. Step 2. On this step, we write down explicitly the component ( A z,z ′ ,w,w ′ ϕ κ ) of (8.2). It collects the contribution from the terms with ε = ε and ε = ε . Because δ = ε − ε , we have pδ + ε = ( p + 1) δ + ε . Using this relation one can write ( A z,z ′ ,w,w ′ ϕ κ ) in the following form:( A z,z ′ ,w,w ′ ϕ κ ) = X + X , (8.4)where X := r ( κ + ε , κ ) ϕ κ + ε + r ( κ + ε , κ ) ϕ κ + ε + (cid:2) r ( κ + δ + ε , κ + δ ) − r ( κ + ε , κ ) (cid:3) ϕ κ + δ + ε (8.5) and X := ∞ X p =1 (cid:2) r ( κ + pδ + ε , κ + pδ ) − r ( κ + ( p − δ + ε , κ + ( p − δ )+ r ( κ + ( p + 1) δ + ε , κ + ( p + 1) δ ) − r ( κ + pδ + ε , κ + pδ ) (cid:3) ϕ κ + pδ + ε . (8.6)To proceed further we need the explicit values of the jump rates: if λ = ( l , l )with l ≥ l , then r ( λ + ε , λ ) = ( w + l + 2)( w ′ + l + 2) , (8.7) r ( λ + ε , λ ) = ( ( w + l + 1)( w ′ + l + 1) , if l > l , if l = l . (8.8)Let us substitute this in (8.6). Then λ = κ + pδ or λ = κ + ( p + 1) δ with p ≥ l > l . After a simple computation one finds X = 2 ∞ X p =1 (2 p + 1 + k − k ) ϕ κ + pδ + ε . It is convenient to extend the summation to p = 0 and, to compensate this, subtractfrom X the term 2( k − k + 1) ϕ κ + ε .Then we rewrite the decomposition (8.4) in a modified form:( A z,z ′ ,w,w ′ ϕ κ ) = X ′ + X ′ , (8.9)where X ′ = 2 ∞ X p =0 (2 p + 1 + k − k ) ϕ κ + pδ + ε = 2 ∞ X p =0 (2 p + 1 + k − k ) ϕ k + p +1 ϕ k − m (8.10)and X ′ = X − k − k + 1) ϕ κ + ε . Finally, using again (8.7) and (8.8) one can check that X ′ = ( w + k + 1)( w ′ + k + 1) ϕ k +1 ϕ k + ( w + k + 1)( w ′ + k + 1) ϕ k ϕ k +1 (8.11)8.4. Step 3. Now let us turn to ( D z,z ′ ,w,w ′ ϕ κ ) . This quantity results from anappropriate truncation of the operator D . Namely, we replace it by D (1) z,z ′ ,w,w ′ := X n ∈ Z A (1) nn ∂ ∂ϕ n + 2 X n ,n ∈ Z n >n A (1) n n ∂ ∂ϕ n ∂ϕ n + X n ∈ Z B (1) n ∂∂ϕ n , where, for any indices n ≥ n , A (1) n n = ∞ X p =0 ( n − n + 2 p + 1) ϕ n + p +1 ϕ n − p and, for any n ∈ Z , B (1) n = ( n + w + 1)( n + w ′ + 1) ϕ n +1 . We represent ( D z,z ′ ,w,w ′ ϕ κ ) = D (1) z,z ′ ,w,w ′ ϕ κ as the sum of two components, the onecoming from the action of the first order derivatives and the other coming from theaction of the second order derivatives. From the explicit expressions above one canreadily check that these two components coincide with X ′ and X ′ , respectively.This completes the proof of the identity ( A z,z ′ ,w,w ′ ϕ κ ) = ( D z,z ′ ,w,w ′ ϕ κ ) , whichis the first equality in (8.3). Now we apply similar arguments to prove the thirdequality in (8.3).8.5. Step 4 (cf. Step 2 above). Here we compute ( A z,z ′ ,w,w ′ ϕ κ ) . From (8.2) weobtain( A z,z ′ ,w,w ′ ϕ κ ) = r ( κ , κ ) ϕ κ + ∞ X p =1 (cid:2) r ( κ + pδ, κ + pδ ) − r ( κ + ( p − δ, κ + ( p − δ ) (cid:3) ϕ κ + pδ . (8.12)Recall that r ( λ, λ ) := q ( λ, λ ). By (6.7), for λ = ( l , l ) with l ≥ l , r ( λ, λ ) = − ( z − l )( z ′ − l ) − ( w + l + 1)( w ′ + l + 1) − ( z − l + 1)( z ′ − l + 1) − ( w + l )( w ′ + l )+ z + z ′ + w + w ′ + 2 . We substitute this into (8.12) and obtain( A z,z ′ ,w,w ′ ϕ κ ) = Y + Y , where Y = (cid:8) − ( z − l )( z ′ − l ) − ( w + l + 1)( w ′ + l + 1) − ( z − l + 1)( z ′ − l + 1) − ( w + l )( w ′ + l )+ z + z ′ + w + w ′ + 2 (cid:9) ϕ κ (8.13)and Y = − ∞ X p =1 ( k − k + p ) ϕ κ + pδ . (8.14)8.6. Step 5 (cf. Step 3 above). Let us turn to ( D z,z ′ ,w,w ′ ϕ ) . We write( D z,z ′ ,w,w ′ ϕ ) = D (0) z,z ′ ,w,w ′ ϕ κ with D (0) z,z ′ ,w,w ′ being the following truncated operator: D (0) z,z ′ ,w,w ′ = X n ∈ Z A (0) nn ∂ ∂ϕ n + 2 X n ,n ∈ Z n >n A (0) n n ∂ ∂ϕ n ∂ϕ n + X n ∈ Z B (0) n ∂∂ϕ n , where, for any indices n ≥ n , A (0) n n = − ( n − n ) ϕ n ϕ n and, for any n ∈ Z , B (0) n = − (cid:0) ( n − z )( n − z ′ ) + ( n + w )( n + w ′ ) (cid:1) ϕ n . It is readily seen that the result of the action on ϕ κ of the first derivatives in D (0) z,z ′ ,w,w ′ coincides