Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A
aa r X i v : . [ m a t h . QA ] N ov TWISTED TRACES AND POSITIVE FORMS ONQUANTIZED KLEINIAN SINGULARITIES OF TYPE A
PAVEL ETINGOF, DANIIL KLYUEV, ERIC RAINS, AND DOUGLAS STRYKER
Abstract.
Following [BPR] and [ES], we undertake a detailed study oftwisted traces on quantizations of Kleinian singularities of type A n − .In particular, we give explicit integral formulas for these traces and usethem to determine when a trace defines a positive Hermitian form on thecorresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed in [BPR] in connectionwith 3-dimensional superconformal field theory. In particular, we confirmthe conjecture from [BPR] that for n ≤ a unitary short star-product isunique and compute its parameter as a function of the quantization pa-rameters, giving exact formulas for the numerical functions from [BPR].If n = 2 , this, in particular, recovers the theory of unitary sphericalHarish-Chandra bimodules for sl . Thus the results of this paper maybe viewed as a starting point for a generalization of the theory of uni-tary Harish-Chandra bimodules over enveloping algebras of reductive Liealgebras ([V]) to more general quantum algebras. Finally, we derive re-currences to compute the coefficients of short star-products correspondingto twisted traces, which are generalizations of discrete Painlev´e systems. To Vitaly Tarasov and Alexander Varchenko with admiration
Contents
1. Introduction 22. Filtered quantizations and twisted traces 52.1. Filtered quantizations 52.2. Even quantizations 72.3. Quantizations with a conjugation and a quaternionic structure. 72.4. Twisted traces 92.5. The formal Stieltjes transform 103. An analytic construction of twisted traces 11
The work of P. E. was also partially supported by the NSF grant DMS-1502244. P ( x ) satisfy | Re α | < . 113.2. Relation to orthogonal polynomials 133.3. Conjugation-equivariant traces 163.4. Construction of traces when all roots of P ( x ) satisfy | Re α | ≤ . 163.5. Twisted traces in the general case 174. Positivity of twisted traces 194.1. Analytic lemmas 194.2. The case when all roots of P ( x ) satisfy | Re α | < . 204.3. The case of a closed strip 304.4. The general case 325. Explicit computation of the coefficients a k , b k of the 3-termrecurrence for orthogonal polynomials and discrete Painlev´esystems 35References 431. Introduction
The notion of a short star-product for a filtered quantization A of a hy-perK¨ahler cone was introduced by Beem, Peelaers and Rastelli in [BPR] mo-tivated by the needs of 3-dimensional superconformal field theory (under thename “star-product satisfying the truncation condition"); this is an algebraicincarnation of non-holomorphic SU (2) -symmetry of such cones. Roughlyspeaking, these are star-products which have fewer terms than expected (infact, as few as possible). The most important short star-products are non-degenerate ones, i.e., those for which the constant term CT( a ∗ b ) of a ∗ b defines a nondegenerate pairing on A = gr A . Moreover, physically the mostinteresting ones among them are those for which an appropriate Hermitianversion of this pairing is positive definite; such star-products are called uni-tary . Namely, short star-products arising in 3-dimensional SCFT happen tobe unitary, which is a motivation to take a closer look at them.In fact, in order to compute the parameters of short star-products arisingfrom 3-dimensional SCFT, in [BPR] the authors attempted to classify unitaryshort star-products for even quantizations of Kleinian singularities of type A n − for n ≤ . Their low-degree computations suggested that in these casesa unitary short star-product should be unique for each quantization. Whilethe A case is easy (as it reduces to the representation theory of SL ), in WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 3 the A case the situation is already quite interesting. Namely, in this casean even quantization depends on one parameter κ , and Beem, Peelaers andRastelli showed that (at least in low degrees) short star-products for sucha quantization are parametrized by another parameter α . Moreover, theycomputed numerically the function α ( κ ) expressing the parameter of theunique unitary short star-product on the parameter of quantization ([BPR],Fig. 2), but a formula for this function (even conjectural) remained unknown.These results were improved upon by Dedushenko, Pufu and Yacoby in[DPY], who computed the short star-products coming from 3-dimensionalSCFT in a different way. This made the need to understand all nondegenerateshort star-products and in particular unitary ones less pressing for physics,but it remained a very interesting mathematical problem.Motivated by [BPR], the first and the last author studied this problem in[ES]. There they developed a mathematical theory of nondegenerate shortstar-products and obtained their classification. As a result, they confirmedthe conjecture of [BPR] that such star-products exist for a wide class of hy-perK¨ahler cones and are parametrized by finitely many parameters. Themain tool in this paper is the observation, due to Kontsevich, that nonde-generate short star-products correspond to nondegenerate twisted traces onthe quantized algebra A , up to scaling. The reason this idea is effective isthat traces are much more familiar objects (representing classes in the zerothHochschild homology of A ), and can be treated by standard techniques ofrepresentation theory and noncommutative geometry. However, the specificexample of type A n − Kleinian singularities and in particular the classifica-tion of unitary short star-products was not addressed in detail in [ES].The goal of the present paper is to apply the results of [ES] to this ex-ample, improving on the results of [BPR]. Namely, we give an explicit clas-sification of nondegenerate short star-products for type A n − Kleinian singu-larities, expressing the corresponding traces of weight elements (i.e., poly-nomials P ( z ) in the weight zero generator z ) as integrals R i R P ( x ) w ( x ) | dx | of P ( x ) against a certain weight function w ( x ) . As a result, the correspond-ing quantization map sends monomials z k to p k ( z ) , where p k ( x ) are monicorthogonal polynomials with weight w ( x ) which belong to the class of semi-classical orthogonal polynomials . If n = 1 , or n = 2 with special parameters,they reduce to classical hypergeometric orthogonal polynomials, but in gen-eral they do not. We also determine which of these short star-products areunitary, confirming the conjecture of [BPR] that for even quantizations of A n − , n ≤ a unitary star product is unique. Moreover, we find the exact P. ETINGOF, D. KLYUEV, E. RAINS, AND D. STRYKER formula for the function α ( κ ) whose graph is given in Fig. 2 of [BPR]: α ( κ ) = 14 − κ + − cos( π q κ + ) . In particular, this recovers the value α ( − ) = − π predicted in [BPR] andconfirmed in [DPY, DFPY].It would be very interesting to develop a similar theory of positive tracesfor higher-dimensional quantizations, based on the algebraic results of [ES].It would also be interesting to extend this analysis from the algebra A tobimodules over A (e.g., Harish-Chandra bimodules, cf. [L]). Finally, itwould be interesting to develop a q -analogue of this theory. These topics arebeyond the scope of this paper, however, and are subject of future research.For instance, the q -analogue of our results for Kleinian singularities of typeA will be worked out by the second author in a forthcoming paper [K2]. Remark 1.1.
