q-deformed Character Theory for Infinite-Dimensional Symplectic and Orthogonal Groups
aa r X i v : . [ m a t h . R T ] D ec q -DEFORMED CHARACTER THEORY FOR INFINITE-DIMENSIONALSYMPLECTIC AND ORTHOGONAL GROUPS CESAR CUENCA AND VADIM GORIN
Abstract.
The classification of irreducible, spherical characters of the infinite-dimensional uni-tary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized,irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity.We solve a q -deformed version of the latter problem for orthogonal and symplectic groups, extend-ing previously known results for the unitary group. The proof is based on novel determinantal anddouble-contour integral formulas for the q -specialized characters. Contents
1. Introduction 21.1. Preface 21.2. q –deformed characters: main results 31.3. Further results and outlook 41.4. Methodology 51.5. Terminology and conventions 51.6. Acknowledgements 62. Characters of U ( N ), SO ( N ) and Sp ( N ) 62.1. Type A characters 62.2. Type B-C-D characters 73. Asymptotics of q -deformed characters 103.1. Type A characters: statements of results 103.2. Type A characters: proofs of results 113.3. Type B-C-D characters: statements of results 173.4. Integral representations and multivariate formulas for B-C-D type characters 183.5. Type B-C-D characters: proofs of results 234. Branching graphs 244.1. Generalities on Branching Graphs 244.2. Symmetric q -Gelfand-Tsetlin graph 264.3. BC type q -Gelfand-Tsetlin graph 265. Boundary of the symmetric q -Gelfand-Tsetlin graph 275.1. A family of coherent probability measures 275.2. Concentration bound 295.3. Law of Large Numbers and the Martin boundary 325.4. Characterization of the minimal boundary 336. Boundary of the BC type q -Gelfand-Tsetlin graph 336.1. A family of coherent probability measures 346.2. Concentration bound 356.3. Law of Large Numbers and the Martin boundary 396.4. Characterization of the minimal boundary 39References 39 Introduction
Preface.
For each of the three series of classical compact Lie groups: unitary U ( N ), or-thogonal SO ( N ), and symplectic Sp ( N ), one can naturally embed groups of the smaller rankinto the larger ones and form inductive limits U ( ∞ ) = S ∞ N =1 U ( N ), SO ( ∞ ) = S ∞ N =1 SO ( N ), Sp ( ∞ ) = S ∞ N =1 Sp ( N ). The study of such infinite–dimensional or “big” groups has been a centraltopic of the asymptotic representation theory during the last 40 years. These groups are wild, whichmeans that one needs to restrict the class of representations, in order to get a meaningful theory.One point of view is to deal with characters, i.e. central, positive–definite, continuous functions onthe group. The extreme characters then correspond to finite factor representations of the group; for U ( ∞ ) all of them were classified by Voiculescu [Vo], while for SO ( ∞ ) and Sp ( ∞ ) similar results wereobtained by Boyer [Bo2]. More general characters can be identified with spherical representationsof the Gelfand pairs ( U ( ∞ ) × U ( ∞ ) , U ( ∞ )), ( SO ( ∞ ) × SO ( ∞ ) , SO ( ∞ )), ( Sp ( ∞ ) × Sp ( ∞ ) , Sp ( ∞ )),and such a theory was introduced and thoroughly studied by Olshanski [O1], [O2]. From anotherdirection, as first noticed in [VK2],[Bo1], one can identify extreme characters with totally–positiveToeplitz matrices, and then their classification theorem becomes equivalent to the earlier Edrei’stheorem [E] from classical analysis.Yet another approximative approach was suggested by Vershik and Kerov [VK2]; in this approach,the parameters of the characters of U ( ∞ ), SO ( ∞ ), Sp ( ∞ ) become limits of normalized lengths as N → ∞ of rows and columns in the Young diagrams parameterizing the irreducible representationsof finite–dimensional groups U ( N ), SO ( N ), Sp ( N ). This idea can be used to produce severaldistinct proofs of the character classification theorems of Voiculescu and Boyer, see [OO2], [OO3],[BO], [Pe], [GP], [O6].Classical Lie groups admit a q –deformation to quantum groups , cf. [BK], [CP], which leads toa natural question of whether a similar deformation is possible for the character theory of theirinfinite–dimensional versions. This question for the unitary groups U ( N ) was first addressed bythe second author [G] and based on the following observation. In the Vershik–Kerov approach, thecharacters of U ( ∞ ) can be treated as the limits of normalized Schur polynomials (characters ofirreducible representation of U ( N )) as the number of variables goes to infinitylim N →∞ s λ ( x , x , . . . , x k , N − k ) s λ (1 N ) , k = 1 , , . . . , | x | = | x | = · · · = | x k | = 1 , (1.1)where λ = λ ( N ) changes in an appropriate way (as N → ∞ ) to guarantee the existence of thelimit, and 1 N means N variables all equal to 1. Then the q –deformation of [G] is based on thereplacement of 1 N − k and 1 N in (1.1) by the geometric series with ratio q . Such a point of viewturned out to be fruitful: in [G] a classification theorem for the new q –characters was obtained(they are parameterized by nondecreasing sequences of integers) and in [BG], [C], [Pe], [GP], [GO],[O5] the topic was further developed. It was noticed in [G] that the q –characters have a link tothe quantum traces for the representations of the quantized universal enveloping algebra U q ( gl N );however, no infinite–dimensional object was constructed. A more elaborate representation–theoreticinterpretation for the q –characters of U ( ∞ ) was presented recently by Sato [S] in the language ofcompact quantum groups.After [G] appeared, an immediate question arose: can the results be extended to other rootsystems, i.e. to orthogonal and symplectic groups? Despite several other approaches to q –charactersof unitary groups appearing in the subsequent years, it remained unclear whether the existence ofa “good” q –deformation of the character theory for the infinite–dimensional group is an artifact ofthe root system of type A , or if it exists for all classical series of Lie groups? In the present articlewe resolve this question by constructing a rich q –deformed character theory for SO ( ∞ ) and Sp ( ∞ ). -DEFORMED CHARACTER THEORY · · · q –deformed characters: main results. The Vershik–Kerov approach [VK2], [OO1] to theasymptotic representation theory makes the classification of all extreme characters of U ( ∞ ) equiv-alent to the following problem. Find all sequences of signatures (i.e. highest weights of irreduciblerepresentations) λ ( N ) = ( λ ( N ) ≥ λ ( N ) ≥ · · · ≥ λ ( N ) N ) ∈ Z N , N = 1 , , . . . , such that for each k = 1 , , . . . , the limit (1.1) exists uniformly on the torus { ( x , . . . , x k ) ∈ C k : | x i | = 1 , ≤ i ≤ k } .The limit itself is then identified with the value of the character on a unitary matrix from the group U ( ∞ ) with non-trivial (i.e. different from 1) eigenvalues x , . . . , x k .The q –deformation of [G] suggests to fix a parameter 0 < q < λ ( N ), such that for each k = 1 , , . . . there exists a limitlim N →∞ s λ ( N ) ( x , x , . . . , x k , q − k , q − k − , . . . , q − N ) s λ (1 , q − , q − , . . . , q − N ) . (1.2)A priori in (1.2) the convergence is assumed to be uniform over | x i | = q − i , 1 ≤ i ≤ k , i.e. onthe torus of q –growing radius. However, a posteriori, (1.2) converges for all x i = 0. The limit of(1.2) is then the desired q –character. There are several reformulations of this asymptotic problem:one of them is the identification of the minimal boundary (=extreme Gibbs measure on the spaceof paths) of a certain branching graph, called the q –Gelfand–Tsetlin graph. The vertices of thisgraph are labels of irreducible representations of U ( N ), N = 1 , , . . . , i.e. signatures, and the edgesencode the branching rules upon restrictions from U ( N ) onto U ( N − q –dependent weight, and the whole combinatorics can be linked to the notion of the quantumdimension for the representations of the quantized unversal envelopping algebra U q ( gl N ). We referto [G] and Section 4 for the details. Another reformulation deals with q –Toeplitz matrices, see [G,Section 1.5].The limits in (1.2) turn out to be parameterized by infinite sequences of integers ν ≤ ν ≤ ν ≤ . . . , ν i ∈ Z , which are identified with the last rows of λ ( N ): ν i = lim N →∞ λ ( N ) N +1 − i , i = 1 , , . . . . (1.3)One clear feature of (1.2) is its asymmetry under the change q q − . The properties ofSchur polynomials imply that such change is equivalent to the transformation λ ( N ) e λ ( N ) :=( − λ ( N ) N ≥ · · · ≥ − λ ( N ) ) and therefore, when q >
1, as N → ∞ we would need to consider thefirst rows of λ ( N ) instead of the last rows in (1.3).If we now switch to orthogonal and symplectic groups, then the symmetry becomes important.Indeed, the eigenvalues for the matrices of these groups come in pairs { z i , z − i } and the charactersare invariant under inversion of the variables. Thus, the first step towards the q –deformation for SO ( ∞ ), Sp ( ∞ ) is to make the setting of (1.2) symmetric by changing it tolim N →∞ s λ ( N ) ( q − N , q − N , . . . , q − k − , x − k , . . . , x k , q k +1 , q k +1 , . . . , q N ) s λ ( N ) ( q − N , q − N , . . . , q N − , q N ) , x , . . . , x k ∈ C ∗ , (1.4)where λ ( N ) now has 2 N + 1 rows: λ ( N ) − N ≥ λ ( N ) − N ≥ · · · ≥ λ ( N ) N . Note that, since the Schurpolynomials are homogeneous, there is some further freedom, as we can multiply all its variablesby arbitrary q M ; the most general setup is described in Section 3.1.Our first main result (Theorem 3.3) proves that the limits of (1.4) are parameterized by two–sided sequences of integers · · · ≥ ν − ≥ ν ≥ ν ≥ . . . and the limit exists if and only iflim N →∞ λ ( N ) i = ν i for all i ∈ Z . (1.5)The limiting functions (“symmetric q –characters”) are given by explicit contour integral formulas. CESAR CUENCA AND VADIM GORIN
Proceeding to the orthogonal and symplectic cases, let G ( N ) denote either SO (2 N +1), or Sp ( N ),or SO (2 N ), corresponding to the root systems B N , C N , D N , respectively. The irreducible represen-tations are still parameterized by signatures λ ≥ λ ≥ · · · ≥ λ N , but this time all the coordinatesare required to be positive . The characters χ Gλ of the representations can be then identified withsymmetric Laurent polynomials in N variables z , . . . , z N , invariant under the inversions z i z − i ,see Section 2.2 for more details. Set ǫ = 1 / , , B, C, D , respectively, and consider thelimits lim N →∞ χ Gλ ( N ) ( x , . . . , x k , q k + ǫ , . . . , q N − ǫ ) χ Gλ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) , x , . . . , x k ∈ C ∗ , (1.6)where λ ( N ) = ( λ ( N ) ≥ λ ( N ) ≥ · · · ≥ λ ( N ) N ≥ ≤ ν ≤ ν ≤ . . . and the limit exists if and only iflim N →∞ λ ( N ) N +1 − i = ν i , i = 1 , , . . . . (1.7)The limit functions are given by explicit contour integral formulas.One might be wondering about our choice of normalization in (1.6). It has several explanations.First, the numbers ǫ, ǫ + 1 , . . . , N − ǫ are precisely the coordinates of ρ — the half–sum of thepositive roots in the corresponding root system. The value of χ Gλ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) is typicallycalled the quantum dimension in the quantum groups literature — this matches the normalizationin (1.1) by the ordinary dimension Dim( λ ) = s λ (1 N ) of the representation. From a more technicalpoint of view, under the normalization of (1.6), the characters possess the label–variable duality,see Lemma 2.8; this duality is one of the key ingredients of our proofs.Another point worth emphasizing is that we get two different q –deformed character theoriesfor SO ( ∞ ): one from the approximation by odd-dimensional groups and another one from even–dimensional groups. In the q = 1 case there was no difference, however the answers (e.g. thefunctional form of the limit in (1.6)) look different in our q –deformation. From the representation–theoretic point of view this can be linked to the fact that quantum groups related to SO (2 N ) and SO (2 N + 1) are also very different, and no embeddings of one into another are known. In theinfinite–dimensional setting related to quantum groups, the distinction between B and D seriesalso appeared before, e.g. in [O3].1.3. Further results and outlook.
There is another reformulation of the classification theoremfor characters of U ( ∞ ), SO ( ∞ ), Sp ( ∞ ) in the language of combinatorial probability. Then oneneeds to describe all sequences of measures P N , N = 1 , , . . . on the labels of irreducible represen-tations of groups of rank N , which would agree through certain coherency relations , obtained fromthe branching rules for the representations. Such a reformulation is also possible in our q –deformedsetting and we describe it in Section 4. In particular, the stabilization property of (1.5), (1.7) thenturns into the Law of Large Numbers as explained in Sections 5.3 and 6.3.We believe that there should be a way to transform our constructions from the language ofspecial functions theory (as study of limits in (1.4), (1.6)) and from the language of combina-torial probability (as classifying the coherence family of probability measures in Section 4) into atruly representation–theoretic framework dealing with certain infinite–dimensional analogues of thequantum groups. Development of such a framework is an important open problem.In [G] it was found that by using certain inhomogeneous analogues of Schur functions, onecan encode the limits to (1.2) through simple multiplicative formulas (in particular, avoiding anycontour integrals). It would be interesting to try to find similar formulas also for the symmetricsetup (1.4) as well as for the
B, C, D series of (1.6), since the conceptual understanding for the For SO (2 N ), λ N is allowed to be negative, but one should have λ ≥ · · · ≥ λ N − ≥ | λ N | . In this case we dealinstead with the direct sum of two twin representations that differ by a flip of the sign of λ N . -DEFORMED CHARACTER THEORY · · · limiting functions in (1.4) and (1.6) (given by the contour integral formulas in Theorems 3.3 and3.13) is currently missing.1.4. Methodology.
Let us indicate one important idea that made the developments of this articlepossible. In [GP] an approach to the study of asymptotics of (1.2) was suggested based on contourintegral formulas for k = 1 and k × k determinantal formulas reducing the general k case to thebase case k = 1. Through label–variable symmetry duality for the normalized Schur functionsand analytic continuation, these formulas are dual to the closed generating function expressionfor complete homogeneous symmetric functions h k and Jacobi–Trudi formulas expressing Schurfunctions through h k .Although certain formulas for symplectic and orthogonal characters were also presented in [GP],they were not suitable for the purpose of the constructions of asymptotic representation theory (thenormalization in an analogue of (1.6) was different and, more importantly, the particular choiceof the geometric progression was k –dependent). The approach of [GP] also did not work for thesymmetric case of (1.4).In the present paper (as well as in the companion article [GS] where similar ideas are used forthe study of products of random matrices) we make the following observation. Through the label–variable duality and analytic continuation, the k = 1 versions of the expressions (1.4), (1.6) arelinked to the characters corresponding to hook signatures . The latter also have explicit generatingfunctions, though more complicated than the ones for h k , and therefore, we need to use double contour integrals. The next step is to reduce general k case to k = 1. Instead of Jacobi–Trudi, ourformulas are now related to Giambelli and Frobenius formulas, expressing characters with arbitrarysignatures as determinants of hooks. We refer to Sections 3.2, 3.4 for the details.1.5. Terminology and conventions.
For convenience of the reader, we collect here some termi-nology that is used throughout the paper.Unless otherwise stated, we assume q ∈ (0 ,
1) is a fixed parameter.We often use the q -Pochhammer symbols( a ; q ) n := n Y i =1 (1 − aq i − ) , n ≥
0; ( a ; q ) ∞ := ∞ Y i =1 (1 − aq i − ) . Moreover,( a , . . . , a m ; q ) n := ( a ; q ) n · · · ( a m ; q ) n ; ( a , . . . , a m ; q ) ∞ := ( a ; q ) ∞ · · · ( a m ; q ) ∞ . We also use the following terminology and conventions: • We denote N := { , , , . . . } and N := N ∪ { } = { , , , . . . } . • For N ∈ N , denote by GT N the set of signatures of length N , i.e., the set of N -tuples λ = ( λ , . . . , λ N ) ∈ Z N such that λ ≥ . . . ≥ λ N . • For N ∈ N , denote by GT + N the set of nonnegative signatures of length N , i.e., the set of N -tuples λ = ( λ , . . . , λ N ) ∈ GT N such that λ N ≥ • For λ ∈ GT N , denote | λ | := P Ni =1 λ i and n ( λ ) := λ + 2 λ + . . . + ( N − λ N . For λ ∈ GT N , µ ∈ GT N − , write λ ≻ µ (or µ ≺ λ ) if λ ≥ µ ≥ λ ≥ · · · ≥ µ N − ≥ λ N .Similarly, when λ, µ ∈ GT + N , write λ ≻ µ if λ ≥ µ ≥ λ ≥ µ ≥ · · · ≥ λ N ≥ µ N . • We use i as an index very often, thus we use the bold letter i := √− • The notation ( a n ), n ∈ N , indicates the string ( a, . . . , a ) ( n entries). This is an emptystring if n = 0. CESAR CUENCA AND VADIM GORIN
Acknowledgements.