with Y (see (8.13)), while the action of the second derivativesleads to Y (see (8.14)).This completes the proof of (8.1). Thus, Claim 7.7 is proved, too.9. Proof of Claim 7.8 Reduction of the problem. Let us fix two nonnegative integers k and l , notequal both to 0. Proposition 9.1 (cf. Proposition 4.3) . If z = k and w = l as above, then theoperator A z,z ′ ,w,w ′ preserves the ideal I ( k, − l ) , the kernel of the canonical map R → R ( k, − l ) .Proof. Let us set S ( k, − l ) = S ∞ N =1 S N ( k, − l ) (recall that the definition of S N ( n + , n − )is given in (5.3)). The ideal I ( k, − l ) is the closed linear span of the basis elements σ ν such that ν / ∈ S ( k, − l ), where the closure is taken in the I -adic topology. Therefore,it suffices to prove the following: if ν / ∈ S N ( k, − l ) and µ ∈ S N ( k, − l ), then thequantity r k,z ′ ,l,w ′ | N ( ν, µ ) vanishes.Next, this claim is readily verified by using the definition of r z,z ′ ,w,w ′ | N ( ν, µ ), see(6.5) and (6.6). (cid:3) By Proposition 5.3, A z,z ′ ,w,w ′ preserves the ideal J (for arbitrary ( z, z ′ , w, w ′ )).Therefore, if z = k and w = l , then A z,z ′ ,w,w ′ preserves the ideal J ( k, − l ) generatedby J and I ( k, − l ), and hence gives rise to an operator on the quotient algebra b R ( k, − l ) = R/J ( k, − l ) = C [ ϕ − l , . . . , ϕ k ] (cid:14) ( ϕ − l + · · · + ϕ k − , (this quotient has already appeared in (3.5)). Let us denote the latter operator by¯ A k,z ′ ,l,w ′ . Proposition 9.2. To prove Claim 7.8 it suffices to show that the operators ¯ A k,z ′ ,l,w ′ have order ≤ .Proof. Indeed, Proposition 7.6 says that Claim 7.8 is equivalent to the relation[ M ψ , [ M ψ , [ M ψ , A z,z ′ ,w,w ′ ]]] ψ = 0 , which has to hold for arbitrary four elements ψ , ψ , ψ , ψ ∈ R . Without loss ofgenerality we may assume that all these elements are homogeneous. Then the left-hand side is homogeneous, too, as it follows from the definition of A z,z ′ ,w,w ′ . Byvirtue of Proposition 3.4, it suffices to prove that the left-hand side belongs to J . Because A z,z ′ ,w,w ′ preserves J (Proposition 7.4), this allows us to pass from from R to its quotient b R = R/J .Next, we want to specify z = k , w = l and to reduce the desired relation modulothe ideal I ( k, − l ). This is possible for the following reasons:1. A z,z ′ ,w,w ′ depends quadratically on the parameters, which allows us to specialize( z, z ′ , w, w ′ ) to any Zariski dense subset of C (or even any subset which is a set ofuniqueness for quadratic polynomials);2. as k, l → + ∞ , the ideals I ( k, − l ) decrease and their intersection is { } ;3. we know that the operator A z,z ′ ,w,w ′ can be reduced modulo I ( k, − l ) providedthat z = k and w = l . (cid:3) As the result of the factorization modulo both J and I ( k, − l ) the algebra R isreduced to the algebra b R ( k, − l ) := C [ ϕ − l , . . . , ϕ k ] (cid:14) ( ϕ − l + · · · + ϕ k − , which is isomorphic to the algebra of polynomials in m := k + l variables (we have m + 1 variables subject to a linear relation). This substantially simplifies our task,because instead of the operators A z,z ′ ,w,w ′ acting on the huge space R we may dealwith the operators ¯ A k,z ′ ,l,w ′ acting on algebras of polynomials.We have a large freedom in the choice of parameters ( z ′ , w ′ ), because the argumentabove allows us to restrict them to an arbitrary set which is a set of uniqueness forquadratic polynomials. For the reasons that will become clear below it is convenientto set z ′ = k + a , w ′ = l + b , where a and b are real numbers > − A k,k + a,l,l + b , which acts on the algebra b R ( k, − l ), is of order ≤ b R ( k, − l ) as the algebra of polyno-mial functions on the simplex Ω( k, − l ). Our aim is to show that in this realization,¯ A k,k + a,l,l + b is given by a second order partial differential operator (the Jacobi oper-ator). This