We show in Example 4.10 that for n = 2 the theory of positivetraces developed here recovers the classification of irreducible unitary spher-ical representations of SL ( C ) ([V]). Moreover, this can be extended to thenon-spherical case if we consider traces on Harish-Chandra bimodules overquantizations (with different parameters on the left and the right, in general)rather than just quantizations themselves. One could expect that a similartheory for higher-dimensional quantizations, in the special case of quotientsof U ( g ) by a central character (i.e., quantizations of the nilpotent cone) wouldrecover the theory of unitary representations of the complex reductive group G with Lie algebra g . This suggests that the theory of positive traces on fil-tered quantizations of hyperK¨ahler cones may be viewed as a generalizationof the theory of unitary Harish-Chandra bimodules for simple Lie algebras.A peculiar but essential new feature of this generalization (which may scareaway classical representation theorists), is that a given simple bimodule mayhave more than one Hermitian (and even more than one unitary) structureup to scaling (namely, unitary structures form a cone, often of dimension > ), and that a bimodule which admits a unitary structure need not besemisimple. Remark 1.2.
The second author studied the existence of unitary star-products for type A n − Kleinian singularities in [K1] and obtained a partialclassification of quantizations that admit a unitary star-product. That pa-per also contains examples of non-semisimple unitarizable bimodules. The
WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 5 present paper has stronger results: it contains a complete description of theset of unitary star-products for any type A n − Kleinian singularity.The organization of the paper is as follows. Section 2 is dedicated tooutlining the algebraic theory of filtered quantizations and twisted traces forKleinian singularities of type A, following [ES]. In Section 3 we introduce ourmain analytic tools, representing twisted traces by contour integrals againsta weight function. In this section we also use this weight function to studythe orthogonal polynomials arising from twisted traces. In Section 4, usingthe analytic approach of Section 3, we determine which twisted traces arepositive. In particular, we confirm the conjecture of [BPR] that a positivetrace is unique up to scaling for n ≤ (for the choice of conjugation as in[BPR]), and find the exact dependence of the parameter of the positive traceon the quantization parameters for n = 3 and n = 4 , which was computednumerically in [BPR]. Finally, in Section 5 we discuss the problem of explicitcomputation of the coefficients a k , b k of the 3-term recurrence for the orthog-onal polynomials arising from twisted traces, which appear as coefficients ofthe corresponding short star-product. Since these orthogonal polynomials aresemiclassical, these coefficients can be computed using non-linear recurrenceswhich are generalizations of discrete Painlev´e systems. Acknowledgements.
The work of P. E. was partially supported by theNSF grant DMS-1502244. P. E. is grateful to Anton Kapustin for introducinghim to the topic of this paper, and to Chris Beem, Mykola Dedushenko andLeonardo Rastelli for useful discussions. E.R. would like to thank NicholasWitte for pointing out the reference [M].2.
Filtered quantizations and twisted traces
Filtered quantizations.
Let X n be the Kleinian singularity of type A n − . Recall that A := C [ X n ] = C [ p, q ] Z /n , where Z /n acts by p e πi/n p, q e − πi/n q . Thus A = C [ u, v, z ] / ( uv − z n ) , where u = p n , v = q n , z = pq. It is curious that, unlike classical representation theory, this dependence is given by atranscendental function.
P. ETINGOF, D. KLYUEV, E. RAINS, AND D. STRYKER
This algebra has a grading defined by the formulas deg( p ) = deg( q ) = 1 ,thus(2.1) deg( u ) = deg( v ) = n, deg( z ) = 2 . The Poisson bracket is given by { p, q } = n and on A takes the form { z, u } = − u, { z, v } = v, { u, v } = nz n − . Also recall that filtered quantizations A of A are generalized Weyl algebras ([B]) which look as follows. Let P ∈ C [ x ] be a monic polynomial of degree n . Then A = A P is the algebra generated by u, v, z with defining relations [ z, u ] = − u, [ z, v ] = v, vu = P ( z − ) , uv = P ( z + ) and filtration defined by (2.1). Thus we have [ u, v ] = P ( z + ) − P ( z − ) = nz n − + ..., i.e., the quasiclassical limit indeed recovers the algebra A with the abovePoisson bracket.Note that we may consider the algebra A P for a polynomial P that isnot necessarily monic. However, we can always reduce to the monic case byrescaling u and/or v . Also by transformations z z + β we can make surethat the subleading term of P is zero, i.e., P ( x ) = x n + c x n − + ... + c n . Thus the quantization A depends on n − essential parameters (the roots of P , which add up to zero).The algebra A decomposes as a direct sum of eigenspaces of ad z : A = ⊕ k ∈ Z A k . If b ∈ A m , we will say that b has weight m . The weight decomposition of A can be viewed as a C × -action; namely, for t ∈ C × let g t = t ad z : A → A bethe automorphism of A given by g t ( v ) = tv, g t ( u ) = t − u, g t ( z ) = z. Then g t ( b ) = t m b if b has weight m . WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 7
Example 2.1.