We would like to thank G. Olshanski for encouraging us to studywhether the extension of [G] to orthogonal and symplectic groups is possible and for a numberof fruitful discussions. V.G. was partially supported by the NSF grant DMS-1664619, by the NECCorporation Fund for Research in Computers and Communications, and by the Sloan ResearchFellowship. The authors also thank the organizers of the Park City Mathematics Institute researchprogram on Random Matrix Theory, where part of this work was carried out.2.
Characters of U ( N ) , SO ( N ) and Sp ( N )2.1. Type A characters.
For any λ ∈ GT N , the Schur polynomial s λ ( x , . . . , x N ) is s λ ( x , . . . , x N ) := det ≤ i,j ≤ N h x λ j + N − ji i V ( x , . . . , x N ) , where V ( x , . . . , x N ) := Y ≤ i 0, and λ = (1 m , N − m ) = (1 , . . . , , , . . . , N ≥ m ≥ 0. In fact, s ( m, N − ) ( x , . . . , x N ) = h m ( x , . . . , x N ) is the m -th complete homogeneous symmetric polynomial, whereas s (1 m , N − m ) ( x , . . . , x N ) = e m ( x , . . . , x N ) is the m -th elementary symmetric polynomial.The reader can refer to [M, Ch. I] for a very thorough combinatorial study of Schur polynomials.Below we only list the properties that will be used in this article. Proposition 2.1 (branching rule; [M], Ch. I, 5.11) . For any N ∈ N , λ ∈ GT N +1 , s λ ( x , . . . , x N , u ) = X µ ∈ GT N µ ≺ λ s µ ( x , . . . , x N ) u | λ |−| µ | . Lemma 2.2 (label-variable duality) . For any λ, µ ∈ GT N , we have s λ ( q µ + N − , . . . , q µ N − +1 , q µ N ) s λ ( q N − , . . . , q, 1) = s µ ( q λ + N − , . . . , q λ N − +1 , q λ N ) s µ ( q N − , . . . , q, . Proof. Obvious from the definition of Schur polynomials. (cid:3) Proposition 2.3 ( q -geometric specialization; [M], Ch. I, Ex. 1) . For any λ ∈ GT N , s λ (1 , q, . . . , q N − ) = q n ( λ ) Y ≤ i For N ∈ N , let G ( N ) be one of the rank N classical compact Liegroups SO (2 N + 1) , Sp ( N ) , SO (2 N ) . In these cases, G ( N ) is of type B, C or D , respectively.We recall that the orthogonal group is realized as a matrix group by O ( N ′ ) := { A ∈ GL ( N ′ , R ) : AA t = A t A = I } , and the special orthogonal group SO ( N ′ ) is the subgroup of O ( N ′ ) consisting ofthose matrices with determinant 1. Similarly, the symplectic group is Sp (2 N ) := (cid:26) A ∈ GL (2 N, C ) : A t BA = B, B = (cid:20) I N I N (cid:21)(cid:27) and the compact symplectic group (which is of our interest) is Sp ( N ) := Sp (2 N ) ∩ U (2 N ) . The irreducible representations of G ( N ) are parametrized by GT + N if G ( N ) is of type B or C . We denote the character that corresponds to λ ∈ GT + N by so N +1 λ and sp Nλ . On the otherhand, if G ( N ) is of type D , then the irreducible representations of G ( N ) are parametrized by N -tuples of integers ( λ , . . . , λ N ) such that λ ≥ . . . ≥ λ N − ≥ | λ N | . For any λ ∈ GT + N , denote by so Nλ the reducible character of SO (2 N ) that is the sum of the twin characters corresponding to λ + := ( λ , . . . , λ N − , λ N ) and λ − := ( λ , . . . , λ N − , − λ N ).The characters so N +1 λ , sp Nλ and so Nλ are central functions of the groups SO (2 N + 1), Sp ( N )and SO (2 N ), respectively. Consequently, if M is a matrix in any of these groups, the value ofa corresponding character on M depends only on the eigenvalues of M . The eigenvalues of anymatrix M ∈ G ( N ) come in pairs { z i , z − i } , | z i | = 1 (any matrix in SO (2 N + 1) has an additionaleigenvalue 1). Therefore all the information about the characters so N +1 λ , sp Nλ , so Nλ , is encoded insome functions of N variables z , . . . , z N , which are written down below in (2.1).To uniformize notation, let T := { ζ ∈ C : | ζ | = 1 } , and for any λ ∈ GT + N , define the functions χ Bλ , χ Cλ , χ Dλ : T N → C by χ Gλ ( z , . . . , z N ) := so N +1 λ ( M ) , if G = B ; M ∈ SO (2 N + 1) has eigenvalues { z i , z − i } ni =1 ∪ { } ; sp Nλ ( M ) , if G = C ; M ∈ Sp ( N ) has eigenvalues { z i , z − i } ni =1 ; so Nλ ( M ) , if G = D ; M ∈ SO (2 N ) has eigenvalues { z i , z − i } ni =1 . They are symmetric with respect to permutation of their N variables and also with respect tothe involutions z i z − i , i.e., they are symmetric with respect to the natural action of the Weylgroup W N = ( Z / Z ) N ⋊ S N . Moreover they are Laurent polynomials in z , . . . , z N , i.e., they belongto the ring C [ z ± , . . . , z ± N ]. In fact, if we denote V s ( z , . . . , z N ) := Y ≤ i Thus we can (and we will) treat χ Gλ ( z , . . . , z N ) either as W N -symmetric Laurent polynomials oras functions on ( C ∗ ) N , via the formulas in (2.1). Proposition 2.5 (branching rule) . For any N ∈ N , λ ∈ GT + N +1 , and G ∈ { B, C, D } , we have χ Gλ ( z , . . . , z N , u ) = X µ ∈ GT + N χ Gλ/µ ( u ) χ Gµ ( z , . . . , z N ) , where χ Gλ/µ ( u ) := X ν ∈ GT + N +1 λ ≻ ν ≻ µ τ G ( u ; λ, ν, µ ) u | ν |−| λ |−| µ | , (2.2)and for λ ≻ ν ≻ µ : τ B ( u ; λ, ν, µ ) := ( , if ν ′ ≤ N, u − , otherwise; τ C ( u ; λ, ν, µ ) := 1; τ D ( u ; λ, ν, µ ) := , if 0 < ν N +1 < min { µ N , λ N +1 } , , if λ ′ = µ ′ = N, , otherwise . Proof. This proposition is a reformulation of [Pr, Prop. 10.2]. More specifically, the branching ofcharacters χ Cλ is given in part (a) of that proposition. This statement appeared first in [Zh]. Thebranching of characters χ Bλ is given in part (b). The branching of characters χ Dλ is given in part(c), which discusses the characters of certain (possibly reducible) tensor representations of SO (2 N ).An equivalent description of the branching for the characters χ Dλ appeared earlier in [KES]. (cid:3) Remark 2.6. The Laurent polynomial χ Gλ/µ ( u ) is symmetric with respect to the inversion u ↔ /u ,even though this is not evident from (2 . χ Gλ/µ ( u ) = X ν ∈ GT + N +1 λ ≻ ν ≻ µ τ G (1 /u ; λ, ν, µ ) u | λ | + | µ |− | ν | . Notation 2.7. For the propositions below, and for the remainder of the paper, we consider thereal parameter ǫ = ǫ ( G ) defined by ǫ := / , if G = B, , if G = C, , if G = D. Also, given nonnegative signatures λ, ν ∈ GT + N , we denote l Gi := λ i + N − i + ǫ, n Gi := ν i + N − i + ǫ, for all i = 1 , , . . . , N. These numbers are integers (when G = C, D ) or half-integers (when G = B ). We will simply write l , . . . , l N (resp. n , . . . , n N ) instead of l G , . . . l GN (resp. n G , . . . , n GN ), as the type G ∈ { B, C, D } willbe clear from the context. Lemma 2.8 (label-variable duality) . For G ∈ { B, C, D } , and any λ, ν ∈ GT + N , we have χ Gλ ( q n , q n , . . . , q n N ) χ Gλ ( q N − ǫ , q N − ǫ , . . . , q ǫ ) = χ Gν ( q l , q l , . . . , q l N ) χ Gν ( q N − ǫ , q N − ǫ , . . . , q ǫ ) . Proof. This is a consequence of Weyl’s formulas (2 . (cid:3) -DEFORMED CHARACTER THEORY · · · Proposition 2.9 ( q -geometric specialization) . For G ∈ { B, C, D } , and any λ ∈ GT + N , we have χ Bλ ( q , q , . . . , q N − ) = N Y i =1 q − l i − q l i q − ( N − i + ) − q ( N − i + ) V s ( q l N , q l N − , . . . , q l ) V s ( q , q , . . . , q N − ) ,χ Cλ ( q, q , . . . , q N ) = N Y i =1 q − l i − q l i q − ( N +1 − i ) − q N +1 − i V s ( q l N , q l N − , . . . , q l ) V s ( q, q , . . . , q N ) ,χ Dλ (1 , q, . . . , q N − ) = 2 · V s ( q l N , q l N − , . . . , q l ) V s (1 , q, . . . , q N − ) . Proof. The formulas for q -geometric specializations can be derived from Weyl’s denominator for-mulas; for instance, for types C and D, see [JN, (4.4), (4.5)]. One also can give direct proofs; letus do it for type B. From (2 . χ Bλ ( q , . . . , q N − ) equals Q Ni =1 ( q ( i − ) − q − ( i − ) ) − times V s ( q , q , . . . , q N − ) − times the determinantdet (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y − y − y − y − . . . y N − y − N y − y − y − y − . . . y N − y − N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .y N − − y − ( N − )1 y N − − y − ( N − )2 . . . y N − N − y − ( N − ) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where we denoted y i := q l i for all i . By elementary row operations, we see that the determinantabove equals det ≤ i,j ≤ N (cid:20) ( y j − y − j ) i − (cid:21) = Q Ni =1 ( y i − y − i ) × det ≤ i,j ≤ N h ( y j + y − j − i − i = Q Ni =1 ( y i − y − i ) × Q ≤ i For any integers N, a, b such that N ≥ b + 1 and a, b ≥ 0, we have χ B ( a +1 , b , N − b − ) = ( H Ba +1 − H Ba − ) E Bb − . . . + ( − b ( H Ba + b +1 − H Ba − b − ) E B ; χ C ( a +1 , b , N − b − ) = H Ca +1 E Cb − ( H Ca +2 + H Ca ) E Cb − + . . . + ( − b ( H Ca + b +1 + H Ca − b +1 ) E C ; χ D ( a +1 , b , N − b − ) = (cid:0) − { b = N − } (cid:1) × { ( H Da +1 − H Da − ) E Db − . . . + ( − b ( H Da + b +1 − H Da − b − ) E D } . Proof. This is a well known statement, see for example [ESK, (3.14), (3.15)]. The idea for theproof is simple: for hook Young diagrams λ = ( a + 1 , b ), expand the following Jacobi-Trudi •••• - π ln qπ ln q - π qπ q ℜ z ℑ z Figure 1. Contours for Lemma 3 . . u -contour is red (dashedlines) and the v -contour is blue (thick line).formulas (which can be found, for instance, also in [JN, Sec. 4] or [SV]) for symplectic/orthogonalpolynomials along the first row: χ Cλ = det ≤ i,j ≤ b +1 (cid:2) H Cλ i + j − i + (1 − δ j, ) H Cλ i − i − j +2 (cid:3) ; (2.3) χ Gλ = det ≤ i,j ≤ b +1 (cid:2) H Gλ i + j − i − H Gλ i − i − j (cid:3) , for G = B, D ; (2.4)and then identify the b × b minors with the Laurent polynomials E Gm by another application of theJacobi-Trudi formulas. (cid:3) Asymptotics of q -deformed characters In this section, we study the asymptotic behavior of characters as N → ∞ and all but finitelymany variables are specialized in a q -geometric series.3.1. Type A characters: statements of results.Definition 3.1. We denote by X the set of all doubly infinite, nondecreasing integer sequences,i.e., X := { t = ( . . . , t − , t − , t , t , t , . . . ) ∈ Z ∞ | · · · ≤ t − ≤ t ≤ t ≤ · · · } . Let b = ( b (1) , b (2) , . . . ) ∈ N ∞ be any sequence of nonnegative integers such that lim N →∞ b ( N ) =lim N →∞ ( N − b ( N )) = + ∞ . A sequence of signatures { λ ( N ) ∈ GT N } N ≥ b - stabilizes to t ∈ X iflim N →∞ λ ( N ) b ( N )+1 − i = t i , for all i ∈ Z . For t ∈ X , letΦ t ( x ; q ) := x (ln q ) ( q ; q ) ∞ ( qx, q/x ; q ) ∞ × Z π i ln q − π i ln q dv π i Z L du π i x u − q v − u ∞ Y i =1 − q t i + i − − v − q t i + i − − u ∞ Y j =1 − q j + v − t − j − q j + u − t − j − Z π i q − π i q x v ln q dv π i , (3.1)where the u -contour L consists of the directed lines (+ ∞ − π i q , −∞ − π i q ) and ( −∞ + π i q , + ∞ + π i q ), see Figure 1. The expressions x u and x v stand for e u ln x and e v ln x , where ln x is taken overthe branch of the logarithm with the cut along the negative real axis. Lemma 3.2. For any t ∈ X , the function Φ t ( x ; q ) defined by (3 . 1) is analytic on the domain C \ (( −∞ , ∪ { q n : n ∈ Z } ). Moreover, it admits an analytic continuation to C ∗ . -DEFORMED CHARACTER THEORY · · · We denote the analytic continuation of Φ t ( x ; q ) to C ∗ also by Φ t ( x ; q ). For m ∈ Z , let A m : X → X be the map A m t = t ′ := ( · · · ≤ t ′− ≤ t ′ ≤ t ′ ≤ . . . ), t ′ n := t n +1 − m . Define also the multivariatefunction Φ t ( x , . . . , x k ; q ) := q k ( k − k − V ( x k , . . . , x ) det ≤ i,j ≤ k Φ A j t ( x i q − j ; q ) Y ≤ s ≤ ks = j ( x i q − s − . (3.2)By Lemma 3 . 2, the functions Φ A j t ( x ; q ) are analytic on C ∗ , therefore Φ t ( x , . . . , x k ; q ) defines ananalytic function on ( C ∗ ) k . In fact, the determinant in (3 . 2) has zeroes at the diagonals x i = x j ,which cancel the poles coming from the Vandermonde determinant V ( x k , . . . , x ). Theorem 3.3. Let k ∈ N , and let b = ( b (1) , b (2) , . . . ) ∈ N ∞ be such that lim N →∞ b ( N ) =lim N →∞ ( N − b ( N )) = + ∞ . Also let { λ ( N ) ∈ GT N } N ≥ be a sequence of signatures that b -stabilizes to some t ∈ X . Thenlim N →∞ s λ ( N ) (1 , q, . . . , q b ( N ) − , q b ( N ) x , . . . , q b ( N ) x k , q b ( N )+ k , . . . , q N − ) s λ ( N ) (1 , q, . . . , q N − ) = Φ t ( x , . . . , x k ; q ) (3.3)holds uniformly over ( x , . . . , x k ) in compact subsets of ( C ∗ ) k . Conversely, if { λ ( N ) ∈ GT N } N ≥ is a sequence of signatures such that the limit in the left hand side of (3 . 3) exists and is uniformon compact subsets of ( C ∗ ) k and any k ∈ N , then there exists t ∈ X such that { λ ( N ) ∈ GT N } N ≥ b -stabilizes to t , and (3 . 