will evidently imply that it has order ≤ A z,z ′ ,w,w ′ behaves exactly as D z,z ′ ,w,w ′ with respect to the shift of variables ϕ n ϕ n +const (see Proposition 4.3). Thisallows us to assume, without loss of generality, that l = 0, which slightly simplifiesthe notation.Thus, in what follows we assume that z = m, z ′ = m + a, w = 0 , w ′ = b, (9.1)where m = 1 , , . . . and a, b > − 1, and we are dealing with the operator ¯ A m,m + a, ,b acting on b R ( m, The Jacobi differential operators. As in Subsection 3.3 we introduce newvariables t , . . . , t m related to ϕ , . . . , ϕ m in the following way: m X n =0 ϕ n u n = m Y i =1 ( t i + (1 − t i ) u ) , where u is a formal variable. In other words, we substitute for ϕ , . . . , ϕ m certainsymmetric polynomials in t , . . . , t m . Then we may identify b R ( m, 0) with the algebraof symmetric polynomials in variables t , . . . , t m (see Proposition 3.8). We alsoregard ( t , . . . , t m ) as coordinates on Ω( m, 0) with the understanding that1 ≥ t ≥ · · · ≥ t m ≥ . Let us introduce the Jacobi differential operator on [0 , D ( a,b ) = t (1 − t ) d dt + [ b + 1 − ( a + b + 2) t ] ddt . Its connection with the classic Jacobi orthogonal polynomials is explained below(Subsection 9.5). Let us observe that D ( a,b ) t n = − n ( n + a + b + 1) t n + lower degree terms , n = 0 , , , . . . . (9.2)Let V m = V m ( t , . . . , t m ) := Y ≤ i As explained above, we identify b R ( m, with the algebra of symmet-ric polynomials in m variables t , . . . , t m . Then the action of the operator ¯ A m,m + a, ,b on this algebra is implemented by the m -variate Jacobi differential operator D ( a,b ) m . The proof occupies the rest of the section. Here is the scheme of proof.As explained in Subsection 5.4, we dispose of finite stochastic matrices Λ N +1 N : S N +1 ( m, S N ( m, 0) and the links Λ ∞ N : Ω( m, S N ( m, C ( S N ( m, S N ( m, ∞ N maps C ( S N ( m, C (Ω( m, b R ( m, ⊂ C (Ω( m, N grows, this image enlarges (because of the relationΛ ∞ N = Λ ∞ N +1 Λ N +1 N ) and in the limit as N → ∞ it exhausts the whole space b R ( m, A z,z ′ ,w,w ′ was defined through the difference operators D z,z ′ ,w,w ′ | N . In the special case when z = m and w = 0, the N th difference operatoris well defined on the subset S N ( m, A m,m + a, ,b itfollows that it is characterized by the commutation relations¯ A m,m + a, ,b Λ ∞ N = Λ ∞ N D m,m + a, ,b | N , where N = 1 , , . . . and the both sides are viewed as operators from the finite-dimensional space C ( S N ( m, b R ( m, A m,m + a, ,b can be replaced by the Jacobi operator D ( a,b ) m . That is, one has D ( a,b ) m Λ ∞ N = Λ ∞ N D m,m + a, ,b | N , (9.7) This will imply the desired equality ¯ A m,m + a, ,b = D ( a,b ) m .The signatures λ ∈ S N ( m, 0) can be viewed as Young diagrams contained in therectangular diagram ( m N ) := ( m, . . . , m | {z } N ) . Given such a diagram λ , we associate with it the complementary diagram κ ⊆ ( N m ):it is obtained from the shape ( m N ) \ λ by rotation and conjugation.The proof (9.7) is divided into three steps: Step 1. We express Λ ∞ N in terms of ( t , . . . , t m ) and κ (Proposition 9.6). Step 2. We show that under the correspondence λ ↔ κ , the difference operator D m,m + a, ,b | N in the right-hand side of (9.7) turns into the m -variate Hahn differenceoperator (Proposition 9.8). As the result, (9.7) takes the form D ( a,b ) m Λ ∞ N = Λ ∞ N ∆ ( a,b,N + m − m , N = 1 , , . . . , (9.8)where ∆ ( a,b,N + m − m is the Hahn difference operator in question. Step 3. We prove that Λ ∞ N transforms the m -variate symmetric Hahn polynomi-als into the respective m -variate symmetric Jacobi polynomials (Proposition 9.10).Then the proof is readily completed.We proceed to the detailed proof of the theorem.9.3. Step 1: transformation of the link Λ ∞ N . Let λ range over the set of Youngdiagrams