1. Let n = 1 , P ( x ) = x . Then A is the Weyl algebragenerated by u, v with [ u, v ] = 1 , and z = vu + = uv − .2. Let n = 2 and P ( x ) = x − C . Then setting e = v , f = − u , h = 2 z ,we get [ h, e ] = 2 e, [ h, f ] = − f, [ e, f ] = h, f e = − ( h +12 ) + C, i.e., A is the quotient of the universal enveloping algebra U ( sl ) by the rela-tion f e + ( h +12 ) = C , where f e + ( h +12 ) is the Casimir element.2.2. Even quantizations.
Let s be the automorphism of A given by s ( u ) = ( − n u, s ( v ) = ( − n v, s ( z ) = z ; in other words, we have s = g ( − n . Thus gr s : A → A equals ( − d ,where d is the degree operator. Recall ([ES], Subsection 2.3) that a filteredquantization A is called even if it is equipped with an antiautomorphism σ such that σ = s and gr σ = i d , and that σ is unique if exists ([ES], Remark2.10). This means that σ ( z ) = − z , σ ( u ) = i n u , σ ( v ) = i n v . It is easy to seethat σ exists if and only if ( − n P ( z − ) = ( − n vu = σ ( v ) σ ( u ) = σ ( uv ) = σ ( P ( z + )) = P ( − z + ) . This is equivalent to P ( − x ) = ( − n P ( x ) , i.e., P contains only terms x n − i . Thus even quantizations of A are parametrizedby [ n/ essential parameters, and all quantizations for n ≤ are even.2.3. Quantizations with a conjugation and a quaternionic structure.
Recall ([ES], Subsection 3.6) that a conjugation on A is an antilinear filtrationpreserving automorphism ρ : A → A that commutes with s . We will considerconjugations on A given by(2.2) ρ ( v ) = λu, ρ ( u ) = λ ∗ v, ρ ( z ) = − z, where λ, λ ∗ ∈ C × . The automorphism u γ − u , v γv rescales λ by | γ | − and λ ∗ by | γ | , so we may assume that | λ | = | λ ∗ | = 1 , i.e., λ = ± i − n e − πic , P. ETINGOF, D. KLYUEV, E. RAINS, AND D. STRYKER where c ∈ [0 , . Then P ( − z + ) = ρ ( P ( z + )) = ρ ( uv ) = ρ ( u ) ρ ( v ) = λλ ∗ vu = λλ ∗ P ( z − ) , i.e., P ( − x ) = λλ ∗ P ( x ) . Thus λ ∗ = ( − n λ − = ± i − n e πic and P ( − x ) = ( − n P ( x ) , i.e., i n P is real on i R . We also have ρ ( u ) = λ ∗ λu, ρ ( v ) = λλ ∗ v, ρ ( z ) = z, so ρ = g t , where t = ( − n λλ − = ( − n λ − = e πic , i.e., | t | = 1 . Thus we see that for every t there are two non-equivalentconjugations, corresponding to the two choices of sign for λ , which we denoteby ρ + and ρ − .In particular, consider the special case t = ( − n , i.e., g t = s . Then c = for n odd and c = 0 for n even. Thus λ = ± , so the conjugation ρ on A isgiven by ρ ( v ) = ± u, ρ ( u ) = ± ( − n v, ρ ( z ) = − z. Now assume in addition that A is even, i.e., P ( − x ) = ( − n P ( x ) . Thenwe have ρσ = σ − ρ , so ρ and σ give a quaternionic structure on A (cf. [ES],Subsection 3.7). So this quaternionic structure exists if and only if P ∈ R [ x ] , P ( − x ) = ( − n P ( x ) . Example 2.2.
Let n = 2 , so A is the quotient of the enveloping algebra U ( g ) , g = sl , by the relation f e + ( h +1) = C , where C ∈ R . Since e = v, f = − u, h = 2 z, we have ρ ± ( e ) = ± f, ρ ± ( f ) = ± e, ρ ± ( h ) = − h. So g + := g ρ + has basis x = e + f , y = i ( e − f )2 , z = ih . Thus, [ x , y ] = − z , [ z , x ] = y , [ y , z ] = x . Hence, setting E := y − z , F := y + z , H := 2 x , we have [ H, E ] = 2 E, [ H, F ] = − F, [ E, F ] = H, so g + = sl ( R ) .On the other hand, g − := g ρ − has basis i x , i y , z , hence g − = so ( R ) = su .So ρ + and ρ − correspond to the split and compact form of g , respectively. WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 9
Twisted traces.
Let A = A P be a filtered quantization of A . Recall([ES], Subsection 3.1) that a g t - twisted trace on A is a linear map T : A → C such that T ( ab ) = T ( bg t ( a )) . It is shown in [ES], Section 3, that ( s -invariant)nondegenerate twisted traces, up to scaling, correspond to ( s -invariant) non-degenerate short star-products on A .Let us classify g t -twisted traces T on A . The answer is given by thefollowing proposition. Proposition 2.3. T : A → C is a g t -twisted trace on A if and only if(1) T ( A j ) = 0 for j = 0 .(2) T ( S ( z − ) P ( z − )) = tT ( S ( z + ) P ( z + )) for all S ∈ C [ x ] .In particular, any twisted trace is automatically s -invariant.Proof. Suppose T satisfies (1),(2). It is enough to check that T ( ub ) = t − T ( bu ) , T ( vb ) = tT ( bv ) , T ( zb ) = T ( bz ) for b ∈ A .The equality T ( zb ) = T ( bz ) says that T ( A j ) = 0 for j = 0 , which iscondition (1).By (1), it is enough to check the equality T ( ub ) = t − T ( bu ) for b ∈ A − .In this case b = vS ( z + ) for some polynomial S . We have T ( ub ) = T ( uvS ( z + )) = T ( P ( z + ) S ( z + )) ,T ( bu ) = T ( vS ( z + ) u ) = T ( vuS ( z − )) = T ( P ( z − ) S ( z − )) , which yields the desired identity using (2).Similarly, it is enough to check the equality T ( vb ) = tT ( bv ) for b ∈ A .In this case b = uS ( z − ) . We have T ( vb ) = T ( vuS ( z − )) = T ( P ( z − ) S ( z − )) ,T ( bv ) = T ( uS ( z − ) v ) = T ( uvS ( z + )) = T ( P ( z + ) S ( z + )) , which again gives the desired identity using (2).Conversely, the same argument shows that if T is a g t -twisted trace then(1),(2) hold. (cid:3) Thus we get
Corollary 2.4.