3) holds.3.2. Type A characters: proofs of results. We need the following formulas for q -specializationsof Schur polynomials. Theorem 3.4. Let λ ∈ GT N , b ∈ { , , . . . , N − } , and a ∈ C ; then s λ (1 , q, . . . , q b − , q a , q b +1 , . . . , q N − ) s λ (1 , . . . , q N − ) = N − b − Y i =1 − q i q a − b − q i b Y i =1 − q i − q a − b + i × I { q λi + N − i } dz π i I {∞} dw π i · z a w − b − z − w · N Y i =1 w − q λ i + N − i z − q λ i + N − i . (3.4)In (3 . z -contour encloses all poles q λ i + N − i , i = 1 , . . . , N , but not the origin, while the w -contour encloses the origin and the z -contour. Remark 3.5. There are some apparent singularities in the right side of (3 . a ,coming from factors in the denominators. However, the left side is a Laurent polynomial in q a andthus an entire function of a . Therefore the double integral vanishes at the apparent singularities. Remark 3.6. For the special case b = N − 1, a similar formula was proved in [GP]; see also [C]. Proof. The theorem was stated in [GS, (3.10)] and is similar to Theorem 3.14. We will go into thedetails of the proof of Theorem 3.14, so we only present a sketch of proof here.Since both sides of (3.4) are rational functions on q a (for the right hand side, do a residueexpansion), it suffices to prove this identity for positive integers a ≥ N . But then we can useLemma 2.2 to the left hand side of (3.4) and find that it is proportional (up to a factor notdepending on λ ) to s ( a +1 − N, N − b − , b ) ( q λ + N − , . . . , q λ N ). Apply Proposition 2.4 to express this as N − b − X i =1 h a +1 − N + i ( { q λ i + N − i } ) e N − b − − i ( { q λ i + N − i } ) . (3.5) Then use the following well–known generating functions for the symmetric polynomials h n and e m (see [M, Ch. I.2, (2.2) & (2.5)]): ∞ X n =0 h n ( x , . . . , x N ) z n = N Y j =1 (1 − zx j ) − ; N X n =0 e n ( x , . . . , x N ) w n = N Y j =1 (1 + wx j ) . Equipped with these generating functions, write each term of (3.5) as a product of two contourintegrals, whose contours are small counterclockwise oriented circles. After swapping the sum andthe integration signs, we find that (3.5) equals Z { } dz π i Z { } dw π i N Y j =1 wq λ j + N − j − zq λ j + N − j N − b − X i =1 z − ( a +2 − N + i ) w − ( N − b − i ) . After some simplifications, one then arrives at the double contour integral in the right hand side of(3.4). To obtain the precise factor before the integral, one also needs Proposition 2.3. (cid:3) Theorem 3.7. Let b ∈ N , k, N ∈ N , be such that b + k ≤ N , and also let λ ∈ GT N . Then s λ (1 , q, . . . , q b − , q b x , . . . , q b x k , q b + k , . . . , q N − ) s λ (1 , q, . . . , q N − ) = q k ( k − k − V ( x k , . . . , x ) × det ≤ i,j ≤ k s λ (1 , q, . . . , q b + j − , q b x i , q b + j , . . . , q N − ) s λ (1 , q, . . . , q N − ) Y ≤ s ≤ ks = j ( x i q − s − . (3.6) Proof. This theorem is similar to [GS, Prop. 3.1] and also to Theorem 3.18. We go into details forthe proof of Theorem 3.18, but here we only give a sketch.Since both sides are rational functions on x , . . . , x k , it suffices to prove the identity for x = q N + a − b − , . . . , x k = q N + a k − b − k , for any integers a ≥ a ≥ · · · ≥ a k > k . But then we can applyLemma 2.2 to the left hand side of (3.6), and conclude that it is proportional (up to a factor indepen-dent of λ ) to s ( a ,...,a k ,k N − b − k , b ) ( q λ + N − , . . . , q λ N − +1 , q λ N ). Since the partition ( a , . . . , a k , k N − b − k )has Frobenius coordinates ( a − , . . . , a k − k | N − b − , . . . , N − b − k ), we can use Giambelliformula (see [M, Ch. I.3, Example 9]) to express s ( a ,...,a k ,k N − b − k , b ) ( q λ + N − , . . . , q λ N − +1 , q λ N ) asthe determinant det ≤ i,j ≤ k h s ( a i − i +1 , N − b − j , b + j − ) ( q λ + N − , . . . , q λ N − +1 , q λ N ) i . (3.7)Multiply and divide the ( i, j )–entry of by s ( a i − i +1 , N − b − j , b + j − ) (1 , q, . . . , q N − ), so that we canapply Lemma 2.2 again; then the determinant (3.7) becomesdet ≤ i,j ≤ k (cid:20) s λ (1 , q, . . . , q b + j − , q a i + N − i , q b + j , . . . , q N − ) s λ (1 , q, . . . , q N − ) s ( a i − i +1 , N − b − j , b + j − ) (1 , q, . . . , q N − ) (cid:21) . Note that the ( i, j )–entry here contains the factor s λ (1 , q, . . . , q b + j − , q a i + N − i , q b + j , . . . , q N − ), whichis precisely the Schur polynomial in the ( i, j )–entry of the right hand side of (3.6) after specializing x i = q N + a i − b − i . Finally, after some simplifications and using Proposition 2.3, we arrive at thedesired (3.6). (cid:3) The following equivalent version of Theorem 3 . . Lemma 3.8. Let b, N ∈ N be such that b ∈ { , , . . . , N − } . Let x ∈ C \ (( −∞ , ∪ { q n : n ∈ Z } )be such that q N − b < | x | < q − ( b +1) . Also let λ ∈ GT N and denote t Ni := λ b +1 − i , for all b + 1 − N ≤ i ≤ b. -DEFORMED CHARACTER THEORY · · · Then s λ (1 , q, . . . , q b − , q b x, q b +1 , . . . , q N − ) s λ (1 , . . . , q N − ) = x (ln q ) ( q ; q ) b ( q ; q ) N − b − ( qx ; q ) b ( q/x ; q ) N − b − × Z π i ln q − π i ln q dv π i Z L du π i x u − q v − u b Y i =1 − q t Ni + i − − v − q t Ni + i − − u N − b Y j =1 − q − t N − j + j + v − q − t N − j + j + u − Z π i q − π i q x v ln q dv π i . (3.8)The expressions x u and x v stand for e u ln x and e v ln x , where ln x is taken over the branch of thelogarithm with the cut along the negative real axis. The u -contour L consists of the directed lines(+ ∞ − π i q , −∞ − π i q ) and ( −∞ + π i q , + ∞ + π i q ), see Figure 1. Remark 3.9. The condition q N − b < | x | < q − ( b +1) is in place to assure that the double integralin (3 . 8) converges. In fact, as ℜ u → ∞ , u ∈ L , the integrand is of order | xq b +1 | ℜ u , whereas if ℜ u → −∞ , u ∈ L , then the integrand is of order | x − q N − b | −ℜ u . Proof of Lemma . . We massage the formula from Theorem 3 . 4. Let a = b + c , and make thechange of variables z q N − b z , w q N − b w . We obtain s λ (1 , q, . . . , q b − , q b + c , q b +1 , . . . , q N − ) s λ (1 , . . . , q N − ) = q c N − b − Y i =1 − q i − q i q − c b Y i =1 − q i − q i q c × I { q λi + b − i } dz π i I | w | = q R dw π i · z b + c w − b − z − w · N Y i =1 w − q λ i + b − i z − q λ i + b − i , (3.9)where R is chosen to be any negative real number with very large absolute value. Next we makethe change of variables z = q u and w = q v , or equivalently, u = ln z/ ln q and v = ln w/ ln q , wherethe principal branch of the logarithm is used. Then (3 . 9) becomes s λ (1 , q, . . . , q b − , q b + c , q b +1 , . . . , q N − ) s λ (1 , . . . , q N − ) = q c N − b − Y i =1 − q i − q i q − c b Y i =1 − q i − q i q c × (ln q ) I { λ i + b − i } du π i Z R + i π/ ln qR − i π/ ln q dv π i · q ub − vb q uc − q v − u · N Y i =1 q v − q λ i + b − i q u − q λ i + b − i . (3.10)In the integral formula (3 . u -contour is closed, counter-clockwise oriented, encloses all points λ i + b − i , i = 1 , . . . , N , and is within the strip { z ∈ C : |ℑ z | < π i / ln q } , whereas the v -contour isthe line [ R − π i / ln q, R + π i / ln q ] going downwards, and is to the left of the u -contour.Let x ∈ C \ ( { } ∪ { q n : n ∈ Z } ) and set c := ln x/ ln q , so that q c = x . Exchange the order ofintegration for the double integral in (3 . q -Pochhammer symbols. Then (3 . 10) is rewritten as s λ (1 , q, . . . , q b − , q b x, q b +1 , . . . , q N − ) s λ (1 , . . . , q N − ) = x (ln q ) ( q ; q ) b ( q ; q ) N − b − ( qx ; q ) b ( q/x ; q ) N − b − × Z R + i π/ ln qR − i π/ ln q dv π i I { λ i + b − i } du π i · x u − q v − u · b Y i =1 − q t Ni + i − − v − q t Ni + i − − u N − b Y j =1 − q − t N − j + j + v − q − t N − j + j + u . (3.11)Next we modify the contours in (3 . u , has polesat u = v + 2 kπ i / ln q , k ∈ Z , and at the points of the set P N := { t Ni + i − i = b + 1 − N, . . . , , , . . . , b } = { λ i + b − i : i = 1 , . . . , N } , and their shifts by some integral multiple of 2 π i / ln q . However, the only poles enclosed by the u -contour are those in P N . Thus we can deform the u -contour to be the counter-clockwise oriented rectangle with vertices at the points A ± π i q and B ± π i q , for any real numbers A, B such that R < A < λ N + b − N < λ + b − < B . Call such contour L ( A, B ).Let L be the infinite contour which is described in the statement of the lemma. We want toreplace L ( A, B ) by L in the integral formula (3 . | u | → ∞ along L . In fact, if ℜ u is verylarge, then Q N − bj =1 (1 − q − t N − j + j + u ) ≈ x u (1 − q v − u ) − Q bi =1 (1 − q t Ni + i − − u ) − decaysexponentially fast as ℜ u → ∞ along L , because | xq b +1 | < 1. One can obtain the same conclusionif ℜ u → −∞ along L , by using instead | x − q N − b | < L ( A, B ) by L , we pick up residues at the singularities u = v ∈ [ R − i π q , R + i π q ].The residue is simply x v / ln q because all factors in the products of (3 . 11) equal 1 when u = v . Weconclude s λ (1 , q, . . . , q b − , q b x, q b +1 , . . . , q N − ) s λ (1 , . . . , q N − ) = x (ln q ) ( q ; q ) b ( q ; q ) N − b − ( qx ; q ) b ( q/x ; q ) N − b − × Z R + π i ln q R − π i ln q dv π i Z L du π i x u − q v − u b Y i =1 − q t Ni + i − − v − q t Ni + i − − u N − b Y j =1 − q − t N − j + j + v − q − t N − j + j + u − Z R + π i q R − π i q x v ln q dv π i . (3.12)Finally we need to show that (3 . 12) does not depend on R , and thus we can set R = 0, concludingthe proof of the lemma. For that we subtract the second line of (3 . 12) with R = R from the sameexpression with R = R and show that the result is zero; without loss of generality, assume R < R .If we do this for the second term in the second line of (3 . − Z R + π i q R − π i q + Z R + π i q R − π i q ! x v ln q dv π i = Z R + π i q R + π i q − Z R − π i q R − π i q ! x v ln q dv π i . (3.13)For the first term in the second line of (3 . Z R + π i ln q R − π i ln q dv π i Z L du π i − Z R + π i ln q R − π i ln q dv π i Z L du π i ! F ( u, v )= Z L du π i Z R − π i ln q R − π i ln q dv π i − Z R + π i ln q R + π i ln q dv π i ! F ( u, v ) + Z L Res v = u F ( u, v ) du π i , (3.14)where F ( u, v ) := x u − q v − u b Y i =1 − q t Ni + i − − v − q t Ni + i − − u N − b Y j =1 − q − t N − j + j + v − q − t N − j + j + u is a function of u, v . Observe that F ( u, v ) depends on v via the variable q v . If ℑ v = ± π i / ln q , wehave q v = − q ℜ v , thus the integral of F ( u, v ) with respect to v along the contour [ R − π i ln q , R − π i ln q ]is equal to its integral along the contour [ R + π i ln q , R + π i ln q ]; consequently the first term in thesecond line of (3 . 14) vanishes. On the other hand, it is clear that Res v = u F ( u, v ) = − x u / ln q if R < ℜ u < R and Res v = u F ( u, v ) = 0 otherwise. Therefore the second line of (3 . 14) equals Z u ∈L∩{ R < ℜ z 15) cancels exactly (3 . . 12) does not depend on R . (cid:3) We prove Theorem 3 . . -DEFORMED CHARACTER THEORY · · · Proofs of Theorem . and Lemma . . We proceed in several steps. Step 1. We prove the first part of Lemma 3 . 2, which says that the formula (3 . 1) defines an analyticfunction on C \ (( −∞ , ∪ { q n : n ∈ Z } ). The first line of (3 . 1) is clearly analytic on C \ { q n : n ∈ Z } .As for the second line of (3 . x on C \ ( −∞ , Q ∞ i =1 (1 − q t i + i − − u ) − Q ∞ j =1 (1 − q j + u − t − j ) − ensure that the integrand is exponentially small, as |ℜ u | → ∞ , u ∈ L , and in particular theintegral converges uniformly for x in compact subsets of C \ ( −∞ , t ( x ; q )follows. Step 2. The general case k > . k = 1 case and the multivariateformula in Theorem 3 . 7. We are left to prove the second part of Lemma 3 . t ( x ; q )), Theorem 3 . k = 1, and the last statement of Theorem 3 . . 3) for k = 1, uniformly for x in compact subsets of C \ ( −∞ , t ( x ; q ) admits an analytic continuation to C ∗ and that the limit (3 . 3) for k = 1continues to hold uniformly for x in compact subsets of C ∗ . Finally, step 6 shows the conversestatement of Theorem 3 . Step 3. We begin with formula (3 . 8) from Lemma 3 . 8, which is valid for x in the domain C \ (( −∞ , ∪ { q n : n ∈ Z } ) and N large enough so that | xq b ( N )+1 | , | x − q N − b ( N ) | < 1. We showthat the prefactor of (3 . 8) converges to the prefactor of (3 . . 8) converges pointwise to the integrand (of the first integral) of (3 . N →∞ x (ln q ) ( q ; q ) b ( N ) ( q ; q ) N − b ( N ) − ( qx ; q ) b ( N ) ( q/x ; q ) N − b ( N ) − = x (ln q ) ( q ; q ) ∞ ( qx, q/x ; q ) ∞ , whereas the statement about the integrands is equivalent to the pointwise limitlim N →∞ b ( N ) Y i =1 − q t Ni + i − − v − q t Ni + i − − u N − b ( N ) Y j =1 − q − t N − j + j + v − q − t N − j + j + u = ∞ Y i =1 − q t i + i − − v − q t i + i − − u ∞ Y j =1 − q − t − j + j + v − q − t − j + j + u . (3.16)Both are obvious: the first comes from the assumption lim N →∞ b ( N ) = lim N →∞ ( N − b ( N )) = + ∞ ,and the second from the limits lim N →∞ t Ni = lim N →∞ λ ( N ) b ( N )+1 − i = t i . Step 4. From the claim about the integrands in the previous step, we want to conclude thatlim N →∞ Z π i ln q − π i ln q dv π i Z L du π i x u (1 − q v − u ) b ( N ) Y i =1 − q t Ni + i − − v − q t Ni + i − − u N − b ( N ) Y j =1 − q − t N − j + j + v − q − t N − j + j + u = Z π i ln q − π i ln q dv π i Z L du π i x u (1 − q v − u ) ∞ Y i =1 − q t i + i − − v − q t i + i − − u ∞ Y j =1 − q − t − j + j + v − q − t − j + j + u , (3.17)uniformly for x on compact subsets of C \ ( −∞ , − q t Ni + i − − v and1 − q t Nj + j + v has a modulus that can be upper bounded by the constant Q ∞ i =1 (1+ q t + i − )(1+ q − t + i ),at least when N is large enough so that t N = t and t N = t . Because u ∈ L , we have | x u | = | x | ℜ u e ± π arg( x ) / q ≤ | x | ℜ u e − π / (4 ln q ) .Because q v − u is purely imaginary for u ∈ L and v ∈ [ − π i / ln q, π i / ln q ], then | / (1 − q v − u ) | ≤ q u is purely imaginary for u ∈ L , the products of factors 1 / (1 − q − t N − j + j + u ) andthe factors 1 / (1 − q − t N − j + j + u ) have moduli that are bounded above by 1. Moreover, if u ∈ L and ℜ u → + ∞ , then the product of factors 1 / (1 − q t Ni + i − − u ) decays exponentially and overcomes the factor | x | ℜ u of the previous paragraph, whereas if u ∈ L and ℜ u → −∞ , then the product of factors1 / (1 − q t N − j + j + u ) is the one that decays exponentially and overcomes the factor | x | ℜ u .All the bounds above show that the integrand in the left side of (3 . 