contained in the rectangle ( m N ), and κ ⊆ ( N m ) be the complementarydiagram to λ . In more detail, κ = ( N − λ ′ m , . . . , N − λ ′ ) , where the diagram λ ′ is conjugate to the diagram λ . Next, we set l i := λ i + N − i, i = 1 , . . . , N ; k j = κ j + m − j, j = 1 , . . . , m. Evidently, l > · · · > l N and k > · · · > k m . Lemma 9.4. The set { , . . . , N + m − } is the disjoint union of the sets L := { l , . . . , l N } and K := { k , . . . , k m } .Proof. This is a well-known fact, see e.g. Macdonald [25, ch. I, (1.7)]. (cid:3) Introduce a notation: M := N + m − , I M = { , . . . , M } . Next, for a finite collection of numbers X = { x > · · · > x n } we set V ( X ) = V n ( x , . . . , x n ) = Y ≤ i One has V ( l , . . . , l N ) = 0!1! . . . M ! V ( k , . . . , k m ) m Q j =1 k j !( M − k j )! (9.9) Proof. By the preceding lemma, I M = L ⊔ K , whence V ( I M ) = V ( K ⊔ L ) = V ( K ) · V ( L ) · Y x ∈ K Y y ∈ L | x − y | . (9.10)For x ∈ K , set f ( x ) := Y z ∈ I M \{ x } | x − z | and observe that Y x ∈ K Y y ∈ L | x − y | = Q x ∈ K f ( x )( V ( K )) . Substituting this into (9.10) gives V ( L ) = V ( I M ) V ( K ) Q x ∈ K f ( x ) . On the other hand, it is readily checked that f ( x ) = x !( M − x )!and V ( I M ) = 0!1! . . . M !. This completes the proof. (cid:3) Proposition 9.6. Let ω = ω ( t , . . . , t m ) be the point of the simplex Ω( m, withcoordinates ( t , . . . , t m ) . In the notation introduced above, Λ ∞ N ( ω ; λ ) = const m,M V ( k , . . . , k m ) V ( t , . . . , t m ) det (cid:20)(cid:18) Mk j (cid:19) t k j i (1 − t i ) M − k j (cid:21) mi,j =1 , where const m,M = m Y i =1 ( M − i + 1)! M ! . In particular, in the simplest case m = 1, there is a single coordinate t = t ∈ [0 , λ has a single column, the complementary diagram has a singlerow whose length equals κ = k ∈ { , . . . , N } , and Λ ∞ N is represented as the link[0 , { , . . . , N } that assigns to a point t ∈ [0 , 1] the binomial distribution on { , . . . , N } with parameter t . Proof. (i) By the very definition of the link Λ ∞ N (see (5.2) and the comment afterit), Λ ∞ N ( ω, λ ) = Dim N λ · { coefficient of s λ ( u , . . . , u N ) in Φ( u ; ω ) . . . Φ( u N ; ω ) } . We haveΦ( u ; ω ) . . . Φ( u N ; ω ) = m Y i =1 N Y j =1 (1 + β + i ( u j − m Y i =1 N Y j =1 ( t i + (1 − t i ) u j )= m Y i =1 (1 − t i ) N · m Y i =1 N Y j =1 (cid:18) t i − t i + u j (cid:19) = m Y i =1 (1 − t i ) N · X λ : λ ⊆ ( m N ) s κ (cid:18) t − t , . . . , t m − t m (cid:19) s λ ( u , . . . , u N ) , where the last equality follows from the dual Cauchy identity, see [25, Chapter I,Section 4, Example 5]. Therefore,Λ ∞ N ( ω, λ ) = Dim N λ · m Y i =1 (1 − t i ) N · s κ (cid:18) t − t , . . . , t m − t m (cid:19) . (ii) By the definition of the Schur polynomials, m Y i =1 (1 − t i ) N · s κ (cid:18) t − t , . . . , t m − t m (cid:19) = m Q i =1 (1 − t i ) N · det "(cid:18) t i − t i (cid:19) k j V (cid:18) t − t , . . . , t m − t m (cid:19) , where the determinant in the numerator is of order m .The denominator of this expression is equal to m Y i =1 (1 − t i ) − m +1 · V ( t , . . . , t m ) . Therefore, the whole expression is m Q i =1 (1 − t i ) M · det "(cid:18) t i − t i (cid:19) k j V ( t , . . . , t m ) = det h t k j i (1 − t i ) M − k j i V ( t , . . . , t m ) , so that Λ ∞ N ( ω, λ ) = Dim N λV ( t , . . . , t m ) det h t k j i (1 − t i ) M − k j i . (iii) It remains to handle Dim N λ . By Weyl’s dimension formula,Dim N λ = V ( L ) V ( N − , N − , . . . , 0) = V ( l , . . . , l N )0!1! . . . ( N − ∞ N ( ω, λ ) = 0!1! . . . M !0!1! . . . ( N − V ( k , . . . , k m ) V ( t , . . . , t m ) det (cid:20) k j !