The space of g t -twisted traces on A is naturally isomorphicto the space (cid:0) C [ z ] / { S ( z − ) P ( z − ) − tS ( z + ) P ( z + ) | S ∈ C [ z ] } (cid:1) ∗ and has dimension n if t = 1 and dimension n − if t = 1 . The formal Stieltjes transform.
There is a useful characterizationof the space g t -twisted traces in terms of generating functions. Given a linearfunctional T on C [ z ] , its formal Stieltjes transform is the generating function F T ( x ) := X n ≥ x − n − T ( z n ) ∈ C [[ x − ]] , or equivalently F T ( x ) = T (( x − z ) − ) , with ( x − z ) − itself expanded as aformal power series in x − . Proposition 2.5.
The formal Stieltjes transform of a g t -twisted trace on A satisfies P ( x )( F T ( x + ) − tF T ( x − )) ∈ C [ x ] , and this establishes an isomorphism of the space of g t -twisted traces with thespace of polynomials of degree ≤ n − (for t = 1 ) or ≤ n − (for t = 1 ).Proof. We may write P ( x )( F T ( x + ) − tF T ( x − ))= T (cid:18) P ( x ) x + − z − t P ( x ) x − − z (cid:19) = T (cid:18) P ( z − ) x + − z − t P ( z + ) x − − z (cid:19) + T (cid:18) P ( x ) − P ( z − ) x − ( z − ) − t P ( x ) − P ( z + ) x − ( z + ) (cid:19) . In the final expression, the second term is the image under T of a polynomialin z and x , while the first term expands as X n ≥ x − n − T ( P ( z − )( z − ) n − tP ( z + )( z + ) n ) = 0 . Since the map F F ( x + 1 / − tF ( x − / is injective on x − k [[1 /x ]] , thisestablishes an injective map from g t -twisted traces to polynomials of degree WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 11 < deg( P ) . This establishes the conclusion for t = 1 , while for t = 1 , weneed simply observe that for any F ∈ x − C [[ x − ]] , F T ( x + ) − F T ( x − ) ∈ x − C [[ x − ]] , and thus the polynomial has degree < deg( P ) − . (cid:3) Remark 2.6.
It is easy to see that F ( x ) F ( x + ) − tF ( x − ) actstriangularly on x − C [[ x − ]] , of degree (with nonzero leading coefficients) if t = 1 and degree − (ditto) if t = 1 , letting one see directly that there is aunique solution of P ( x )( F ( x + ) − tF ( x − )) = R ( x ) for any polynomial R satisfying the degree constraint. Remark 2.7.
A similar argument establishes an isomorphism between lin-ear functionals satisfying T ( P ( q − z ) S ( q − z ) − qtP ( q z ) S ( q z )) = 0 andelements F ∈ x − C [[ x − ]] such that P ( x )( F T ( q x ) − tF T ( q − x )) ∈ C [ x ] , or, for t = 1 , between linear functionals satisfying T ( z − ( P ( q − z ) S ( q − z ) − P ( q z ) S ( q z ))) = 0 and formal series satisfying P ( x ) x − ( F T ( q x ) − F T ( q − x )) ∈ C [ x ] . An analytic construction of twisted traces
Construction of twisted traces when all roots of P ( x ) satisfy | Re α | < . As before, let t = exp(2 πic ) where ≤ Re c < (clearly, such c exists and is unique).Let P ( x ) = Q nj =1 ( x − α j ) . Define P ( X ) := n Y j =1 ( X + e πiα j ) . When P ( x ) satisfies the equation P ( − x ) = ( − n P ( x ) (the condition forexistence of a conjugation ρ ) the polynomial P ( X ) has real coefficients.Assume that every root α of P ( x ) satisfies | Re α | < . Also suppose firstthat t does not belong to R > \ { } , i.e., Re c ∈ (0 , or c = 0 . Then we canconstruct all g t -twisted traces in the following way. Proposition 3.1.
Every g t -twisted trace is given by T ( R ( z )) = Z i R R ( x ) w ( x ) | dx | , where w is the weight function defined by the formula w ( x ) = w ( c, x ) := e πicx G ( e πix ) P ( e πix ) , where G is a polynomial of degree ≤ n − and G (0) = 0 if c = 0 .Proof. It is easy to see that the function w ( x ) enjoys the following properties:(1) w ( x + 1) = tw ( x ) ;(2) | w ( x ) | decays exponentially and uniformly when Im x tends to ±∞ ;(3) w ( x + ) P ( x ) is holomorphic when | Re x | ≤ .Let T ( R ( z )) := R i R R ( x ) w ( x ) | dx | . We should check that T ( tS ( z + ) P ( z + ) − S ( z − ) P ( z − )) = 0 . We have T ( tS ( z + ) P ( z + ) − S ( z − ) P ( z − )) = Z i R tS ( x + ) P ( x + ) w ( x ) | dx | − Z i R S ( x − ) P ( x − ) w ( x ) | dx | = Z + i R tS ( x ) P ( x ) w ( x − ) | dx | − Z i R − S ( x ) P ( x ) w ( x + ) | dx | = Z + i R S ( x ) P ( x ) w ( x + ) | dx | − Z i R − S ( x ) P ( x ) w ( x + ) | dx | = Z ∂ ([ − , × R ) S ( x ) P ( x ) w ( x + ) | dx | . But this integral vanishes by the Cauchy theorem since S ( x ) P ( x ) w ( x + ) isholomorphic when | Re x | ≤ and decays exponentially as Im x → ± i ∞ .It is easy to see that the space of polynomials G ( X ) has the same dimen-sion as the space of g t -twisted traces, so we have described all traces. (cid:3) WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 13
Now consider the remaining case t ∈ R \ { } , i.e., c ∈ i R \ { } . In thiscase the function w ( x ) does not decay at + i ∞ , so the integral in Proposition3.1 is not convergent. However, we can write the formula for T ( R ( z )) asfollows, so that it makes sense in this case: T ( R ( z )) = lim δ → Z i R R ( x ) w ( c + δ, x ) | dx | . Alternatively, one may say that T ( R ( z )) is the value of the Fourier transformof the distribution R ( − iy ) w (0 , − iy ) at the point ic (it is easy to see that thisFourier transform is given by an analytic function outside of the origin). Wethen have the following easy generalization of Proposition 3.1: Proposition 3.2.