17) converge uniformly (on u and x ) to the integrand in the right side of (3 . . 17) follows, and the limit (3 . . k = 1 and for x belonging to compact subsets of C \ (( −∞ , ∪ { q n : n ∈ Z } ). Step 5. We show that Φ t ( x ; q ) admits an analytic continuation to C ∗ and that (3 . 3) continuesto hold uniformly on compact subsets of C ∗ . It will be convenient to use the notationΦ λ ( x ; q, b, N ) := s λ (1 , q, . . . , q b − , q b x, q b +1 , . . . , q N − ) s λ (1 , q, . . . , q N − ) . Let R > R / ∈ { q n : n ∈ Z } . For any x ∈ C ∗ with R − ≤ | x | ≤ R , we have | x | n < R n + R − n for any n ∈ Z . By the triangle inequality and thepositivity of branching coefficients of the Schur polynomials, we deduce (cid:12)(cid:12) Φ λ ( N ) ( x ; q, b ( N ) , N ) (cid:12)(cid:12) ≤ Φ λ ( N ) ( | x | ; q, b ( N ) , N ) < Φ λ ( N ) ( R ; q, b ( N ) , N ) + Φ λ ( N ) ( R − ; q, b ( N ) , N ) . From our choice of R and steps 1–2, the limits lim N →∞ Φ λ ( N ) ( R ± ; q, b ( N ) , N ) exist and thus, both { Φ λ ( N ) ( R ; q, b ( N ) , N ) } N ≥ , { Φ λ ( N ) ( R − ; q, b ( N ) , N ) } N ≥ , are bounded sequences. Then the sequence of functions { Φ λ ( N ) ( z ; q, b ( N ) , N ) } N ≥ is uniformlybounded on { z ∈ C : R − < | z | < R } . Montel’s theorem implies that any subsequence of { Φ λ ( N ) ( x ; q, b ( N ) , N ) } N ≥ has a subsequential limit, and the convergence is uniform on compactsubsets of { z ∈ C : R − < | z | < R } . Since R > { Φ λ ( N ) ( x ; q, b ( N ) , N ) } N ≥ has subsequential limits, uniformly on compact subsets of C ∗ ; say oneof the limiting functions is the analytic function e Φ( x ; q ) on C ∗ . But we already proved (3 . 1) in arestricted domain so that e Φ( x ; q ) = Φ( x ; q ), for x ∈ C \ (( −∞ , ∪ { q n : n ∈ Z } ) and such analyticcontinuation e Φ( x ; q ) is unique. Step 6. Assume that { λ ( N ) ∈ GT N } is a sequence of signatures such that the limit in the lefthand side of (3 . 3) exists for all k ∈ N . The prelimit expression is a symmetric Laurent polynomial in x , . . . , x k , therefore it can be written as a linear combination of Schur polynomials s µ ( x , . . . , x k ), µ ∈ GT k . Write the expansion as s λ ( N ) (1 , q, . . . , q b ( N ) − , q b ( N ) x , . . . , q b ( N ) x k , q b ( N )+ k , . . . , q N − ) s λ ( N ) (1 , q, . . . , q N − , q N − )= X µ ∈ GT k Λ Nk ( λ ( N ) , µ ) s µ ( x , . . . , x k ) s µ (1 , q, . . . , q k − ) . (3.18)Theorem 2 . Nk ( λ ( N ) , µ ) of the expansion are nonnegative. Moreover,by plugging x = 1 , x = q, . . . , x k = q k − into (3 . P µ ∈ GT k Λ Nk ( λ ( N ) , µ ) = 1, i.e.,Λ Nk ( λ ( N ) , · ) is a probability measure on GT k . The proof of Proposition 5 . . 3) on compact subsets of ( C ∗ ) k ) impliesthe weak convergence of the sequence Λ Nk ( λ ( N ) , · ), as N goes to infinity, to a probability measure M k on GT k . The proof of Theorem 5 . N →∞ λ ( N ) b ( N )+ m (3.19)exist for all m ∈ Z . If we denote by t − m the limit (3 . · · · ≤ t − ≤ t ≤ t ≤ · · · . Inother words, t := ( . . . , t − , t , t , . . . ) belongs to X and { λ ( N ) ∈ GT N } N ≥ b -stabilizes to t . Wecan now apply the first part of the proof (Steps 1–5) to show that the limit (3 . 3) holds uniformlyon compact subsets of ( C ∗ ) k and we are done. (cid:3) -DEFORMED CHARACTER THEORY · · · Type B-C-D characters: statements of results.Definition 3.10. Let Y be the set of nondecreasing, nonnegative integer sequences, i.e., Y := { y = ( y , y , y , . . . ) ∈ N ∞ | ≤ y ≤ y ≤ · · · } . We say that a sequence { λ ( N ) ∈ GT + N } N ≥ stabilizes to y ∈ Y iflim N →∞ λ ( N ) N +1 − i = y i , for all i ≥ . To state our results, we need to define some functions parametrized by elements y ∈ Y , by atype G ∈ { B, C, D } , and by a nonnegative integer m ∈ N :Φ y ,Bm ( x ; q ) := (ln q ) · q ( m + ) − q − ( m + ) x − x − ( q m +1 , q − m ; q ) m ( q, q m +2 ; q ) ∞ ( q x, q /x ; q ) m ( q m + x, q m + /x ; q ) ∞ × (Z π i ln q − π i ln q dv π i Z L du π i x u ( q u − ( m +1) v − q − u +( m +1) v )( q v − q − v )( q v − u − q u − v )( q u − q − u ) ∞ Y i =1 (1 − q y i − v )(1 − q y i + v )(1 − q y i − u )(1 − q y i + u ) − q Z π i q − π i q x v (cid:16) q ( m + ) v − q − ( m + ) v (cid:17) dv π i ) ; (3.20)Φ y ,Cm ( x ; q ) := (ln q ) · q m +1 − q − ( m +1) x − x − ( q m +2 , q − m ; q ) m ( q, q m +3 ; q ) ∞ ( qx, q/x ; q ) m ( q m +2 x, q m +2 /x ; q ) ∞ × (Z π i ln q − π i ln q dv π i Z L du π i x u q ( m +1) v − q − ( m +1) v q u − v − ∞ Y i =1 (1 − q y i − v )(1 − q y i + v )(1 − q y i − u )(1 − q y i + u ) − q Z π i q − π i q x v (cid:16) q ( m +1) v − q − ( m +1) v (cid:17) dv π i ) ; (3.21)Φ y ,Dm ( x ; q ) := (cid:0) − { m =0 } (cid:1) (ln q ) q m , q − m ; q ) m ( q, q m +1 ; q ) ∞ ( x, /x ; q ) m ( q m +1 x, q m +1 /x ; q ) ∞ × (Z π i ln q − π i ln q dv π i Z L du π i x u q u − ( m + ) v + q − u +( m + ) v q v − u − q u − v ∞ Y i =1 (1 − q y i − v )(1 − q y i + v )(1 − q y i − u )(1 − q y i + u )+ 1ln q Z π i q − π i q x v (cid:0) q mv + q − mv (cid:1) dv π i ) . (3.22)In the expressions above, x u , x v and x (for G = B ) stand for e u ln x , e v ln x and e ln x/ , where ln x isdefined over the branch of the logarithm with the cut along the negative real axis. The u -contour L consists of the directed lines (+ ∞ − π i q , −∞ − π i q ) and ( −∞ + π i q , + ∞ + π i q ), see Figure1. Lemma 3.11. The functions Φ y ,Bm ( x ; q ), m ∈ N , defined by (3 . C \ (( −∞ , ∪{ } ∪ { q n + : n ∈ Z } ). The functions Φ y ,Gm ( x ; q ), m ∈ N , G ∈ { C, D } , defined by (3 . . C \ (( −∞ , ∪ { q n : n ∈ Z } ).Moreover, all of these functions admit analytic continuations to C ∗ , and satisfy Φ y ,Gm ( x ; q ) =Φ y ,Gm (1 /x ; q ). Theorem 3.12. Let { λ ( N ) ∈ GT + N } N ≥ be a sequence that stabilizes to some y ∈ Y , and let m ∈ N . Then lim N →∞ χ Gλ ( N ) ( q ǫ , . . . , q m − ǫ , x, q m +1+ ǫ , . . . , q N − ǫ ) χ Gλ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) = Φ y ,Gm ( x ; q ) (3.23)holds uniformly on compact subsets of C ∗ .For any k ∈ N and G ∈ { B, C, D } , define the constant c Gk ( q ) := 1 Q ki =1 ( q i − ǫ , q − i , q, q k − i +2 ǫ +1 ; q ) i − , and the functionsΦ y ,G ( x , . . . , x k ; q ) := c Gk ( q ) · V s ( q ǫ , q ǫ , . . . , q k − ǫ ) V s ( x , x , . . . , x k ) × det ≤ i,j ≤ k h Φ y ,Gj ( x i ; q ) · ( q ǫ x i , q ǫ /x i ; q ) j − ( q j + ǫ x i , q j + ǫ /x i ; q ) k − j i . (3.24)The determinant in (3 . 24) vanishes when x i = x j or x i = 1 /x j , for some i = j . Then the poles of V s ( x , . . . , x k ) are cancelled out so that the functions Φ y ,G ( x , . . . , x k ; q ) are analytic on ( C ∗ ) k . Theorem 3.13. Let k ∈ N , G ∈ { B, C, D } , and { λ ( N ) ∈ GT + N } N ≥ be a sequence of nonnegativesignatures that stabilizes to y ∈ Y . Thenlim N →∞ χ Gλ ( N ) ( x , . . . , x k , q k + ǫ , . . . , q N − ǫ ) χ Gλ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) = Φ y ,G ( x , . . . , x k ; q ) (3.25)holds uniformly on compact subsets of ( C ∗ ) k . Conversely, if { λ ( N ) ∈ GT + N } N ≥ is a sequence ofnonnegative signatures such that the limit in the left hand side of (3 . 25) exists and is uniform oncompact subsets of ( C ∗ ) k , for any k ∈ N , then there exists y ∈ Y such that { λ ( N ) ∈ GT N } N ≥ stabilizes to y , and the limit relation (3 . 25) holds.3.4. Integral representations and multivariate formulas for B-C-D type characters.Theorem 3.14. Let 0 ≤ m ≤ N − λ ∈ GT + N and a ∈ C ; then χ Bλ ( q , . . . , q m − , q a , q m + , . . . , q N − ) χ Bλ ( q , q , . . . , q N − ) = q ( m + ) − q − ( m + ) q a − q − a × ( q m +1 , q − m ; q ) m ( q, q m +2 ; q ) N − m − ( q + a , q − a ; q ) m ( q m + + a , q m + − a ; q ) N − m − × I dz π i I dw π i z a + N − ( z − w m +2 ) w N + m +2 ( w − z ) ( w − z − N Y i =1 ( w − q λ i + N + − i )( w − q − ( λ i + N + − i ) )( z − q λ i + N + − i )( z − q − ( λ i + N + − i ) ) ; (3.26) χ Cλ ( q, . . . , q m , q a , q m +2 , . . . , q N ) χ Cλ ( q, q , . . . , q N ) = ( q m +1 − q − ( m +1) )( q m +2 , q − m ; q ) m ( q, q m +3 ; q ) N − m − ( q a − q − a )( q a , q − a ; q ) m ( q m +2+ a , q m +2 − a ; q ) N − m − × I dz π i I dw π i z a + N − (1 − w m +2 ) w N + m +1 ( w − z ) N Y i =1 ( w − q λ i + N +1 − i )( w − q − ( λ i + N +1 − i ) )( z − q λ i + N +1 − i )( z − q − ( λ i + N +1 − i ) ) ; (3.27) -DEFORMED CHARACTER THEORY · · · χ Dλ (1 , . . . , q m − , q a , q m +1 , . . . , q N − ) χ Dλ (1 , q, . . . , q N − ) = (cid:0) − { m =0 } (cid:1) ( q m , q − m ; q ) m ( q, q m +1 ; q ) N − m − q a , q − a ; q ) m ( q m +1+ a , q m +1 − a ; q ) N − m − × I dz π i I dw π i z a + N − ( z + w m +1 ) w N + m +1 ( w − z ) N Y i =1 ( w − q λ i + N − i )( w − q − ( λ i + N − i ) )( z − q λ i + N − i )( z − q − ( λ i + N − i ) ) . (3.28)In the right hand sides of (3 . . 27) and (3 . z -contour encloses the smallest closed interval I ⊆ R that contains all the singularities { q ± ( λ i + N − i + ǫ ) } ≤ i ≤ N (where ǫ = , , B, C, D ,respectively), but it does not enclose the origin, whereas the w -contour encloses the z -contour andthe origin. Remark 3.15. There are some apparent singularities in the right sides of (3 . . 27) and (3 . a , coming from Pochhammer symbols in the denominators. However, the left sidesof these identities are entire functions of a . Therefore the integrals in the identities vanish at theapparent singularities. Remark 3.16. If the limit q → 1, these integral representations recover [GP, Thm. 3.18], for type C , and the integral representations coming from [BG, Props. 7.3, 7.4] for types B , D . Proof. The proofs of the three integral representations are very similar. We give full details for thecase G = B , and leave the cases G = C, D to the reader. Step 1. From Lemma 2 . . 9, for any half-integer a ≥ N + ( a − ∈ Z ), and0 ≤ m ≤ N − 1, we obtain χ Bλ ( q , . . . , q m − , q m + , . . . , q N − , q a ) χ Bλ ( q N − , . . . , q ) = χ B ( a − N + , N − m − , m ) ( { q λ i + N − i + } ) χ B ( a − N + , N − m − , m ) ( q N − , . . . , q ) = ( − N − m +1 ×× ( q ( m + ) − q − ( m + ) )( q m +1 , q − m ; q ) m ( q, q m +2 ; q ) N − m − ( q a − q − a )( q + a , q − a ; q ) m ( q m + + a , q m + − a ; q ) N − m − χ B ( a − N + , N − m − , m ) ( { q λ i + N − i + } ) . (3.29)In the next two steps, we find a double integral representation for χ B ( b +1 , c , N − c − ) ( y , . . . , y N ), for b, c ∈ N with b is large enough, N ≥ c +1, and y , . . . , y N > 0. Then we specialize these parametersto obtain a double integral representation for (3 . a is very large. Step 2. Let us work temporarily with any b, c ∈ N such that b > c + 1, N ≥ c + 1, and anypositive real numbers y , . . . , y N > 0. Let us denote H n := h n (1 , y , y − , . . . , y N , y − N ) , n ≥ E n := e n (1 , y , y − , . . . , y N , y − N ) , N + 1 ≥ n ≥ . We shall use the following generating functions: ∞ X n =0 H n v n = 11 − v N Y i =1 − vy i )(1 − vy − i ) ; (3.30) N +1 X n =0 E n v n = (1 + v ) N Y i =1 (1 + vy i )(1 + vy − i ) . (3.31)From Proposition 2 . 10, and (3 . χ B ( b +1 , c , N − c − ) ( y , . . . , y N ) = ( H b +1 − H b − ) E c − . . . + ( − c ( H b + c +1 − H b − c − ) E = I { } dv π i (cid:8) ( v − b − − v − b ) E c − . . . + ( − c ( v − b − c − − v − b + c ) E (cid:9) (1 − v ) Q Ni =1 (1 − vy i )(1 − vy − i ) . (3.32) The sum in brackets in (3 . 32) can be written as two sums, with c + 1 terms each. The first of thosesums is v − b − E c − . . . + ( − c v − b − c − E = ( − c v − b − c − ( E − vE + . . . + ( − v ) c E c )= ( − c v − b − c − I { } du π i ( u − v ) N Y i =1 ( u − vy i )( u − vy − i ) · (cid:0) u − N − + u − N − + . . . + u − N − c (cid:1) = ( − c v − b − c − I { } du π i ( u − v ) N Y i =1 ( u − vy i )( u − vy − i ) u − N − · − u c +1 − u . To give an integral representation for the second sum, we use: E n = E N +1 − n , for all 0 ≤ n ≤ N + 1 . The second sum is − v − b E c − . . . + ( − c +1 v − b + c E = − v − b E N +1 − c − . . . + ( − c +1 v − b + c E N +1 = ( − c v − b + c − N − (cid:0) ( − v ) N +1 − c E N +1 − c − . . . + ( − v ) N +1 E N +1 (cid:1) = ( − c v − b + c − N − I { } du π i ( u − v ) N Y i =1 ( u − vy i )( u − vy − i ) · ( u − c − + u − c + . . . + u − )= ( − c v − b + c − N − I { } du π i ( u − v ) N Y i =1 ( u − vy i )( u − vy − i ) u − c − · − u c +1 − u . As a result, χ B ( b +1 , c ) ( y , . . . , y N ) is the sum of the following two double contour integrals:( − c I { } dv π i I { } du π i u − v − v N Y i =1 ( u − vy i )( u − vy − i )(1 − vy i )(1 − vy − i ) u − N − v − b − c − · − u c +1 − u ; (3.33)( − c I { } dv π i I { } du π i u − v − v N Y i =1 ( u − vy i )( u − vy − i )(1 − vy i )(1 − vy − i ) u − c − v − b + c − N − · − u c +1 − u . (3.34) Step 3. Now we rewrite the contour integral representations in (3 . 