( M − k j )! t k j i (1 − t i ) M − k j (cid:21) . The constant factor in front equals Q mj =1 ( M − j + 1)!. Dividing it by ( M !) m and introducing the same quantity inside the determinant we finally get the desiredexpression. (cid:3) Step 2: transformation of the difference operator D m,m + a, ,b . We con-tinue to deal with two mutually complementary point configurations L = ( l > · · · > l N ) and K = ( k > · · · > k m ) on the lattice interval I M = { , . . . , M } .Our next aim is to derive a convenient expression for the jump rates introduced inSubsection 6.2. So far they were denoted as q ( ν, ν ± ε i ). Now we rename ν to λ andnext we pass from λ to the corresponding point configuration L . In terms of L ,the transition λ → λ ± ε i can be written as x → x ± 1, where x = l i . Accordingto this we change the former notation for the jump rates and will denote them by q ( x → x ± x ∈ L .Taking into account the values of the parameters (see (9.1)), the formulas ofSubsection 6.2 can be rewritten as follows q ( x → x + 1) = V ( L − { x } + { x + 1 } ) V ( L ) ( M − x )( M + a − x ) , (9.11) q ( x → x − 1) = V ( L − { x } + { x − } ) V ( L ) x ( b + x ) . (9.12)Here L − { x } + { x ± } denotes the configuration obtained from L by removing x and inserting x ± x → x + 1 is forbidden if the corresponding vector λ + ε i is not a signature, which happens when λ i − = λ i . In terms of L , this means x + 1 ∈ L , in which case the configuration L − { x } + { x + 1 } contains the point x + 1 twice, and then V ( L − { x } + { x + 1 } ) should be understood as 0. Likewise,if x → x − V ( L − { x } + { x − } ) vanishes. Thus, (9.11) and(9.12) formally assign rate 0 to forbidden transitions, which is reasonable. Lemma 9.7. In terms of the complementary configuration K , the jump rates takethe form e q ( y → y − 1) = V ( K − { y } + { y − } ) V ( K ) y ( M + 1 + a − y ) , e q ( y → y + 1) = V ( K − { y } + { y + 1 } ) V ( K ) ( M − y )( b + y + 1) . Proof. A jump x → x +1 in L is possible if and only if x ∈ L and x +1 / ∈ L . This isequivalent to saying that x +1 ∈ K and x / ∈ K , which in turn means the possibilityof the jump y → y − 1, where y = x + 1. Therefore, e q ( y → y − 1) = q ( x → x + 1).Now we have to express the quantity q ( x → x + 1) given by (9.11) in terms of K .Lemma 9.5 tell us that V ( L ) = const V ( K ) Q y ∈ K y !( M − y )! . It follows that V ( L − { x } + { x + 1 } ) V ( L ) = V ( K − { y } + { y − } ) V ( K ) yM + 1 − y Next, ( M − x )( M + a − x ) = ( M + 1 − y )( M + 1 + a − y ) . Multiplying out these two quantities we get the desired expression for e q ( y → y − x → x − y → y + 1, where y = x − 1, so werewrite the expression for q ( x → x − 1) given by (9.12). We have V ( L − { x } + { x − } ) V ( L ) = V ( K − { y } + { y + 1 } ) V ( K ) M − yy + 1 . Next, x ( b + x ) = ( y + 1)( b + y + 1) . Multiplying out these two quantities we get the desired expression for e q ( y → y +1). (cid:3) We introduce the Hahn difference operator ∆ ( a,b,M ) by(∆ ( a,b,M ) F )( y ) = ( y + b + 1)( M − y )[ F ( y + 1) − F ( y )]+ y ( M + a − y + 1)[ F ( y − − F ( y )] , (9.13)where F is a function in variable y . Note that ∆ ( a,b,M ) is well defined on I M . Indeed,the coefficient in front of [ F ( y + 1) − F ( y )] vanishes at the point y = M , the rightend of the interval; likewise, the coefficient in front of [ F ( y − − F ( y )] vanishes atthe left end y = 0. The difference operator ∆ ( a,b,M ) is associated with the classic Hahn polynomi-als: see Koekoek-Swarttouw [20, (1.5.5)] and the next subsection. Note that ourparameters ( a, b, M ) correspond to parameters ( β, α, N ) from [20, Section 1.5].It is directly verified that∆ ( a,b,M ) y n = − n ( n + a + b + 1) y n + lower degree terms , n = 0 , , , . . . . Note that the factor in front of y n is exactly the same as in (9.2). In particular, itdoes not depend on the additional parameter M that enters the definition of thedifference operator.Now we introduce the m -variate Hahn difference operator in the same way as wedefined above the m -variate Jacobi