With this modification, Proposition 3.1 is valid for all t . Consider now the special case of even quantizations. Recall ([ES], Sub-section 3.3) that nondegenerate even short star-products on A correspond tonondegenerate s -twisted σ -invariant traces T on various even quantizations A of A , up to scaling. So let us classify such traces. As shown above, s -twisted traces T correspond to w ( x ) such that w ( x + 1) = ( − n w ( x ) . Also,it is easy to see that such T is σ -invariant if and only if T ( R ( z )) = T ( R ( − z )) .We have T ( R ( − z )) = Z i R R ( − x ) w ( x ) | dx | = Z i R R ( x ) w ( − x ) | dx | . So T is σ -invariant of and only if w ( x ) = w ( − x ) . Thus we have the followingproposition. Proposition 3.3.
Suppose that A is an even quantization of A . Then s -twisted σ -invariant traces T are given by the formula T ( R ( z )) = Z i R R ( x ) w ( x ) | dx | , where w is as in Proposition 3.1 and w ( x ) = w ( − x ) = ( − n w ( x + 1) . Relation to orthogonal polynomials.
Let φ : A → A be the quan-tization map defined by a trace T (see [ES], Section 3). Then φ ( z k ) = p k ( z ) ,where p k are monic orthogonal polynomials for the inner product ( f , f ) ∗ := Z i R f ( x ) f ( x ) w ( x ) | dx | . Recall ([Sz]) that these polynomials satisfy a 3-term recurrence p k +1 ( x ) = ( x − b k ) p k ( x ) − a k p k − ( x ) , for some numbers a k , b k , i.e., xp k ( x ) = p k +1 ( x ) + b k p k ( x ) + a k p k − ( x ) . Thus the corresponding short star-product z ∗ z k has the form z ∗ z k = φ − ( φ ( z ) φ ( z k )) = φ − ( zp k ( z )) == φ − ( p k +1 ( z ) − ib k p k ( z ) − a k p k − ( z )) = z k +1 + b k z k + a k z k − . Thus the numbers a k , b k are the matrix elements of multiplication by z inweight for the short star-product attached to T . More general matrixelements of multiplication by u, v, z for this short star-product are computedsimilarly. In other words, to compute the short star-product attached to T ,we need to compute explicitly the coefficients a k , b k and their generalizations.This problem is addressed in Section 5.It is more customary to consider orthogonal polynomials on the real(rather than imaginary) axis, so let us make a change of variable x = − iy .Then we see that the monic polynomials P k ( y ) := i k p k ( − iy ) are orthogonalunder the inner product ( f , f ) := Z ∞−∞ f ( y ) f ( y )w( y ) dy, where w( y ) := w ( − iy ) . Then the 3-term recurrence looks like P k +1 ( y ) = ( y − ib k ) P k ( y ) + a k P k − ( y ) (so for real parameters we’ll have a k ∈ R , b k ∈ i R ). Example 3.4.
Let n = 1 , P ( x ) = x , so P ( X ) = X + 1 . Then a nonzerotwisted trace exists if and only if c = 0 , in which case it is unique up toscaling, and the corresponding weight function is w ( x ) = e πicx e πix + 1 = e πi ( c − ) x πx , w( y ) = e π ( c − ) y πy . The corresponding orthogonal polynomials P k ( y ) are the (monic) Meixner-Pollaczek polynomials with parameters λ = , φ = πc ([KS], Subsection1.7). WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 15
Example 3.5.
Let n = 2 , P ( x ) = x + β , so P ( X ) = ( X + e πβ )( X + e − πβ ) . The space of twisted traces is -dimensional if c = 0 and -dimensional if c = 0 . So for c = 0 the traces up to scaling are defined by the weightfunction w ( x ) = e πi ( c − ) x cos π ( x − iα )2 cos π ( x − iβ ) cos π ( x + iβ ) , w( y ) = e π ( c − ) y cosh π ( y − α )2 cosh π ( y − β ) cosh π ( y + β ) , and the limiting cases α → ±∞ along the real axis, which yield w ( x ) = e πi ( c − ± ) x π ( x − iβ ) cos π ( x + iβ ) , w( y ) = e π ( c − ± ) y π ( y − β ) cosh π ( y + β ) . These formulas for the plus sign also define the unique up to scaling tracefor c = 0 ; i.e., w ( x ) = 14 cos π ( x − iβ ) cos π ( x + iβ ) , w( y ) = 14 cosh π ( y − β ) cosh π ( y + β ) . In this case, the corresponding orthogonal polynomials P k ( y ) are the con-tinuous Hahn polynomials with parameters + iβ, − iβ, − iβ, + iβ ([KS], Subsection 1.4).Also for c = , α = 0 we have w ( x ) = cos πx π ( x + iβ ) cosh π ( x − iβ ) , w( y ) = cosh πy π ( y + β ) cosh π ( y − β ) , so P k ( y ) are the continuous dual Hahn polynomials with a = 0 , b = − iβ , c = + iβ ([KS], Subsection 1.3). Remark 3.6.
In Example 3.4 ( n = 1 ), the only even short star-productcorresponds to w( y ) =
12 cosh πy . This is the Moyal-Weyl star-product. InExample 3.5 ( n = 2 ), the only even short star-product corresponds to w( y ) =
14 cosh π ( y − β ) cosh π ( y + β ) . This is the unique SL -invariant star-product. Example 3.7.