33) and (3 . . O ( | v | − ), as | v | → ∞ . Then we can deform the v -contour,pass it through ∞ , and have the new v -contour enclosing the singularities 1 , y , y − , . . . , y N , y − N ;a minus sign appears in making this deformation. Deform also the u -contour and make it verylarge (in particular, it encloses both 0 and 1). Next, break the integral (3 . 33) into two integralsby writing u − N − (1 − u c +1 ) / (1 − u ) as the difference u − N − / (1 − u ) − u − N + c − / (1 − u ). Thefirst integral, corresponding to u − N − / (1 − u ), is of order O ( | u | − ), when | u | → ∞ . Thus thereis no pole outside the contour, meaning that the first integral vanishes. From these considerations,(3 . 33) equals( − c I { ,y i ,y − i } dv π i I {∞} du π i u − v − v N Y i =1 ( u − vy i )( u − vy − i )(1 − vy i )(1 − vy − i ) u − N + c − v − b − c − − u , where the v -contour encloses 1 , y , y − , . . . , y N , y − N , but not the origin, while the u -contour enclosesboth 0 and 1. After making the change of variables z = 1 /v, w = u/v, (3.35) -DEFORMED CHARACTER THEORY · · · we can have the new z -contour enclosing the singularities 1 , y , . . . , y − N , but not around the origin,whereas the new w -contour encloses the z -contour and 0. Then the integral (3 . 33) equals( − c +1 I { ,y i ,y − i } dz π i I {∞} dw π i w − z − N Y i =1 ( w − y i )( w − y − i )( z − y i )( z − y − i ) z N + b +1 w − N + c − z − w . (3.36)For the integral (3 . v -contour though infinity so that itencloses the singularities 1 , y , . . . , y − N , but not the origin; a negative sign appears. Break theintegral into two by using u − c − (1 − u c +1 ) / (1 − u ) = u − c − / (1 − u ) − / (1 − u ), and get rid of thesecond integral because no singularity at 0 remains. Then do the change of variables (3 . 35) again.After these simplifications, (3 . 34) equals( − c I { ,y i ,y − i } dz π i I { } dw π i w − z − N Y i =1 ( w − y i )( w − y − i )( z − y i )( z − y − i ) · z N + b w − c − z − w . (3.37)The integrand, as a function of w , has only w = z as a singularity. Thus by expanding the w -contour,so that it swallows the z -contour, we pick up the residue w = z . In enlarging the w -contour, thetotal residue that we pick up is (when w = z , the residue of the integrand in (3 . 37) is − z N + b − c − ):( − c +1 I { ,y i ,y − i } z N + b − c − dz π i = 0 . Thus (3 . 37) does not change if the w -contour in that formula is replaced by one that encloses boththe origin and the z -contour. We conclude that χ B ( b +1 , c , N − c − ) ( y , . . . , y N ) equals the sum of (3 . . w -contour instead. Step 4. In step 3, specialized at b = a − N − , c = N − m − 1, and y i = q λ i + N − i + , we found χ B ( a − N + , N − m − , m ) ( { q λ i + N − i + } i =1 ,...,N ) = ( − N − m +1 I { ,q ± ( λi + N − i + 12 ) } dz π i I { } dw π i ( w − z − N Y i =1 ( w − q λ i + N − i + )( w − q − ( λ i + N − i + ) )( z − q λ i + N − i + )( z − q − ( λ i + N − i + ) ) · z N + a − w − N + m − z N + a + w − N − m − z − w ) . Plugging the above formula into (3 . . a was a very largepositive half-integer. However, observe that both sides of our identity are rational functions of q a (for the right side, this is seen after a residue expansion), so the identity (3 . 26) follows for all a ∈ C . (cid:3) Recall the Frobenius coordinates for a partition λ . Let d be the length of the main diagonal ofthe Young diagram corresponding to λ . The Frobenius coordinates ( a , . . . , a d | b , . . . , b d ) of λ are a i := λ i − i, b i := λ ′ i − i, for i = 1 , , . . . , d, where λ ′ := ( λ ′ , λ ′ , . . . ) is the conjugate partition of λ . For example, the “hook-shaped” partition λ = ( a + 1 , b ) has Frobenius coordinates ( a | b ). Proposition 3.17 (Frobenius identities) . Let N ∈ N , and λ ∈ GT + N be a partition with Frobeniuscoordinates ( a , . . . , a d | b , . . . , b d ). Denote χ Gµ = χ Gµ ( x , . . . , x N ), for any µ ∈ GT + N . Also, for any a, b ∈ N , N ≥ b + 1, denote ( a | b ) := ( a + 1 , b , N − b − ) ∈ GT + N . Then χ Gλ = 1(1 + { G = D } ) d − × det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ G ( a | b ) χ G ( a | b ) . . . χ G ( a | b d ) χ G ( a | b ) χ G ( a | b ) . . . χ G ( a | b d ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .χ G ( a d | b ) χ G ( a d | b ) . . . χ G ( a d | b d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Proof. This is a well-known fact in representation theory, which was derived in [ESK]. The samestatement holds for a more general class of functions, called generalised Schur functions , see [SV,Thm 3.1]. That theorem specializes to our Proposition in three special cases, as described in [SV,Section 4.2]. (cid:3) Theorem 3.18. For any 1 ≤ k ≤ N , G ∈ { B, C, D } , let c Gk,N ( q ) := k Y i =1 ( q i , q k + i − ǫ ; q ) N − k ( q i − ǫ , q − i ; q ) i − ( q, q i − ǫ ; q ) N − i . Then for any λ ∈ GT + N , and x , . . . , x k ∈ C ∗ , we have χ Gλ ( x , . . . , x k , q k + ǫ , . . . , q N − ǫ ) χ Gλ ( q ǫ , q ǫ , . . . , q N − ǫ ) = c Gk,N ( q ) · V s ( q ǫ , q ǫ , . . . , q k − ǫ ) V s ( x , x , . . . , x k ) × det ≤ i,j ≤ k (cid:20) χ Gλ ( q ǫ , . . . , q j − ǫ , x i , q j + ǫ , . . . , q N − ǫ ) χ Gλ ( q ǫ , q ǫ , . . . , q N − ǫ ) ( q ǫ x i , q ǫ /x i ; q ) j − ( q j + ǫ x i , q j + ǫ /x i ; q ) k − j (cid:21) . (3.38) Proof. Let us give details for G = B (where ǫ = ), leaving the cases G = C, D to the reader. Let m ≥ · · · ≥ m k be any integers such that m k ≥ k . The partition ( m , . . . , m k , k N − k ) has Frobeniuscoordinates a i = m i − i, b i = N − i, for all i = 1 , . . . , k. By Lemma 2 . . 17, we have χ Bλ ( q m + N − , . . . , q m k + N − k + , q N − , . . . , q k + ) χ Bλ ( q , q , . . . , q N − ) = χ B ( m ,...,m k ,k N − k ) ( q λ + N − , . . . , q λ N + ) χ B ( m ,...,m k ,k N − k ) ( q , q , . . . , q N − )= 1 χ B ( m ,...,m k ,k N − k ) ( q , q , . . . , q N − ) det ≤ i,j ≤ k h χ B ( m i − i | N − j ) ( q λ + N − , . . . , q λ N + ) i . (3.39)By Lemma 2 . χ B ( m i − i | N − j ) ( q λ + N − , . . . , q λ N + ) = χ B ( m i − i +1 , N − j , j − ) ( q λ + N − , . . . , q λ N + )= χ B ( m i − i +1 , N − j , j − ) ( q λ + N − , . . . , q λ N + ) χ B ( m i − i +1 , N − j , j − ) ( q , q , . . . , q N − ) χ B ( m i − i +1 , N − j , j − ) ( q , . . . , q N − )= χ Bλ ( q m i + N − i + , q N − , . . . , q j + , q j − , . . . , q ) χ Bλ ( q , q , . . . , q N − ) χ B ( m i − i +1 , N − j , j − ) ( q , . . . , q N − ) . From Proposition 2 . 9, we obtain χ B ( m i − i +1 , N − j , j − ) ( q , . . . , q N − ) = ( − N − j ( q m i + N − i + ) − q j − · − q m i + N − i + − q j − × ( q q m i + N − i + , q /q m i + N − i + ; q ) j − ( q j , q − j ; q ) j − , -DEFORMED CHARACTER THEORY · · · as well as χ B ( m ,...,m k ,k N − k ) ( q , q , . . . , q N − ) = ( − k ( N +1) k Y i =1 q ( m i + N − i + ) − q − ( m i + N − i + ) q ( i − ) − q − ( i − ) × V s ( q m + N − , . . . , q m k + N − k + ) V s ( q k − , . . . , q , q ) k Y i =1 ( q k + q m i + N − i + , q k + /q m i + N − i + ; q ) N − k ( q i , q k + i ; q ) N − k . Plugging these formulas into (3 . . x i = q m i + N − i + , i = 1 , . . . , k . Since m ≥ · · · ≥ m k ≥ k are arbitrary, and bothsides of (3 . 38) are rational functions on x , . . . , x k with the only poles on both sides being x i = 0, i = 1 , . . . , k , the identity (3 . 38) holds for all x , . . . , x k ∈ C ∗ too. (cid:3) Type B-C-D characters: proofs of results. The following lemma is the analogous toLemma 3 . 8; its proof is also very similar and we omit it. Lemma 3.19. Let 0 ≤ m ≤ N − λ ∈ GT + N , G ∈ { B, C, D } , and x be a complexnumber in the domain C \ (( −∞ , ∪ { } ∪ { q n + ǫ : n ∈ Z } ), such that q N < | x | < q − N . Then χ Gλ ( q ǫ , . . . , q m − ǫ , x, q m +1+ ǫ , . . . , q N − ǫ ) χ Gλ ( q ǫ , q ǫ , . . . , q N − ǫ )admits the following integral representation (recall l , . . . , l N are defined in Notation 2 . q ) · q ( m + ) − q − ( m + ) x − x − ( q m +1 , q − m ; q ) m ( q, q m +2 ; q ) N − m − ( q x, q /x ; q ) m ( q m + x, q m + /x ; q ) N − m − × (Z π i ln q − π i ln q dv π i Z L du π i x u ( q u − ( m +1) v − q − u +( m +1) v )( q v − q − v )( q v − u − q u − v )( q u − q − u ) N Y i =1 (1 − q l i − v )(1 − q l i + v )(1 − q l i − u )(1 − q l i + u ) − q Z π i q − π i q x v (cid:16) q ( m + ) v − q − ( m + ) v (cid:17) dv π i ) , if G = B ;(ln q ) · q m +1 − q − ( m +1) x − x − ( q m +2 , q − m ; q ) m ( q, q m +3 ; q ) N − m − ( qx, q/x ; q ) m ( q m +2 x, q m +2 /x ; q ) N − m − × (Z π i ln q − π i ln q dv π i Z L du π i x u q ( m +1) v − q − ( m +1) v q u − v − N Y i =1 (1 − q l i − v )(1 − q l i + v )(1 − q l i − u )(1 − q l i + u ) − q Z π i q − π i q x v (cid:16) q ( m +1) v − q − ( m +1) v (cid:17) dv π i ) , if G = C ; (cid:0) − { m =0 } (cid:1) (ln q ) q m , q − m ; q ) m ( q, q m +1 ; q ) N − m − ( x, /x ; q ) m ( q m +1 x, q m +1 /x ; q ) N − m − × (Z π i ln q − π i ln q dv π i Z L du π i x u q u − ( m + ) v + q − u +( m + ) v q v − u − q u − v N Y i =1 (1 − q l i − v )(1 − q l i + v )(1 − q l i − u )(1 − q l i + u )+ 1ln q Z π i q − π i q x v (cid:0) q mv + q − mv (cid:1) dv π i ) , if G = D. Remark 3.20. The condition q N < | x | < q − N ensures that the double integrals converge. Proof of Theorems . , . , and Lemma . . It is not difficult to show that the formulas (3 . . 21) and (3 . 22) define analytic functions on C \ (( −∞ , ∪ { } ∪ { q n + ǫ : n ∈ Z } ), proving thefirst part of Lemma 3 . 11. Theorem 3 . 13 is a consequence of Theorems 3 . 12 and 3 . 18, thus itsuffices to prove Theorem 3 . 12. We claim that (3.23) holds for x belonging to compact subsetsof C \ (( −∞ , ∪ { } ∪ { q n + ǫ : n ∈ Z } ). This uses Lemma 3 . 19 and follows closely the proof ofTheorem 3 . x should avoid ( −∞ , 0] because x u , x v and x are only defined on C \ ( −∞ , x should avoid { } ∪ { q n + ǫ : n ∈ Z } to avoid singularitiesin the right hand sides of (3 . . 27) and (3 . χ Gλ ( N ) ( q ǫ , · · · , q m − ǫ , x, q m +1+ ǫ , · · · , q N − ǫ ) are invariant with respect to the inversion x x − ,(3.23) for x ∈ C \ R implies Φ y ,Gm ( x ; q ) = Φ y ,Gm (1 /x ; q ), for x ∈ C \ R .We still have to show: (A) Φ y ,Gm ( x ; q ) admits an analytic continuation to C ∗ and the limit (3 . x in compact subsets of C ∗ ; and (B) the converse statement that ifthe limit in the left side of (3 . 13) exists, then { λ ( N ) ∈ GT + N } N ≥ stabilizes to some y ∈ Y .Observe that (A) would immediately imply Φ y ,Gm ( x ; q ) = Φ y ,Gm (1 /x ; q ), for all x ∈ C ∗ .The proof of (B) is identical to the proof of the analogous result of Theorem 3 . 3, except that ituses the proof of Proposition 6 . . 2) and the proof of Theorem 6 . . R > R / ∈ { q n + ǫ : n ∈ Z } . We claim that the sequence ( χ λ ( N ) ( q ǫ , . . . , q m − ǫ , z, q m +1+ ǫ , . . . , q N − ǫ ) χ λ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) ) N ≥ (3.40)of holomorphic functions on C ∗ is uniformly bounded on { /R < | z | < R } . In fact, from thenonnegativity of the branching coefficients for symplectic/orthogonal characters (see Prop. 2 . /R < | z | < R implies | z | n < R n + 1 /R n , for any n ∈ Z , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ λ ( N ) ( q ǫ , . . . , q m − ǫ , z, q m +1+ ǫ , . . . , q N − ǫ ) χ λ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ χ λ ( N ) ( q ǫ , . . . , q m − ǫ , | z | , q m +1+ ǫ , . . . , q N − ǫ ) χ λ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) ≤ χ λ ( N ) ( . . . , q m − ǫ , R, q m +1+ ǫ , . . . ) χ λ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) + χ λ ( N ) ( . . . , q m − ǫ , /R, q m +1+ ǫ , . . . ) χ λ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) . By our choice of R , the limit (3 . 23) holds pointwise for x = R, /R , implying that χ λ ( N ) ( . . . , q m − ǫ , R, q m +1+ ǫ , . . . ) χ λ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ ) and χ λ ( N ) ( . . . , q m − ǫ , /R, q m +1+ ǫ , . . . ) χ λ ( N ) ( q ǫ , q ǫ , . . . , q N − ǫ )are both uniformly bounded sequences, thus implying our claim. Next we can apply Montel’stheorem: it implies that any subsequence of (3 . 40) has subsequential limits, which are analytic on { /R < | z | < R } . Each such holomorphic function agrees with Φ y ,Gm ( z ; q ) on an open set, so byanalytic continuation, they must all be the same. Since R > (cid:3) Branching graphs Generalities on Branching Graphs. We recall some general facts about branching graphsand their boundaries. Our exposition is similar to that of [O4]; see also [GO]. Similar statements,in equivalent forms, can be also found in [DF, D, W].For a Borel space X , let M ( X ) denote the set of probability measures on X . The set M ( X )is a convex subset of the vector space of finite, signed measures on X . A Markov kernel betweenBorel spaces K : X Y induces an affine map K : M ( X ) → M ( Y ) between convex sets, thatwe denote by the same letter. -DEFORMED CHARACTER THEORY · · · Let X , X , . . . be a sequence of countable sets, equipped with their Borel structure coming fromthe discrete topology. Assume we have, for each N ≥ 1, a Markov kernel Λ N +1 N : X N +1 X N .Then the chain of Markov kernels X L99 X L99 X L99 . . . naturally induces the chain M ( X ) ← M ( X ) ← M ( X ) ← . . . of affine maps of convex sets. Let lim ←− M ( X N ) be the projective limit. As a set, it consists of coherent systems { M N ∈ M ( X N ) } N ≥ , i.e., for each N ≥ 1, the probability measures M N and M N +1 are related by M N +1 Λ N +1 N = M N . More explicitly, the coherency property is X x ∈ X N +1 M N +1 ( x )Λ N +1 N ( x, y ) = M N ( y ) , for all y ∈ X N , N ≥ . The sequence { X N , Λ N +1 N : N ≥ } is called a branching graph .Equip lim ←− M ( X N ) with the Borel structure arising from the embeddinglim ←− M ( X N ) ֒ → Y N ≥ M ( X N ) . (4.1)The product space in (4 . 