operator:∆ ( a,b,M ) m = 1 V m ◦ m X i =1 ∆ ( a,b,M )variable y i ! ◦ V m + const a,b,m . (9.14)Here y , . . . , y m is an m -tuple of variables, V m = V m ( y , . . . , y m ) is the Vandermonde,∆ ( a,b,M )variable y i denotes the one-variate Hahn operator acting on the i th variable, and theconstant is given by (9.4). The same argument as above shows that the operator∆ ( a,b,M ) m is well defined on the space of symmetric polynomials and kills the constants.Alternatively, ∆ ( a,b,M ) m can be interpreted as an operator acting on the space offunctions on m -point configurations K = ( k > · · · > k m ) ⊆ ( N m ) (here we write( k , . . . , k m ) instead of ( y , . . . , y m )). This is just the interpretation that we need.On the other hand, the difference operator D m,m +1 , ,b | N acts on the functionsdefined on set of the diagrams λ or, equivalently, on the set of configurations L .Now we use the correspondence L ↔ K to compare the both operators. Proposition 9.8. Under the correspondence λ ↔ L ↔ K ↔ κ , the operator D m,m + a, ,b | N turns into the operator ∆ ( a,b,M ) m .Proof. Let us regard D m,m + a, ,b | N as an operator on the space of functions F ( K ).Then Lemma 9.7 shows that D m,m + a, ,b | N acts as the following difference operator( D m,m + a, ,b | N F )( K ) = X y ∈ K X ε = ± e q ( y → y + ε )[ F ( K − { y } + { y + ε } ) − F ( K )] . Looking at the explicit expressions for the jump rates e q ( y → y + ε ) given in Proposi-tion 9.8 and comparing them with the definition of ∆ ( a,b,M ) m (see (9.13)) we concludethat D m,m + a, ,b | N = ∆ ( a,b,M ) m . (cid:3) Step 3: The transformation Hahn → Jacobi. Let us collect a few classicformulas about the Hahn and Jacobi orthogonal polynomials. They can be found,e.g., in Koekoek-Swarttouw [20]. The Hahn polynomials with parameters ( a, b, M ), denoted here by H ( a,b,M ) n ( y ), arethe orthogonal polynomials on I M = { , . . . , M } with the weight W ( a,b,M )Hahn ( y ) = (cid:18) b + yy (cid:19)(cid:18) a + M − yM − y (cid:19) , y ∈ I M . The subscript n is the degree; it ranges also over I M . As was already pointed out,our notation slightly differs from that of [20]: our parameters ( a, b ) correspond toparameters ( β, α ) in [20, Section 1.5].The Hahn polynomials form an eigenbasis for the Hahn difference operator ∆ ( a,b,M ) defined in (9.13): ∆ ( a,b,M ) H ( a,b,M ) n = − n ( n + b + a + 1) H ( a,b,M ) n . (9.15)Here is the explicit expression of the Hahn polynomials through a terminatinghypergeometric series of type (3 , 2) at point 1: H ( a,b,M ) n ( y ) = F " − n, n + b + a + 1 , − yb + 1 , − M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , n = 0 , . . . , M. Our notation for the Jacobi polynomials is J ( a,b ) n ( t ); these are the orthogonalpolynomials on the unit interval [0 , 1] with the weight W ( a,b )Jacobi ( t ) = t b (1 − t ) a , ≤ t ≤ . Note that many sources, including [20], take the weight function (1 − x ) a (1 + x ) b with x ranging over [ − , , 1] to [ − , 1] is given by the changeof variable x = 2 t − D ( a,b ) J ( a,b ) n = − n ( n + b + a + 1) J ( a,b ) n , n = 0 , , , . . . . (9.16)The Jacobi polynomials are expressed through the Gauss hypergeometric series: J ( a,b ) n ( t ) = F " − n, n + b + a + 1 b + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t , n = 0 , , , . . . . Note that our normalization of the Jacobi polynomials differs from the conventionalone, but this is convenient for the computation below. Lemma 9.9. The following relation holds M X k =0 (cid:18) Mk (cid:19) t k (1 − t ) M − k H ( a,b,M ) n ( k ) = J ( a,b ) n ( t ) , n = 0 , . . . , M. Proof. This is checked directly using the explicit expressions for the polynomials.Indeed, the sum in the left-hand side equals M X k =0 n X p =0 M ! t k (1 − t ) M − k ( − n ) p ( n + b + a + 1) p ( − k ) p k !( M − k )!