Let t = ( − n , G ( X ) = X [ n/ . Then w ( x ) = n Y j =1
12 cos π ( x − iβ j ) , w( y ) = n Y j =1
12 cosh π ( y + β j ) . which defines an s -twisted trace. The corresponding orthogonal polynomialsare semiclassical but not hypergeometric for n ≥ . Remark 3.8.
The trace of Example 3.7 corresponds to the short star-product arising in the 3-d SCFT, as shown in [DPY], Subsection 8.1.2. Therethe Kleinian singularity of type A n − appears as the Higgs branch, and the pa-rameters β j are the FI parameters. The same trace also shows up in [DFPY],(5.27), where the Kleinian singularity appears as the Coulomb branch, andthe parameters β j are the mass parameters. Conjugation-equivariant traces.
Let now ρ be a conjugation on A (Subsection 2.3). Let us determine which g t -twisted traces are ρ -equivariant(see [ES], Subsection 3.6). A trace T is ρ -equivariant if T ( R ( z )) = T ( R ( − z )) ,which is equivalent to T being real on R [ iz ] . This happens if and only if w ( x ) is real on i R . Since w is meromorphic this means that w ( x ) = w ( − x ) .So we have the following proposition. Proposition 3.9.
Suppose that A is a quantization of A with conjugation ρ .Then ρ -equivariant g t -twisted traces T on A are given by T ( R ( z )) = Z i R R ( x ) w ( x ) | dx | , where w is as in Proposition 3.1 and w ( x ) = w ( − x ) = ( − n w ( x + 1) . Moreover, if A is even then σ -invariant traces among them correspond to thefunctions w with w ( x ) = w ( − x ) . Construction of traces when all roots of P ( x ) satisfy | Re α | ≤ . From now on we suppose that i n P ( x ) is real on i R (so that the conjugations ρ ± are well defined). In particular, the roots of P ( x ) are symmetric withrespect to i R .Suppose that for all roots α of P ( x ) we have | Re α | ≤ , and let us give aformula for twisted traces in this case. There are unique monic polynomials P ∗ ( x ) , Q ( x ) such that P ( x ) = P ∗ ( x ) Q ( x + ) Q ( x − ) , all roots of P ∗ ( x ) belong to the strip | Re x | < and all roots of Q ( x ) belong to i R . Supposethat α , . . . , α k are the roots of P ∗ ( x ) and α k +1 , . . . , α m are the roots of Q ( x + ) . Note that deg Q = n − m . Let P ∗ ( X ) = Q mj =1 ( X + e πiα j ) , w ( x ) = e πicx G ( e πix ) P ∗ ( e πix ) , where G ( X ) is a polynomial of degree at most m − and G (0) = 0 when t = 1 . We have We thank Mykola Dedushenko for this explanation.
WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 17 (1) w ( x + 1) = tw ( x ) ;(2) w ( x ) Q ( x ) is bounded on i R and decays exponentially and uniformlywhen Im x tends to ±∞ .(3) w ( x + ) P ( x ) is holomorphic on | Re x | ≤ .For any R ∈ C [ x ] let R ( x ) = R ( x ) Q ( x ) + R ( x ) , where deg R < deg Q . Proposition 3.10.
A general g t -twisted trace on A has the form T ( R ( z )) = Z i R R ( x ) Q ( x ) w ( x ) | dx | + φ ( R ) , where w ( x ) is as above and φ is any linear functional.Proof. The space of polynomials G has dimension m − δ c , while the spaceof linear functionals φ has dimension deg Q = n − m . So the space of suchlinear functionals T has dimension n − δ c . The space of all g t -twisted traceshas the same dimension, so it is enough to prove that all linear functionals T of this form are g t -twisted traces. In other words, we should prove that T ( S ( z − ) P ( z − ) − tS ( z + ) P ( z + )) = 0 for all S ∈ C [ x ] .We see that S ( x − ) P ( x − ) − tS ( x + ) P ( x + ) is divisible by Q ( x ) ,so T ( S ( z − ) P ( z − ) − tS ( z + ) P ( z + )) = Z i R ( S ( x − ) P ( x − ) − tS ( x + ) P ( x + )) w ( x ) | dx | Since w ( x + ) P ( x ) is holomorphic on | Re x | ≤ , we deduce that this integralis zero similarly to the proof of Proposition 3.1 (cid:3) Twisted traces in the general case.