1) is convex and lim ←− M ( X N ) is a convex subset. Definition 4.1. The set of extreme points of the convex space lim ←− M ( X N ) is called the (minimal)boundary of the branching graph { X N , Λ N +1 N : N ≥ } . Let us denote it by Ω. Given ω ∈ Ω, wedenote by { M ωN } N ≥ the corresponding coherent system. Theorem 4.2 ([O4], Thm. 9.2) . The set Ω is a Borel subset of lim ←− M ( GT N ). Moreover, for everycoherent system { M N } N ≥ , there exists a unique Borel probability measure π ∈ M (Ω) such that M N ( x ) = Z Ω M ωN ( x ) π ( dω ) , for all x ∈ X N , N ≥ . Conversely, every π ∈ M (Ω) gives a coherent system by the formula above. The resulting map M (Ω) → lim ←− M ( X N ) is a bijection.Let us also define the Martin boundary of the branching graph { X N , Λ N +1 N : N ≥ } . If wecompose the Markov kernels Λ n +1 n : X n +1 X n , N > n ≥ k , we obtainΛ Nk := Λ NN − Λ N − N − · · · Λ k +1 k : X N X k . In particular, if N > k and x ( N ) ∈ X N , then Λ Nk ( x ( N ) , · ) is a probability measure on X k . Definition 4.3. Let { M k ∈ M ( X k ) } k ≥ be a coherent system for which there exists a sequence { x ( N ) ∈ X N } N ≥ such that M k ( y ) = lim N →∞ Λ Nk ( x ( N ) , y ) , for all y ∈ X k , k ≥ 1. (4.2)The Martin boundary of the branching graph { X N , Λ N +1 N : N ≥ } is defined as the subset oflim ←− M ( X N ) consisting of the coherent systems { M k ∈ M ( X k ) } k ≥ for which such a sequence { x ( N ) ∈ X N } N ≥ exists. The Martin boundary is denoted Ω Martin . Theorem 4.4 ([OO2], Thm 6.1) . The minimal boundary is contained in the Martin boundary. Inother words, for any ω ∈ Ω, there exists a sequence { x ( N ) ∈ X N } N ≥ such that M ωk ( y ) = lim N →∞ Λ Nk ( x ( N ) , y ) , for all y ∈ X k , k ≥ . Symmetric q -Gelfand-Tsetlin graph. Let σ = ( σ , σ , . . . ) ∈ { +1 , − } ∞ be arbitrary; wewill construct a branching graph associated to σ .For each N ≥ 1, define the matrix [Λ N +1 N ( λ, µ )] λ,µ of format GT N +1 × GT N via the branchingrelations s λ ( x , . . . , x N , q − N ) s λ (1 , q − , . . . , q − N ) = X µ ∈ GT N Λ N +1 N ( λ, µ ) s µ ( x , . . . , x N ) s µ (1 , q − , . . . , q − N ) , if σ N = − ,s λ ( x , . . . , x N , q N ) s λ (1 , q, . . . , q N ) = X µ ∈ GT N Λ N +1 N ( λ, µ ) s µ ( x , . . . , x N ) s µ (1 , q, . . . , q N − ) , if σ N = +1 . (4.3)This means the following. For λ ∈ GT N +1 , the left hand sides of (4.3) are symmetric Laurentpolynomials in N variables. The Schur polynomials s µ ( x , . . . , x N ), µ ∈ GT N , form a basis of thespace of symmetric Laurent polynomials in x , . . . , x N . Thus we are guaranteed the existence anduniqueness of the coefficients Λ N +1 N ( λ, µ ), µ ∈ GT N , in the right hand sides of (4.3).From Proposition (2.1) and the homogeneity of Schur polynomials, we obtainΛ N +1 N ( λ, µ ) = { µ ≺ λ } × s µ ( q, q , . . . , q N ) s λ (1 , q, . . . , q N ) if σ N = − ,s µ ( q − , q − , . . . , q − N ) s λ (1 , q − , . . . , q − N ) if σ N = +1 . (4.4) Lemma 4.5. For each N ≥ 1, the matrix [Λ N +1 N ( λ, µ )] is stochastic. Proof. For any λ ∈ GT N +1 , µ ∈ GT N , let us show Λ N +1 N ( λ, µ ) ≥ 0. From Proposition (2 . s κ (1 , q, . . . , q K ) > q > 0. Then Λ N +1 N ( λ, µ ) ≥ . λ ∈ GT N +1 , set x i = q i − , i = 1 , , . . . , N , in the polynomial equalities(4 . 3) to show 1 = P µ Λ N +1 N ( λ, µ ) for both σ N = 1 and σ N = − (cid:3) Instead of σ = ( σ , σ , . . . ) ∈ { +1 , − } ∞ , we can consider the sequence b = ( b (1) , b (2) , . . . ) ∈ N ∞ given by b ( n ) := { ≤ i ≤ n : σ i = − } , so that n − b ( n ) = { ≤ i ≤ n : σ i = +1 } . (4.5)The sequence b satisfies b (1) ∈ { , } and b ( n + 1) − b ( n ) ∈ { , } for all n ≥ 1; conversely anysequence b with these properties arises from some σ ∈ { +1 , − } ∞ via the relations (4 . Definition 4.6. The sequence { GT N , Λ N +1 N : N ≥ } , defined by (4 . symmetric q -Gelfand-Tsetlin graph associated to σ ∈ {± } ∞ (or associated to b ∈ N ∞ ). The minimal boundaryof the symmetric q -Gelfand-Tsetlin graph associated to σ ∈ {± } ∞ will be denoted Ω q ( σ ), orsimply Ω q , if there is no confusion about σ . Its Martin boundary will be denoted Ω Martin q ( σ ), orjust Ω Martin q .It will be shown later in Theorems 5.6 and 5.7 that whenever σ, σ ′ ∈ {± } ∞ satisfy the genericcondition (5.1), the topological spaces Ω q ( σ ) and Ω q ( σ ′ ) are homeomorphic. The same remarkapplies to the Martin boundaries.4.3. BC type q -Gelfand-Tsetlin graph. For G = B, C and D , set ǫ = , 1, and 0, respectively.For each N ≥ 1, define the matrix [Λ N +1 N ( λ, µ )] λ,µ of format GT + N +1 × GT + N via the branchingrelation χ Gλ ( x , . . . , x N , q N + ǫ ) χ Gλ ( q ǫ , q ǫ , . . . , q N + ǫ ) = X µ ∈ GT + N Λ N +1 N ( λ, µ ) χ Gµ ( x , x , . . . , x N ) χ Gµ ( q ǫ , q ǫ , . . . , q N − ǫ ) . (4.6) -DEFORMED CHARACTER THEORY · · · From Proposition 2 . 5, we haveΛ N +1 N ( λ, µ ) = χ Gµ ( q ǫ , q ǫ , . . . , q N − ǫ ) χ Gλ ( q ǫ , q ǫ , . . . , q N + ǫ ) χ Gλ/µ ( q N + ǫ ) . (4.7)Note that the kernels Λ N +1 N depend on G , but we suppress it from the notation.Repeating the proof of Lemma 4 . 5, we obtain: Lemma 4.7. For each N ≥ 1, the matrix [Λ N +1 N ( λ, µ )] is stochastic. Definition 4.8. The sequence { GT + N , Λ N +1 N : N ≥ } , defined by (4 . BC type q -Gelfand-Tsetlin graph . The (minimal) boundary of the BC type q -Gelfand-Tsetlin graph willbe denoted Ω Gq . The Martin boundary of the BC type q -Gelfand-Tsetlin graph will be denotedΩ G, Martin q .The boundary Ω Gq depends on the type G ∈ { B, C, D } . We show later in Theorems 6.5 and 6.6that, as topological spaces, all three of them are homeomorphic to each other (but they correspondto three distinct families of coherent systems on the BC type q -Gelfand-Tsetlin graph). The sameremark applies to the Martin boundaries Ω G, Martin q .5. Boundary of the symmetric q -Gelfand-Tsetlin graph In this section, let us fix a sequence σ = ( σ , σ , . . . ) ∈ {± } ∞ satisfying the assumptionlim N →∞ { ≤ i ≤ N : σ i = − } = lim N →∞ { ≤ i ≤ N : σ i = +1 } = + ∞ . (5.1)Equivalently, in terms of the sequence b given by (4 . N →∞ b ( N ) = lim N →∞ ( N − b ( N )) = + ∞ . The goal of this section is to characterize the minimal boundary of the symmetric q -Gelfand-Tsetlin graph associated to b . To proceed via the ergodic method of Vershik-Kerov, [VK1, V], weneed to first characterize the Martin boundary of the symmetric q -Gelfand-Tsetlin graph.In subsections 5 . 1, 5 . . 3, we identify the Martin boundary with the set X of all doublyinfinite, nondecreasing integer sequences (recall Definition 3 . . 4, we show thatthe minimal boundary coincides with the Martin boundary. The proof of the latter statement isbased on the Law of Large Numbers of Theorem 5 . A family of coherent probability measures. In this section, we define a map from theset X into the Martin boundary of the symmetric q -Gelfand-Tsetlin graph.Let k ∈ N and let C k be the space of analytic functions f ( z , . . . , z k ) on ( C ∗ ) k for which thereexist a m ,...,m k ∈ C , m , . . . , m k ∈ Z , such that f ( z , . . . , z k ) = X m ,...,m k ∈ Z a m ,...,m k z m · · · z m k k ∀ ( z , . . . , z k ) ∈ ( C ∗ ) k , (5.2)and the right side is an absolutely and uniformly convergent sum on compact subsets of ( C ∗ ) k .Then necessarily a m ,...,m k = I | z | =1 · · · I | z k | =1 f ( z , . . . , z k ) z m +11 · · · z m k +1 k k Y i =1 dz i π i . Let us call C S k k the linear subspace of C k , consisting of functions f ( z , . . . , z k ) for which the coeffi-cients in the expansion (5 . 2) satisfy a m ,...,m k = a m σ (1) ,...,m σ ( k ) , for all σ ∈ S k , m , . . . , m k ∈ Z . For example, finite linear combinations of Schur polynomials s λ ( z , . . . , z k ), λ ∈ GT k , are elementsof C S k k . For each µ ∈ GT k , define the linear functional F µ : C S k k → C by F µ ( f ) := 1 k ! I | z | =1 · · · I | z k | =1 f ( z , . . . , z k ) s µ ( z , . . . , z k ) Y ≤ i The linear functionals { F µ : C S k k → C } µ ∈ GT k , defined by (5 . F µ ( s ν ( z , . . . , z k )) = δ µ,ν , for all µ, ν ∈ GT k .(2) If { f N } N ≥ ⊂ C S k k is such that f N → f uniformly on compact subsets of ( C ∗ ) k , then f ∈ C S k k and F µ ( f N ) → F µ ( f ) for all µ ∈ GT k .(3) Let h ∈ C S k k be such that F µ ( h ) = 0, for all µ ∈ GT k . Then h is identically zero. Proof. The first item of the lemma is well known, see for instance [M, I.4, VI.9].The second item is obvious.For the third item, let h ∈ C S k k be such that F µ ( h ) = 0, for all µ ∈ GT k . The span of theSchur polynomials { s µ ( z , . . . , z k ) } µ ∈ GT k is the linear space of symmetric Laurent polynomials on k variables, and this is dense in the space of continuous, symmetric functions on the k –dimensionaltorus T k . Therefore I | z | =1 · · · I | z k | =1 h ( z , . . . , z k ) g ( z , . . . , z k ) Y ≤ i Take any t ∈ X and let { λ ( N ) ∈ GT N } N ≥ be a sequence that b -stabilizes to t .For each k ≥ µ ∈ GT k , the limit M t k ( µ ) := lim N →∞ Λ Nk ( λ ( N ) , µ )exists and does not depend on the choice of { λ ( N ) } N ≥ . Moreover, M t k is a probability measureon GT k , and { M t k } k ≥ is a coherent system. Proof. Let { λ ( N ) ∈ GT N } N ≥ be any sequence of signatures that b -stabilizes to t . Let k ≥ b k ( N ) := max { , b ( N ) − b ( k ) } , so that b k ( N ) = { k < i ≤ N : σ i = − } whenever N > k . Consider the equality s λ ( N ) (1 , q, . . . , q b k ( N ) − , q b k ( N ) x , . . . , q b k ( N ) x k , q b k ( N )+ k , . . . , q N − , q N − ) s λ ( N ) (1 , q, . . . , q N − , q N − )= X µ ∈ GT k Λ Nk ( λ ( N ) , µ ) s µ ( x , . . . , x k ) s µ (1 , q, . . . , q k − ) , (5.5)which follows from (4 . N − k . Let us look at the first line of (5 . ≤ b ( N ) − b k ( N ) ≤ k and thuslim N →∞ b k ( N ) = lim N →∞ ( N − b k ( N )) = + ∞ . Moreover if we let b k := ( b k (1) , b k (2) , . . . ) , then the sequence { λ ( N ) } N ≥ b k -stabilizes to t ( k ) := ( . . . , t k − , t k , t k , . . . ), t ki := t i + b ( k ) , for all i ∈ Z .Then, from the limit (3.3) in Theorem 3 . 3, the first line of (5 . 5) converges to Φ t ( k ) ( x , . . . , x k ; q ) -DEFORMED CHARACTER THEORY · · · uniformly on compact subsets of ( C ∗ ) k , as N goes to infinity. Then by (5 . 5) and Lemma 5 . N →∞ Λ Nk ( λ ( N ) , µ ) = s µ (1 , q, . . . , q k − ) F µ (Φ t ( k ) ) =: a µ , for any µ ∈ GT k . We show that { a µ : µ ∈ GT k } determines a probability measure on GT k . Since Λ Nk ( λ ( N ) , µ ) ≥ P µ ∈ GT k Λ Nk ( λ ( N ) , µ ) = 1 for all N > k , then a µ ≥ , for all µ ∈ GT k ; X µ ∈ GT k a µ ≤ . (5.6)We need an extra argument to show P µ ∈ GT k a µ = 1. Consider the function g ( z , . . . , z k ) := X µ ∈ GT k a µ s µ ( z , . . . , z k ) s µ (1 , q, . . . , q k − ) . Since (5 . 5) converges uniformly on compact subsets, then in particular, it converges pointwise atthe point ( x , x , . . . , x k ) = ( t, tq, . . . , tq k − ), for any t > 0. But at this point, each summandΛ Nk ( λ ( N ) , µ ) · s µ ( t, tq, . . . , tq k − ) s µ (1 , q, . . . , q k − ) = Λ Nk ( λ ( N ) , µ ) · t | µ | is nonnegative. Then the sum P µ ∈ GT k a µ t | µ | must be convergent, for any t > 0. Let R > z , . . . , z k ) ∈ ( C ∗ ) k such that Rq i − < | z i | < R − q i − , the nonnegativity ofthe branching coefficients for Schur polynomials (see Proposition 2 . 1) and homogeneity of Schurpolynomials imply (cid:12)(cid:12)(cid:12)(cid:12) s µ ( z , . . . , z k ) s µ (1 , . . . , q k − ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ s µ ( | z | , . . . , | z k | ) s µ (1 , . . . , q k − ) ≤ s µ (( R + R − ) , . . . , ( R + R − ) q k − ) s µ (1 , . . . , q k − ) = ( R + R − ) | µ | . As R > g is absolutely and uniformly convergenton compact subsets of ( C ∗ ) k , and g is a well-defined analytic function in that domain. Moreover, g ∈ C S k k because each s µ is a symmetric polynomial.From items (1)–(2) of Lemma 5 . 