( b + 1) p ( − M ) p p ! . Let us change the order of summation and observe that ( − k ) p vanishes unless k ≥ p . Then the above expression can be rewritten as n X p =0 M X k = p M ! t k (1 − t ) M − k ( − n ) p ( n + b + a + 1) p ( − k ) p k !( M − k )!( b + 1) p ( − M ) p p ! . Next, let us set q = k − p and observe that M !( − k ) p k !( M − k )!( − M ) p = M ! k !( M − p )! k !( k − p )! M !( M − k )! = (cid:18) M − pq (cid:19) . It follows that our double sum equals n X p =0 ( − n ) p ( n + b + a + 1) p ( b + 1) p p ! t p M − p X q =0 (cid:18) M − pq (cid:19) t q (1 − t ) M − p − q . The interior sum equals 1, so that we finally get n X p =0 ( − n ) p ( n + b + a + 1) p ( b + 1) p p ! t p = J ( a,b ) n ( t ) , as desired. (cid:3) The m -variate Hahn polynomials are given by H ( a,b,M ) ν ( y , . . . , y m ) = det h H ( a,b,M ) n j ( y i ) i V m ( y , . . . , y m ) . Here ν is an arbitrary Young diagram contained in ( N m ) and n j := ν j + m − j, j = 1 , . . . , m. The definition is correct because the largest index n does not exceed M (recall that M = N + m − 1; therefore, ν ⊆ ( N m ) implies n = ν + m − ≤ M ).Likewise, the m -variate Jacobi polynomials are given by J ( a,b ) ν ( t , . . . , t m ) = det h J ( a,b ) n j ( t i ) i V m ( t , . . . , t m ) . Here ν is an arbitrary Young diagram with at most m nonzero rows. Proposition 9.10. For every N = 1 , , . . . and every Young diagram ν ⊆ ( N m ) , theoperator Λ ∞ N takes the Hahn polynomial H ( a,b,M ) ν to the respective Jacobi polynomial J ( a,b ) ν , within a constant factor.Proof. By virtue of Proposition 9.6,Λ ∞ N H ( a,b,M ) ν )( t , . . . , t m )= const m,M V ( t , . . . , t m ) X M ≥ k > ··· >k m ≥ det (cid:20)(cid:18) Mk j (cid:19) t k j i (1 − t i ) M − k j (cid:21) det (cid:2) H ( a,b,M ) n i ( k j ) (cid:3) . (9.17)Now we apply a well-known identity, which is a consequence of the Cauchy-Binetidentity: X M ≥ k > ··· >k m ≥ det[ f i ( k j )] mi,j =1 det[ g i ( k j )] mi,j =1 = det[ h ij ] mi,j =1 , where h ij := M X k =0 f i ( k ) g j ( k ) . It tells us that the sum in (9.17) equals the determinant of the m × m matrix whose( i, j ) entry is M X k =0 (cid:18) Mk (cid:19) t ki (1 − t i ) M − k H n j ( k ) . By Lemma 9.9, the last sum equals J ( a,b ) n j ( t i ). This completes the proof of theproposition. (cid:3) Completion of proof. As pointed out above (see (9.15) and (9.16)), the clas-sic Hahn and Jacobi polynomials are eigenfunctions of the respective operators, andthe n th eigenvalue in both cases is the same number c ( n ) := − n ( n + a + b + 1).By the very definition of the multivariate polynomials and operators, the similarassertion holds for arbitrary m as well, and the eigenvalue corresponding to a givenlabel ν is equal to m X i =1 [ c ( ν i + m − i ) − c ( m − i )] . Combining this with the result of Step 3 (Proposition 9.10) we obtain the de-sired commutation relation (9.8) which says that the link Λ ∞ N intertwines the Jacobidifferential operator D ( a,b ) m with the Hahn difference operator ∆ ( a,b,N + m − m .Finally, as pointed out in the end of Subsection 9.2, the result of Step 2 (Propo-sition 9.8) reduces Theorem 9.3 to that commutation relation.This completes the proof of Theorem 9.3, which in turn implies Claim 7.8. Thus,the proof of Theorem 7.5 is completed. Appendix: uniform boundedness of multiplicities Here we prove the statement used in the proof of Proposition 2.4, step 1. Weformulate the result in a greater generality, which seems to be more natural.Let e G be a connected reductive complex group and G ⊂ e G be a reductive sub-group. We assume G is spherical, meaning that for any simple e G -module V , thespace V G of G -invariants has dimension at most 1. For a simple G -module W wewrite [ V : W ] := dim Hom G ( W, V ) . Proposition 10.1. Let e G , G , V , and W be as above. If W is fixed, then for themultiplicity [ V : W ] there exists a bound [ V : W ] ≤ const , where the constantdepends only on W but not on V . The fact that we needed in Proposition 2.2 is a particular case of Proposition 10.1corresponding to e G = GL ( M + N, C ) and G = GL ( M, C ) × GL ( N, C ). First proof ( communicated by Vladimir L. Popov ) . Let us fix a Borel subgroup B ⊂ e G and denote by N the unipotent radical of B . Let A = C [ e G/N ] be the algebraof regular functions on e G/N . In other words, A consists of holomorphic functionson e G/N which are e G -finite with respect to the action of e G by left shifts. As a e G -module, A is the multiplicity free direct sum of all simple e G -modules: A = M λ ∈ Λ + A λ , (10.1)where Λ + denotes the additive semigroup of dominant weights with respect to B and A λ denotes the subspace of A carrying the simple e G -module with highest weight λ . We fix a simple G -module W . Given a G -module X , we denote by X ( W ) the W -isotypic component in X . Using this notation, the desired claim can be reformulatedas follows: as λ ranges over Λ + , the quantities dim A ( W ) λ are uniformly bounded fromabove. Step 1. Let A G ⊂ A be the subalgebra of G -invariants. Obviously, A ( W ) is a A G -module. We claim that it is finitely generated.Indeed, this is equivalent to saying that Hom G ( W, A ) is finitely generated as a A G -module.Observe that the expansion (10.1) is a grading of A . That is, A λ ′ A λ ′′ ⊆ A λ ′ + λ ′′ , λ ′ , λ ′′ ∈ Λ + . (10.2)Since the semigroup Λ + is finitely generated, the algebra A is finitely generated.This property together with the fact that G is assumed to be reductive make itpossible to apply the classic trick (used in Hilbert’s theorem on invariants) to the A - G -module Hom( W, A ), see Popov-Vinberg [38, Theorems 3.6 and 3.25]. Then weobtain that (Hom( W, A )) G is a finitely generated A G -module, as desired. Step 2. By virtue of Step 1, there exists a finite collection of weights λ (1) , . . . , λ ( n ) ∈ Λ + such that A ( W ) is generated over A G by the subspace A ( W ) λ (1) + · · · + A ( W ) λ ( n ) . Fromthis and (10.1) we conclude that for every weight λ ∈ Λ + , the subspace A ( W )) λ iscontained in the sum of subspaces of the form A Gλ − λ ( i ) A ( W ) λ ( i ) , where i ∈ { , . . . , n } should be such that λ − λ ( i ) ∈ Λ + .Because G is a spherical subgroup of e G , every subspace A Gλ − λ ( i ) has dimension atmost 1. This gives us the desired bounddim A ( W ) λ ≤ n X i =1 dim A ( W ) λ ( i ) , uniform on λ ∈ Λ + . (cid:3) Second proof ( sketch ) . Given a finite-dimensional G -module Y , we can define theinduced e G -module Ind( Y ): its elements are holomorphic vector-functions f : e G → Y , which are e G -finite with respect to right shifts and such that f ( g e g ) = gf ( e g ) forany g ∈ G and e g ∈ e G .As above, we fix a simple G -module W . The desired claim is equivalent to theexistence of a uniform bound for [Ind( W ) : V ], the multiplicity of an arbitrary simple e G -module V in the decomposition of Ind( W ).Given a finite-dimensional e G -module X , let us denote by X G the same spaceregarded as a G -module. One can choose X in such a way that X G contained W .Then we obviously have [Ind( W ) : V ] ≤ [Ind( X G ) : V ].The key observation is that Ind( X G ) is isomorphic to Ind( C ) ⊗ X , where C standsfor the trivial one-dimensional G -module.Now let V be an arbitrary simple e G -module. We have[Ind( C ) ⊗ X : V ] = dim Hom e G ( V ⊗ X ∗ , Ind( C )) , where X ∗ is the dual module to X . Observe that in the decomposition of V ⊗ X ∗ on simple components, every multiplicity does not exceed dim X ∗ = dim X (thisfollows from a well-known formula describing the decomposition of tensor products,see Zhelobenko [48, end of § § 24, Exercise 9] or else can beeasily proved directly). 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