Let m ( α ) be the multiplicity of α as a root of P ( x ) . Any linear functional φ on the space C [ x ] /P ( x ) C [ x ] canbe written as φ ( S ) = P α, ≤ i Since φ i,α are linearly independent for different i, α , we deduce that Φ = 0 if and only if T is a trace for P ◦ .So we have proved the following proposition: Proposition 3.11. Suppose that P is any polynomial, e P is obtained from P by the minimal integer shift of roots into the strip | Re x | ≤ , and P ◦ isobtained from P by throwing out roots not in the strip | Re x | ≤ . Then anytwisted trace T on A P can be represented as T = Φ + e T , where Φ( R ) = X a/ ∈ i R ,k ≥ c ak R ( k ) ( a ) , and e T is a trace for e P . Furthermore, if Φ = 0 then T is a trace for P ◦ . Remark 3.12. We may think about Proposition 3.11 as follows. When theroots of P lie inside the strip | Re x | < , the trace of R ( z ) is given bythe integral of R against the weight function w along the imaginary axis.However, when we vary P , as soon as its roots leave the strip | Re x | < ,poles of w start crossing the contour of integration. So for the formula toremain valid, we need to add the residues resulting from this. These residuesgive rise to the linear functional Φ .4. Positivity of twisted traces Analytic lemmas. We will use the following classical result: Lemma 4.1. Suppose that w ( x ) ≥ is a measurable function on the realline such that w ( x ) < ce − b | x | for some c, b > . Then polynomials are densein the space L p ( R , w ( x ) dx ) for all ≤ p < ∞ .Proof. Changing x to bx we can asssume that b = 1 .Fix p . Let p + q = 1 . Since L p ( R , w ) ∗ = L q ( R , w ) , it suffices to show thatany function f ∈ L q ( R , w ) such that R R f ( x ) x n w ( x ) dx = 0 for all nonnegativeintegers n must be zero.Choose < a < p . We have e a | x | ∈ L p ( R , w ) . Therefore f ( x ) e a | x | w ( x ) ∈ L ( R ) . Denote f ( x ) w ( x ) by F ( x ) . Let b F be the Fourier transform of F .Since F ( x ) e a | x | ∈ L ( R ) , b F extends to a holomorphic function in the strip | Im x | < a .Since R R f ( x ) x n w ( x ) dx = 0 , we have R R F ( x ) x n dx = 0 , so b F ( n ) (0) = 0 .Since b F is a holomorphic function and all derivatives of b F at the origin are zero, we deduce that b F = 0 . Therefore F = 0 , so f = 0 as an element of L q ( R , w ) , as desired. (cid:3) We get the following corollaries: Lemma 4.2. (1) Suppose that H ( x ) is a continuous complex-valued func-tion on R with finitely many zeros and at most polynomial growth atinfinity. Then the set { H ( x ) S ( x ) | S ( x ) ∈ C [ x ] } is dense in the space L p ( R , w ) .(2) Suppose that M ( x ) is a nonzero polynomial nonnegative on the realline. Then the closure of the set { M ( x ) S ( x ) S ( x ) | S ( x ) ∈ C [ x ] } in L p ( R , w ) is the subset of almost everywhere nonnegative functions.Proof. (1) The function w ( x ) | H ( x ) | p satisfies the assumptions of Lemma 4.1.Therefore polynomials are dense in the space L p ( R , w | H | p ) . The map g gH is an isometry between L p ( R , w | H | p ) and L p ( R , w ) . Thestatement follows.(2) Suppose that f ∈ L p ( R , w ) is positive almost everywhere. Then √ f is an element L p ( R , w ) . Using (1), we find a sequence S n ∈ C [ x ] suchthat √ M S n tends to √ f in L p ( R , w ) . It is then easy to deduce fromthe Cauchy-Schwarz inequality that M S n S n tends to f in L p ( R , w ) .The statement follows. (cid:3) The case when all roots of P ( x ) satisfy | Re α | < . Let A be afiltered quantization of A with conjugations ρ ± such that ρ ± = g t . We wantto classify positive definite Hermitian ρ ± -invariant forms on A , i.e. positivedefinite Hermitian forms ( · , · ) on A such that ( aρ ( y ) , b ) = ( a, yb ) for all a, b, y ∈ A , where ρ = ρ ± .It is easy to see that Hermitian ρ -invariant forms are in one-to-one cor-respondence with g t -twisted ρ -invariant traces. The correspondence is asfollows: ( a, b ) = T ( aρ ( b )) , T ( a ) = ( a, . Therefore it is enough to classify g t -twisted traces T such that the Hermitianform ( a, b ) = T ( aρ ( b )) is positive definite. This means that T ( aρ ( a )) > for WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 21 all nonzero a ∈ A . Recall that ad z acts on A diagonalizably, A = ⊕ d ∈ Z A d .Thus it is enough to check the condition T ( aρ ( a )) > for homogeneous a . Lemma 4.3. (1) T gives a positive definite form if and only if one has T ( aρ ( a )) > for all nonzero a ∈ A of weight or .(2) T gives a positive definite form if and only if T ( R ( z ) R ( − z )) > and λT ( R ( z − ) R ( − z ) P ( z − )) > for all nonzero R ∈ C [ x ] .Proof. (1) Suppose that T ( aρ ( a )) > for all nonzero a ∈ A of weight or . Let a be a nonzero homogeneous element of A with positiveweight. There exists b of weight or and nonnegative integer k suchthat a = v k bv k . In the first case we have T ( aρ ( a )) = λ k T ( v k bv k u k ρ ( b ) u k ) = λ k T ( g − t ( u k ) v k bv k u k ρ ( b )) = λ k t k T ( u k v k bv k u k ρ ( b )) =( − nk T ( u k v k bv k u k ρ ( b )) = T ( u k v k bρ ( u k v k b )) > since u k v k b is a homogeneous element of weight or .Suppose that a is a nonzero homogeneous element of A with negativeweight. Then a = ρ ( b ) , where b is a homogeneous element withpositive weight. We get T ( aρ ( a )) = T ( ρ ( b ) ρ ( b )) = T ( ρ ( b ) g t ( b )) = T ( bρ ( b )) > . (2) Suppose that a is an element of A . Then a = R ( z ) for some R ∈ C [ x ] .We have T ( aρ ( a )) = T ( R ( z ) R ( − z )) .Suppose that a is an element of A . Then a = R ( z − ) v for some R ∈ C [ x ] . We have T ( aρ ( a )) = λT ( R ( z − ) vR ( − z − ) u ) = λT ( R ( z − ) vuR ( − z + )) = λT ( R ( z − ) R ( − z ) P ( z − )) . The statement follows. (cid:3) Proposition 4.4. Suppose that T ( R ( z )) = R i R R ( x ) w ( x ) | dx | . Then T