1, we have F µ ( g ) = a µ /s µ (1 , q, . . . , q k − ), for all µ ∈ GT k . Butwe also know F µ (Φ t ( k ) ) = a µ /s µ (1 , q, . . . , q k − ). Therefore h := g − Φ t ( k ) ∈ C S k k is such that F µ ( h ) = F µ ( g ) − F µ (Φ t ( k ) ) = 0, for all µ ∈ GT k . Item (3) of Lemma 5 . h = 0, i.e., g = Φ t ( k ) . In particular, evaluating this equality at the point (1 , q, . . . , q k − ) and using the limit(3.3) of Theorem 3 . { λ ( N ) } N ≥ is any sequence that b -stabilizes to t ): X µ ∈ GT k a µ = g (1 , q, . . . , q k − ) = Φ t ( k ) (1 , q, . . . , q k − ) =lim N →∞ s λ ( N ) (1 , . . . , q b k ( N ) − , q b k ( N ) x , . . . , q b k ( N ) x k , q b k ( N )+ k , . . . , q N − ) s λ ( N ) (1 , q, . . . , q N − , q N − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x i = q i − ∀ i = lim N →∞ . Thus if we let M t k ( µ ) := a µ , for any µ ∈ GT k , we have that M t k is a probability measure on GT k .Moreover, because Φ t ( k ) does not depend on the choice of b -stabilizing sequence { λ ( N ) } N ≥ , neitherdoes each a µ = s µ (1 , q, . . . , q k − ) F µ (Φ t ( k ) ). The coherency is immediate from the definitions. (cid:3) Concentration bound. We keep the notations from the previous subsection. Proposition 5.3. Let k ∈ N and { λ ( N ) ∈ GT N } N ≥ be arbitrary. Further, let { µ ( N ) ∈ GT N } N ≥ be random variables such that each µ ( N ) is distributed according to the probability distribution Λ N +1 N ( λ ( N + 1) , · ). Then there exists a constant c > 0, independent of N , such thatProb (cid:0) µ ( N ) b ( N )+ i = λ ( N + 1) b ( N +1)+ i , for all − k ≤ i ≤ k (cid:1) > ( − cq b ( N ) , if σ N +1 = − − cq N − b ( N ) , if σ N +1 = +1 . We need some preparations for the proof. Given M ∈ N , ν ∈ GT M +1 , define two probabilitydistributions on GT M , to be denoted P + ( ·| ν ) and P − ( ·| ν ), and given by P + ( κ | ν ) := { κ ≺ ν } s κ ( q, q , . . . , q M ) s ν (1 , q, . . . , q M ) ; P − ( κ | ν ) := { κ ≺ ν } s κ ( q − , q − , . . . , q − M ) s ν (1 , q − , . . . , q − M ) . There is an explicit closed formula for the evaluation of a Schur polynomial on a q -geometric series,see Proposition 2 . 3; it gives P + ( κ | ν ) = { κ ≺ ν } q | κ | + n ( κ ) Y ≤ i Let m ∈ N be fixed. There exists a constant c ′ > M ∈ N , any ν ∈ GT M +1 and any integer 1 ≤ i ≤ M − m + 1, the following holds:Let κ ∈ GT M be distributed according to P + ( · | ν ). ThenProb ( κ j = ν j +1 , for all j = i, i + 1 , . . . , i + m − > − c ′ q i . Proof. The proof is a generalization of the proof of [GO, Lem. 3.13].From Boole’s inequality, it is clear that we only need to consider the case m = 1. In other words,take any 1 ≤ i ≤ M and let us find a constant c ′ > κ i = ν i +1 ) > − c ′ q i . Set a := ν i , b := ν i +1 , so that a ≥ κ i ≥ b almost surely. If a = b , then Prob ( κ i = b = ν i +1 ) = 1and there is nothing to prove. Then assume a ≥ b + 1, so there exist values m ∈ Z such that a − b ≥ m ≥ 1. For each such value, we estimate the ratioProb( κ i = b + m )Prob( κ i = b ) (5.8)and show that it is of order O ( q mi ). Let us actually estimate the ratio (5 . κ , . . . , κ i − , κ i +1 , . . . , κ M , which are fixed, but arbitrary, and satisfy ν j ≥ κ j ≥ ν j +1 for all relevant j . Under such conditional probability, and from the formula (5 . P + ( ·| ν ), we haveProb( κ i = d ) ∝ q id i − Y r =1 (1 − q κ r − d + i − r ) M Y s = i +1 (1 − q d − κ s + s − i ) , for any a ≥ d ≥ b , where the hidden constant is independent of d . Then for any a − b ≥ m ≥ q ∈ (0 , κ i = b + m )Prob( κ i = b ) = q mi i − Y r =1 − q κ r − b − m + i − r − q κ r − b + i − r M Y s = i +1 − q b + m − κ s + s − i − q b − κ s + s − i ≤ q mi Q i − r =1 (1 − q κ r − b + i − r ) Q Ms = i +1 (1 − q b − κ s + s − i ) . (5.9)Next, since κ i − i is strictly decreasing on i and 1 + b ≤ a = ν i ≤ κ i − :2 ≤ κ i − − b + 1 < κ i − − b + 2 < . . . < κ − b + i − -DEFORMED CHARACTER THEORY · · · and thus i − Y r =1 (1 − q κ r − b + i − r ) > (1 − q )(1 − q ) · · · = ( q ; q ) ∞ . (5.10)Similarly, we can deduce 1 ≤ b + 1 − κ i +1 < b + 2 − κ i +2 < . . . < b + M − i − κ M and M Y s = i +1 (1 − q b − κ s + s − i ) > (1 − q )(1 − q ) · · · = ( q ; q ) ∞ . (5.11)From the bounds (5 . . 10) and (5 . κ i = b + m )Prob( κ i = b ) ≤ q mi ( q ; q ) ∞ ( q ; q ) ∞ ∀ ≤ m ≤ a − b. (5.12)By adding the inequalities (5 . 12) over 1 ≤ m ≤ a − b , and using 1 + q + . . . + q a − b − < / (1 − q ):Prob( κ i ≥ b + 1)Prob( κ i = b ) ≤ q i ( q ; q ) ∞ . (5.13)Since Prob( κ i ≥ b + 1) = 1 − Prob( κ i = b ), we finally obtainProb( κ i = b ) ≥ (cid:18) q i ( q ; q ) ∞ (cid:19) − ≥ − q i ( q ; q ) ∞ = 1 − c ′ q i , with positive constant c ′ := 1 / ( q ; q ) ∞ . Note that the bound just proven was uniform over κ , . . . , κ i − , κ i +1 , . . . , κ M that we were conditioning over. Thus the bound also holds without the conditioning, and we aredone. (cid:3) Proof of Proposition . . If σ N +1 = − 1, note that the probability measures Λ N +1 N ( λ ( N + 1) , · ) and P + ( ·| λ ( N + 1)) are the same. We can apply Lemma 5 . µ ( N ) is Λ N +1 N ( λ ( N + 1) , · )-distributed, thenProb( µ ( N ) b ( N )+ i − = λ ( N ) b ( N )+ i for all − k ≤ i ≤ k ) > − cq b ( N ) , (5.14)for some c > N . As σ N +1 = − 1, then b ( N + 1) = b ( N ) + 1, so the left side of(5 . 14) equals Prob( µ b ( N )+ i − = λ ( N ) b ( N +1)+ i − for all − k ≤ i ≤ k ) , giving us the desired result.Let us proceed to the case σ N +1 = +1, which will be deduced from the case σ N +1 = − 1. Fromthe determinantal definition of Schur polynomials, we have s κ ( x , . . . , x M ) = s κ − (1 /x , . . . , /x M ) , κ ∈ GT M , where κ − := ( − κ M , . . . , − κ ) ∈ GT M . Therefore P − ( κ | ν ) := P + ( κ − | ν − ), for any κ ∈ GT M , ν ∈ GT M +1 . In other words, if κ is distributed according to P − ( ·| ν ), then κ − is distributedaccording to P + ( ·| ν − ).Next, let µ ( N ) − := ( − µ ( N ) N , . . . , − µ ( N ) ), λ ( N + 1) − := ( − λ ( N + 1) N +1 , . . . , − λ ( N + 1) ); if c > σ N +1 = − (cid:0) µ ( N ) b ( N )+ i = λ ( N + 1) b ( N +1)+ i , for all − k ≤ i ≤ k (cid:1) = Prob (cid:16) µ ( N ) − N +1 − b ( N ) − i = λ ( N + 1) − N +2 − b ( N +1) − i , for all − k ≤ i ≤ k (cid:17) > − cq N +2 − b ( N +1) = 1 − ( cq ) q N − b ( N ) , where the last equality comes from b ( N + 1) = b ( N ), in the case σ N +1 = +1. (cid:3) Law of Large Numbers and the Martin boundary.Theorem 5.5. Let { µ ( N ) ∈ GT N } N ≥ be a sequence of random signatures such that each µ ( N )is M t N -distributed (see Proposition 5 . M t N ). Then,for each k ∈ Z , the probability of the event µ ( L ) b ( L )+ k = t − k tends to 1, as L goes to infinity. Proof. Let { λ ( N ) ∈ GT N } N ≥ be a sequence of signatures that b -stabilizes to t . From Proposition5 . 2, the probability measures Λ NK ( λ ( N ) , · ) converge weakly to M t K , for all K ≥ 1. So if we let { µ N,K } N ≥ K ⊂ GT K be a sequence of random signatures with µ N,K being Λ NK ( λ ( N ) , · )-distributed,and let µ K ∈ GT K be random M t K -distributed, then µ N,K converges weakly to µ K .Let ǫ > c > . k ∈ N inthe statement of the Theorem. There exists N ∈ N large enough so that ∞ Y i = N (1 − cq i ) > − ǫ. As lim N →∞ b ( N ) = lim N →∞ ( N − b ( N )) = + ∞ , there exists a large enough L ∈ N such thatmin { b ( N ) , N − b ( N ) } ≥ N , for all N > L. Then Proposition 5 . (cid:16) µ N,Lb ( L )+ k = λ ( N ) b ( N )+ k (cid:17) ≥ N − Y i = L (1 − cq x ( i ) ) ∀ N > L, where x ( i ) := ( b ( i ) , if σ i +1 = − i − b ( i ) , if σ i +1 = +1 . By our choice of L , x ( i ) ≥ N for all i ≥ L . Also, the sequence ( x ( L ) , x ( L + 1) , . . . ) does notcontain the same number more than twice. As a result,Prob (cid:16) µ N,Lb ( L )+ k = λ ( N ) b ( N )+ k (cid:17) ≥ ∞ Y i = N (1 − cq i ) > − ǫ ∀ N > L. (5.15)Now since λ ( N ) b -stabilizes to t , we have λ ( N ) b ( N )+ k = t − k , for large enough N ∈ N . Also, asmentioned above, µ N,L converges weakly to µ L , as N goes to infinity. Therefore, from (5 . 15) andthese observations, we obtain the inequalityProb (cid:16) µ Lb ( L )+ k = t − k (cid:17) > − ǫ. (5.16)Since ǫ > . 16) implies the theorem. (cid:3) Theorem 5.6. The Martin boundary of the symmetric q -Gelfand-Tsetlin graph associated to σ isin bijection with the set X , under the map t 7→ { M t N } N ≥ of Proposition 5 . Proof. By Proposition 5 . . 5, each coherent system { M t N } N ≥ belongs to the Martinboundary, and all of them are distinct. It remains to show that there are no other points in theMartin boundary.Let { M k } k ≥ be an element of Ω Martin q ; then there exists a sequence { λ ( N ) ∈ GT N } N ≥ suchthat M k ( µ ) = lim N →∞ Λ Nk ( λ ( N ) , µ ) , for all µ ∈ GT k , k ≥ . (5.17) -DEFORMED CHARACTER THEORY · · · Let m ∈ Z be fixed, but arbitrary, and let c > . k = | m | + 1.Let N ∈ N be such that ∞ Y i = N (1 − cq i ) > / . Since lim N →∞ b ( N ) = lim N →∞ ( N − b ( N )) = + ∞ , there exists L ∈ N such that min { b ( N ) , N − b ( N ) } ≥ N , for all N ≥ L . Fix L for the moment. For any N ≥ L , let µ N ∈ GT L be a randomsignature which is Λ NL ( λ ( N ) , · )-distributed. Also let ξ N := µ Nb ( L )+ m ∈ Z , so it is a random integer.From Proposition 5 . ξ N = λ ( N ) b ( N )+ m ) ≥ Prob (cid:16) µ Nb ( L )+ i = λ ( N ) b ( N )+ i , for all − k ≤ i ≤ k (cid:17) ≥ N − Y i = L (1 − cq x ( i ) ) , where x ( i ) := b ( i ), if σ i +1 = − x ( i ) := i − b ( i ), if σ i +1 = +1. By our choice of L , x ( i ) ≥ N for i ≥ L . Moreover, the sequence ( x ( L ) , x ( L + 1) , . . . ) does not contain the same number morethan twice. As a result, Prob( ξ N = λ ( N ) b ( N )+ m ) ≥ ∞ Y i = N (1 − cq i ) > / . (5.18)By assumption, M L is the weak limit of Λ NL ( λ ( N ) , · ), as N tends to infinity. Therefore, if we let M be the pushforward of M L from GT L to the coordinate b ( N ) + m , then M is a probabilitymeasure and the weak limit of the laws of the random variables ξ N . From (5 . M ( { p } ) > / 3, for any p ∈ Z which is a subsequential limit of the sequence { λ ( N ) b ( N )+ m } N ≥ .This cannot occur for more than one p ∈ Z , or the total measure of M would be at least than 4 / { λ ( N ) b ( N )+ m } N ≥ , sinceotherwise the total measure of M would be at most 1 / 3. Therefore the sequence { λ ( N ) b ( N )+ m } N ≥ converges as N goes to infinity. Since m ∈ Z was arbitrary, the limitlim N →∞ λ ( N ) b ( N )+ m exists for any m ∈ Z . This implies that { λ ( N ) } N ≥ must b -stabilize to some t ∈ X . From (5 . . 2, it follows that M k = M t k for any k ∈ N , i.e., { M k } k coincides with the coherentsystem { M t N } N ≥ , concluding the proof. (cid:3) Characterization of the minimal boundary.Theorem 5.7. The minimal boundary Ω q of the symmetric q -Gelfand-Tsetlin graph associated to σ is equal to the Martin boundary Ω Martin q . Proof. This is a Corollary of Theorems 5 . 5, 5 . . 4. Indeed, by the latter one, the minimalboundary is contained in the Martin boundary and it remains to show that the coherent systems { M t N } N ≥ are extreme points of the convex space lim ← M ( GT N ) of coherent systems on the sym-metric q -Gelfand-Tsetlin graph. By Theorem 5 . 5, all the measures { M t N } N ≥ have pairwise disjointsupports, hence one cannot be a convex combination of the others, which finishes the proof. Seealso [GO, proof of Thm. 3.12] and [O5, Step 3 in proof of Thm. 6.2] for more detailed expositionsof similar proofs. (cid:3) Boundary of the BC type q -Gelfand-Tsetlin graph In this section, we characterize the Martin and minimal boundaries of the BC type q -Gelfand-Tsetlin graph. We suppress the type G ∈ { B, C, D } from most notations, for simplicity. Theparameter ǫ is , , 0, for G = B, C, D , respectively.The reader should compare Section 5 with this one. Both follow the same approach and yieldsimilar results. A family of coherent probability measures. In Proposition 6 . 2, we define a map from Y into construct the measures of the Martin boundary of the BC type q -Gelfand-Tsetlin graph.Let k ∈ N be arbitrary. Recall the linear space C k , defined at the beginning of Section 5 . 1, ofanalytic functions f ( z , . . . , z k ) on ( C ∗ ) k for which an absolutely and uniformly convergent expan-sion f ( z , . . . , z k ) = X m ,...,m k ∈ Z a m ,...,m k z m · · · z m k k ∀ ( z , . . . , z k ) ∈ ( C ∗ ) k , (6.1)exists. Let us call C W k k the linear subspace of C k , consisting of functions f ( z , . . . , z k ) for whichthe coefficients in the expansion (6 . 1) satisfy a m ,...,m k = a ǫ m σ (1) ,...,ǫ k m σ ( k ) , for all σ ∈ S k , m , . . . , m k ∈ Z , ǫ , . . . , ǫ k ∈ { , − } . For example, finite linear combinations of the symplectic/orthogonal polynomials χ Gλ ( z , . . . , z k ), G ∈ { B, C, D } , λ ∈ GT + k , belong to C W k k . For each G ∈ { B, C, D } and µ ∈ GT + k , define thefunctionals F Gµ : C W k k → C via F Gµ ( f ) := e F Gµ ( f ) e F Gµ ( χ Gµ ( z , . . . , z k )) , where e F Gµ ( f ) := I | z | =1 · · · I | z k | =1 f ( z , . . . , z k ) χ Gµ ( z , . . . , z k ) m G ( z , . . . , z k ) k Y i =1 dz i π i ; m G ( z , . . . , z k ) := Y ≤ i The linear functionals { F Gµ : C W k k → C } µ ∈ GT k satisfy(1) F Gµ ( χ Gν ( z , . . . , z k )) = δ µ,ν , for all µ, ν ∈ GT + k .(2) If { f N } N ≥ ⊂ C W k k is such that f N → f uniformly on compact subsets of ( C ∗ ) k , then f ∈ C W k k and F Gµ ( f N ) → F Gµ ( f ) for all µ ∈ GT + k .(3) If h ∈ C W k k and F Gµ ( h ) = 0, for all µ ∈ GT + k , then h is identically zero. Proof. The first item of the lemma is well known, see for instance [OlOs]. The second item isobvious. As for the third item, its proof is similar to the proof of Lemma 5.1 (3). The onlydifference is that now we need the fact that the linear space of W k –symmetric Laurent polynomialson k variables is dense in the space of continuous, W k –symmetric functions on the k -dimensionaltorus T k . Indeed note that, under the change of variables z ∈ T → [ − , ∋ x := ( z + z − ) / 2, the W k –symmetric Laurent polynomials become usual symmetric polynomials. Our statement is thenequivalent to the fact that symmetric polynomials are dense on the space of continuous, symmetricfunctions on [ − , k , and this follows from Weierstrass approximation theorem. (cid:3) Proposition 6.2. Take any y ∈ Y . Let { λ ( N ) ∈ GT + N } N ≥ be a sequence that stabilizes to y .Then for each k ≥ µ ∈ GT + k , the limit M y k ( µ ) = lim N →∞ Λ Nk ( λ ( N ) , µ )exists and does not depend on the choice of { λ ( N ) } N ≥ . For each k ≥ M y k is a probabilitymeasure on GT + k and, moreover, { M y k ∈ M ( GT + k ) } k ≥ is a coherent system. -DEFORMED CHARACTER THEORY · · · Proof. Let { λ ( N ) ∈ GT + N } N ≥ be any sequence of nonnegative signatures that stabilizes to y ∈ Y ;also let k ∈ N be arbitrary. From the definition (4 . 6) of the Markov kernels Λ N +1 N , we have χ Gλ ( x , . . . , x k , q k + ǫ , . . . , q N − ǫ ) χ Gλ ( q ǫ , q ǫ , . . . , q N − ǫ ) = X µ ∈ GT + k Λ Nk ( λ ( N ) , µ ) χ Gµ ( x , . . . , x k ) χ Gµ ( q ǫ , . . . , q k − ǫ ) . (6.3)By virtue of Theorem 3 . 13, the left side of (6 . 3) converges to the analytic function Φ y ,G ( x , . . . , x k ; q ),on compact subsets of ( C ∗ ) k . The rest of the proof is similar to that of Proposition 5 . 2. One showsthat { Λ Nk ( λ ( N ) , µ ) } N ≥ k has a limit, for any µ ∈ GT + k , and moreover each limitlim N →∞ Λ Nk ( λ ( N ) , µ ) = χ Gµ ( q ǫ , q ǫ , . . . , q k − ǫ ) F Gµ (Φ y ,G ) =: a µ (6.4)does not depend on the stabilizing sequence of nonnegative signatures { λ ( N ) } N ≥ . One then shows a µ ≥ P µ ∈ GT + k a µ = 1, and therefore the measure M y k ∈ M ( GT + k ), M y k ( µ ) := a µ , satisfiesthe desired properties. The coherency of the sequence { M y k } k ≥ is evident. In working out theseoutlined steps, one should follow the proof of Proposition 5 . . . (cid:3) Concentration bound.Proposition 6.3. Let k ∈ N and let { λ ( N ) ∈ GT + N } N ≥ be an arbitrary sequence. Let µ ( N ) ∈ GT + N be random variables such that each µ ( N ) is distributed according to the probability distribu-tion Λ N +1 N ( λ ( N + 1) , · ). Then there exists a constant c > 0, independent of N , such thatProb ( µ ( N ) N +1 − i = λ ( N + 1) N +2 − i for all i = 1 , , . . . , k ) > − cq N . Proof. By Boole’s inequality, it suffices to show that for a fixed integer m ≥ 1, there exists aconstant c > 0, independent of N , such thatProb ( µ ( N ) N +1 − m = λ ( N + 1) N +2 − m ) > − cq N . (6.5)Let us do the case m = 1 and then remark on the small differences needed for the argument inthe general case m ≥ 2. For m = 1, it will suffice to show the existence of constants c , c > µ ( N ) N ≤ λ ( N + 1) N +1 − < c q N ; (6.6)Prob( µ ( N ) N ≥ λ ( N + 1) N +1 + 1) < c q N . (6.7)Denote λ ( N + 1) by λ , and µ ( N ) by µ .We begin with (6 . λ N +1 = 0, then Prob( µ N ≤ λ N +1 − 1) = 0 because µ N ≥ λ N +1 ≥ 1. For each 0 ≤ b < λ N +1 , we want to bound the ratioProb( µ N = b )Prob( µ N = b + 1) (6.8)and show it is of order O ( q N ). It is convenient to define a measure M N ( ·| λ ) on GT + N +1 × GT + N via M N ( ν, µ | λ ) := { µ ≺ ν ≺ λ } χ Gµ ( q ǫ , q ǫ , . . . , q N − ǫ ) χ Gλ ( q ǫ , q ǫ , . . . , q N + ǫ ) ( q N + ǫ ) | ν |−| λ |−| µ | · τ G ( q N + ǫ ; λ, ν, µ ) . The formula for τ G ( q N + ǫ ; λ, ν, µ ) is in the statement of Proposition 2 . 5. Using Proposition 2 . 9, andthe fact that 0 < q < 1, we deduce that M N ( ·| λ ) is a positive measure. From Proposition 2 . 5, wealso deduce that the pushforward of M N ( ·| λ ), under the projection π : GT + N +1 × GT + N → GT + N , isΛ N +1 N ( λ, · ). In particular, M N ( ·| λ ) is a probability measure. So instead of estimating the ratio (6 . N +1 N ( λ, · ), we estimate it with respect to the probabilitymeasure M N ( ·| λ ) and show that it is of order O ( q N ). We can do a further simplification. Instead of considering M N ( ·| λ ), let us consider arbitrarynonnegative integers ν , . . . , ν N and µ , . . . , µ N − , such that λ i ≥ ν i ≥ λ i +1 , ν j ≥ µ j ≥ ν j +1 , forrelevant i, j , and let us estimate the ratio (6 . 8) with respect to M N ( ·| λ ) conditioned on ν , . . . , ν N , µ , . . . , µ N − . For any 0 ≤ e ≤ d ≤ λ N +1 (and with respect to this probability measure), we getProb( µ N = d ) = Prob( µ N = d, ν N +1 = d ) + . . . + Prob( µ N = d, ν N +1 = 0) , Prob( µ N = d, ν N +1 = e ) ∝ q ( N + ǫ )(2 e − d ) τ G ( q N + ǫ ; λ, ν, µ ) χ Gµ ( q ǫ , q ǫ , . . . , q N − ǫ );the proportionality constant in the last equation does not depend on d, e .If G = B , τ B ( q N + ǫ ; λ, ν, µ ) = 1 + q − N − ǫ = 1 + q − N − , unless ν N +1 = 0 in which case τ B ( q N + ǫ ; λ, ν, µ ) = 1. Moreover, χ Bµ ( q , q , . . . , q N − ) ∝ q − d (1 − q d + ) N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d +1 ) . As a result, we obtainProb( µ N = d, ν N +1 = e ) ∝ q ( N + )(2 e − d ) − d (1 − q d + )(1 + q − N − − { e =0 } q − N − ) × N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d +1 ) , and thereforeProb( µ N = d ) ∝ q − ( N + ) d − d (1 − q d + ) N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d +1 ) × (1 + q − N − ) 1 − q (2 N +1)( d +1) − q N +1 − q − N − ! , for G = B. If G = C , τ C = 1 always, and χ Cµ ( q, q , . . . , q N ) ∝ q − d (1 − q d +1) ) N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d +2 ) . As a result, we haveProb( µ N = d ) ∝ q − ( N +1) d − d (1 − q d +1) ) N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d +2 ) × − q (2 N +2)( d +1) − q N +2 ! , for G = C. If G = D , τ D = 0 unless ν N +1 ∈ { , µ N } (at least in this case that we assume µ N ≤ λ N +1 ) and τ D = 1 in those two cases. Moreover, χ Dµ (1 , q, . . . , q N − ) ∝ N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d ) . As a result, we haveProb( µ N = d ) ∝ q − Nd (1 + q Nd ) N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d ) , for G = D. -DEFORMED CHARACTER THEORY · · · Then, for any 0 ≤ b < λ N +1 , the ratio Prob( µ N = b ) / Prob( µ N = b + 1) equals q N +1 − q b + − q b + N − Y i =1 (1 − q µ i + N − i − b )(1 − q µ i + N − i + b +1 )(1 − q µ i + N − i − b − )(1 − q µ i + N − i + b +2 ) × − q (2 N +1)( b +1) − q (2 N +1)( b +2) , for G = B ; q N +2 − q b +1) − q b +2) N − Y i =1 (1 − q µ i + N − i − b )(1 − q µ i + N − i + b +2 )(1 − q µ i + N − i − b − )(1 − q µ i + N − i + b +3 ) × − q (2 N +2)( b +1) − q (2 N +2)( b +2) , for G = C ; q N q Nb q N ( b +1) N − Y i =1 (1 − q µ i + N − i − b )(1 − q µ i + N − i + b )(1 − q µ i + N − i − b − )(1 − q µ i + N − i + b +1 ) , for G = D. By following the same analysis as in the proof of Lemma 5 . 4, we can upper bound all threeexpressions above by (the exponentially small) 2 q N / ( q ; q ) ∞ . Let us give, as an example, the detailsfor the type G = B . Use the bounds 0 < − q b + < − q b + , 1 − q (2 N +1)( b +1) < − q (2 N +1)( b +2) and (1 − q µ i + N − i − b )(1 − q µ i + N − i + b +1 ) < ≤ i ≤ N − 1, to obtainProb( µ N = b )Prob( µ N = b + 1) ≤ q N +1 Q N − i =1 (1 − q µ i + N − i − b − )(1 − q µ i + N − i + b +2 ) . (6.9)Since µ ≥ µ ≥ · · · ≥ µ N − , then µ + N − b − > · · · > µ i + N − i − b − > · · · > µ N − − b ≥ µ N − − ( λ N +1 − ≥ µ + N + b + 1 > · · · > µ i + N − i + b + 2 > · · · > µ N − + b + 3 ≥ . So (6 . 9) implies Prob( µ N = b )Prob( µ N = b + 1) ≤ q N +1 q ; q ) ∞ ( q ; q ) ∞ ≤ q N ( q ; q ) ∞ , for type G = B. Similar bounds can be achieved for types C, D . Recall that the probabilities we are dealing with are M N ( ·| λ ) after conditioning over values for ν , . . . , ν N , µ , . . . , µ N − . But the bound achieved didnot take these values into consideration, therefore Prob( µ N = b ) / Prob( µ N = b + 1) < q N / ( q ; q ) ∞ ,if µ N is M N ( ·| λ )-distributed. The bound just shown holds for any 0 ≤ b ≤ λ N +1 . This implies theexistence of a desired constant c > . . . λ N ≥ λ N +1 + 1,since otherwise Prob( µ N = λ N +1 + 1) = 0. For each λ N − ≥ b ≥ λ N +1 , we want to show the ratioProb ( µ N = b + 1)Prob ( µ N = b ) (6.10)is of order O ( q N ). Define the measure M ′ N ( ·| λ ) on GT + N +1 × GT + N via M ′ N ( ν, µ | λ ) := { µ ≺ ν ≺ λ } χ Gµ ( q ǫ , q ǫ , . . . , q N − ǫ ) χ Gλ ( q ǫ , q ǫ , . . . , q N + ǫ ) ( q N + ǫ ) | λ | + | µ |− | ν | · τ G ( q − ( N + ǫ ) ; λ, ν, µ ) . As before, Proposition 2 . . 5, see also Remark 2 . 6, show that M ′ N ( ·| λ ) is aprobability measure whose pushforward under the projection GT + N +1 × GT + N → GT + N is Λ N +1 N ( λ, · ).As we did before, instead of estimating the ratio (6 . 10) with respect to the probability measureΛ N +1 N ( λ, · ), we estimate it when (in both the numerator and denominator) µ is distributed accordingto M ′ N ( ·| λ ) after conditioning over fixed values for ν , ν , . . . , ν N − , ν N +1 and µ , . . . , µ N − . For any λ N +1 ≤ d ≤ e ≤ λ N , with respect to this probability measure, we getProb( µ N = d ) = Prob( µ N = d, ν N = d ) + . . . + Prob( µ N = d, ν N = λ N ) , Prob( µ N = d, ν N = e ) ∝ q ( N + ǫ )( d − e ) τ G ( q − ( N + ǫ ) ; λ, ν, µ ) χ Gµ ( q ǫ , q ǫ , . . . , q N − ǫ ) . (6.11) We know that τ G ( q − ( N + ǫ ) ; λ, ν, µ ) depends only on λ N +1 , ν N +1 and µ N , so let us denote it τ GN ( λ N +1 , ν N +1 , µ N ).The factor τ GN ( λ N +1 , ν N +1 , d ) appears in each term of the first line of (6 . . 9, we obtain that Prob( µ N = d ) isproportional (up to a nonzero factor that does not depend on d ) to: τ BN ( λ N +1 , ν N +1 , d ) · q ( N + )( d − λ N ) − d (1 − q d + ) N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d +1 ) × − q (2 N +1)( λ N − d +1) − q N +1 ! , for G = B ; τ CN ( λ N +1 , ν N +1 , d ) · q ( N +1)( d − λ N ) − d (1 − q d +1) ) N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d +2 ) × − q (2 N +2)( λ N − d +1) − q N +2 ! , for G = C ; τ DN ( λ N +1 , ν N +1 , d ) · q N ( d − λ N ) N − Y i =1 (1 − q µ i + N − i − d )(1 − q µ i + N − i + d ) × − q N ( λ N − d +1) − q N ! , for G = D. Next we note that for any λ N − ≥ b ≥ λ N +1 , we have τ GN ( λ N +1 , ν N +1 , b ) = τ GN ( λ N +1 , ν N +1 , b + 1),except in the case that G = D , b = 0, in which case τ DN ( λ N +1 , ν N +1 , 0) = 1, τ DN ( λ N +1 , ν N +1 , 1) = 2.Therefore the ratio Prob( µ N = b + 1) / Prob( µ N = b ) equals q N (1 − q b + )(1 − q (2 N +1)( λ N − b ) )(1 − q b + )(1 − q (2 N +1)( λ N − b +1) ) N − Y i =1 (1 − q µ i + N − i − b − )(1 − q µ i + N − i + b +2 )(1 − q µ i + N − i − b )(1 − q µ i + N − i + b +1 ) , for G = B ; q N (1 − q b +2) )(1 − q N ( λ N − b ) )(1 − q b +1) )(1 − q N ( λ N − b +1) ) N − Y i =1 (1 − q µ i + N − i − b − )(1 − q µ i + N − i + b +3 )(1 − q µ i + N − i − b )(1 − q µ i + N − i + b +2 ) , for G = C ; q N (1 + { b =0 } ) 1 − q N ( λ N − b ) − q N ( λ N − b +1) N − Y i =1 (1 − q µ i + N − i − b − )(1 − q µ i + N − i + b +1 )(1 − q µ i + N − i − b )(1 − q µ i + N − i + b ) , for G = D. Just as we have done already a few times, we can upper bound all these three expressions by cq N / ( q ; q ) ∞ , for some constant c > 0; the bound is uniform on the values ν , . . . , ν N − , ν N +1 , µ , . . . , µ N − that we conditioned over. Therefore, if µ N is M ′ N ( ·| λ )-distributed, we also have thatProb ( µ N = b + 1) / Prob ( µ N = b ) is upper bounded by cq N / ( q ; q ) ∞ , for any λ N − ≥ b ≥ λ N +1 .The bound (6 . 7) follows.The general case m ≥ . 5) is very similar. It suffices to show the existence of constants c , c > µ ( N ) N +1 − m ≤ λ ( N + 1) N +2 − m − < c q N ; (6.12)Prob( µ ( N ) N +1 − m ≥ λ ( N + 1) N +2 − m + 1) < c q N . (6.13)Denote λ ( N + 1) by λ , and µ ( N ) by µ . For (6 . λ N +2 − m > λ N +3 − m , sinceotherwise Prob( µ N +1 − m ≤ λ N +2 − m − 1) = 0. Then the inequality can be deduced if we show that,for each λ N +3 − m ≤ b < λ N +2 − m , the ratioProb( µ N +1 − i = b )Prob( µ N +1 − i = b + 1) -DEFORMED CHARACTER THEORY · · · is of order O ( q N ). As before, it suffices to consider µ N with law M N ( ·| λ ) and to show the laststatement with respect to this probability measure. In fact, we can even condition over any valuesof ν , . . . , ν N +1 − m , ν N +3 − m , . . . , ν N +1 and µ , . . . , µ N − m , µ N +2 − m , . . . , µ N such that λ i ≥ ν i ≥ λ i +1 , ν j ≥ µ j ≥ ν j +1 , for relevant i, j . For any λ N +3 − m ≤ e ≤ d ≤ λ N +2 − m , we useProb( µ N +1 − m = d ) = Prob( µ N +1 − m = d, ν N +2 − m = d ) + . . . + Prob( µ N +1 − m = d, ν N +2 − m = 0) , Prob( µ N +1 − m = d, ν N +2 − m = e ) ∝ q ( N + ǫ )(2 e − d ) τ G ( q N + ǫ ; λ, ν, µ ) χ Gµ ( q ǫ , q ǫ , . . . , q N − ǫ ) . Then we can repeat the analysis above to prove the desired bound. The only difference is that now τ G ( q N + ǫ ; λ, ν, µ ) does not depend on µ N +1 − m or on ν N +2 − m (it only depends on λ N +1 , ν N +1 , µ N ),so we could even use insteadProb( µ N +1 − m = d, ν N +2 − m = e ) ∝ q ( N + ǫ )(2 e − d ) χ Gµ ( q ǫ , q ǫ , . . . , q N − ǫ ) , which makes the remaining steps of the proof of (6 . 12) even simpler. For the proof of (6 . τ G ( q N + ǫ ; λ, ν, µ ) does not play a role in the proof. (cid:3) Law of Large Numbers and the Martin boundary.Theorem 6.4 (Law of Large Numbers) . Let k ∈ N be arbitrary, let y ∈ Y , and { M y N } N ≥ be thecoherent system of Proposition 6 . 2. Let { µ ( N ) ∈ GT + N } N ≥ be a sequence of random nonnegativesignatures such that each µ ( N ) is M y N -distributed. Then, for any k ∈ N , the probability of theevent µ ( L ) L +1 − k = y k tends to 1, as L goes to infinity. Proof. 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