givespositive definite form if and only if w ( x ) and λw ( x + ) P ( x ) are nonnegativeon i R .Proof. By Lemma 4.3 T gives positive definite form if and only if T ( R ( z ) R ( − z )) > and λT ( R ( z − ) R ( − z ) P ( z − )) > for all nonzero R ∈ C [ x ] . A polynomial S ∈ C [ x ] can be represented as S ( x ) = R ( x ) R ( − x ) if and only if S is nonnegative on i R . So we have T ( R ( z ) R ( − z )) > for all nonzero R ∈ C [ x ] if and only if Z i R S ( x ) w ( x ) | dx | > for all nonzero S ∈ C [ x ] nonnegative on i R . Using Lemma 4.2(2) for M = 1 ,we see that this is equivalent to w ( x ) being nonnegative on i R .We have T ( R ( z − ) R ( − z ) P ( z − )) = Z i R R ( x − ) R ( − x ) P ( x − ) w ( x ) | dx | = Z + i R R ( x ) R ( − x ) P ( x ) w ( x + ) | dx | = Z i R R ( x ) R ( − x ) P ( x ) w ( x + ) | dx | . In the last equality we used the Cauchy theorem and the fact that the function P ( x ) w ( x + ) is holomorphic when | Re x | ≤ . Using Lemma 4.2(2) for M = 1 again, we see that λT ( R ( z − ) R ( − z ) P ( z − )) > for all nonzero R ∈ C [ x ] if and only if λP ( x ) w ( x + ) is nonnegative on i R . (cid:3) Proposition 4.5. (1) If λ = − i − n e − πic (i.e., ρ = ρ − ) then w ( x ) and λP ( x ) w ( x + ) are nonnegative on i R if and only if G ( X ) is nonneg-ative when X > and nonpositive when X < .(2) If λ = + i − n e − πic (i.e., ρ = ρ + ) then w ( x ) and λw ( x + ) P ( x ) arenonnegative on i R if and only if G ( X ) is nonnegative for all X ∈ R .Proof. Recall that w ( x ) = e πicx G ( e πix ) P ( e πix ) . It is easy to see that P ( X ) is positivewhen X > . Therefore w ( x ) is nonnegative on i R if and only if G ( X ) isnonnegative when X > . WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 23 We have λP ( x ) w ( x + ) = ± i − n P ( x ) e πicx G ( − e πix ) P ( − e πix ) . It is clear that i − n P ( x ) P ( − e πix ) belongs to R when x ∈ i R and does not change signon i R . When x tends to − i ∞ , the functions i − n P ( x ) and P ( − e πix ) havesign ( − n . Therefore i − n P ( x ) P ( e − πix ) is positive on i R . We deduce that ± G ( X ) should be nonnegative when X < . So there are two cases:(1) If λ = − i − n e − πic then G ( X ) should be nonnegative when X > andnonpositive when X < .(2) If λ = + i − n e − πic then G ( X ) should be nonnegative for all X ∈ R .This proves the proposition. (cid:3) We deduce the following theorem from Propositions 4.4 and 4.5: Theorem 4.6. Suppose that A is a deformation of A = C [ p, q ] Z /n with con-jugation ρ as above, ρ = g t , t = exp(2 πic ) . Let P ( x ) be the parameter of A , ε = i n e πic λ = ± (so ρ = ρ ε ). Then the cone C + of positive definite ρ -invariant forms on A is isomorphic to the cone of nonzero polynomials G ( X ) of degree ≤ n − with G (0) = 0 if c = 0 such that(1) If ε = − then G ( X ) is nonnegative when X > and nonpositivewhen X < .(2) If ε = 1 then G ( X ) is nonnegative for all X ∈ R . Thus for ρ = ρ − , G ( X ) = XU ( X ) where U ( X ) ≥ is a polynomial ofdegree ≤ n − , and for ρ = ρ + , G ( X ) ≥ is a polynomial of degree ≤ n − with G (0) = 0 if c = 0 ; in the latter case G ( X ) = X U ( X ) where U ( X ) ≥ is a polynomial of degree ≤ n − . Therefore, we get Proposition 4.7. The dimension of C + modulo scaling is • n − for even n and n − for odd n if ρ = ρ − ; • n − for even n and n − for odd n if c = 0 and ρ = ρ + ; • n − for even n and n − for odd n if c = 0 and ρ = ρ + .(Here if the dimension is < , the cone C + is empty). Consider now the special case of even short star-products (i.e., quater-nionic structures). Let A be an even quantization of A , and C even+ the coneof positive σ -stable s -twisted traces (i.e., those defining even short star-products). Then we have Proposition 4.8. The dimensions of C even+ modulo scaling in various casesare as follows: • n − if ρ = ρ − , n odd; • n − if ρ = ρ + , n odd; • n − if ρ = ρ − , n even; • n − if ρ = ρ + , n even. Proposition 4.8 shows that the only cases of a unique positive σ -stable s -twisted trace are ρ = ρ + for n = 1 , and ρ = ρ − for n = 2 , .The paper [BPR] considers the case ρ = ρ + if n = 0 , mod and ρ = ρ − if n = 2 , mod ; this is the canonical quaternionic structure of thehyperK¨ahler cone (see [ES], Subsection 3.8), since it is obtained from ρ + on C [ p, q ] by restricting to Z /n -invariants. Thus for n ≤ the unitary evenstar-product is unique, as conjectured in [BPR]. However, for n ≥ thisis no longer so. For example, for n = 5 (a case commented on at the endof section 6 of [BPR]) by Proposition 4.8 the cone C even+ modulo scaling is2-dimensional (which disproves the most optimistic conjecture of [BPR] thata unitary even star-product is always unique). Example 4.9. Let n = 1 , P ( x ) = x , so P ( X ) = X + 1 . Then for ρ = ρ − there are no positive traces while for ρ = ρ + positive traces exist only if c = 0 . In this case there is a unique positive trace up to scaling given by theweight function w( y ) = e πi ( c − ) y πy . In particular, the only quaternionic case is ρ = ρ + , c = , which gives w( y ) = 12 cos πy . Example 4.10. Let n = 2 , P ( x ) = x + β , β ∈ R so we have P ( X ) =( X + e πβ )( X + e − πβ ) . We assume that β > − so that all roots of P arein the strip | Re x | < . Then ρ = ρ − gives a unique up to scaling positive It is curious that in the case considered in [BPR], the dimension of C even+ modulo scalingis always even. WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 25 trace defined by the weight function w( y ) = e πcy π ( y − β ) cos π ( y + β ) . and ρ = ρ + is possible if and only if c = 0 and gives a unique up to scalingpositive trace defined by the weight function w( y ) = e π ( c − y π ( y − β ) cos π ( y + β ) . In particular, the only quaternionic case is ρ = ρ − , c = 0 , with w( y ) = 14 cos π ( y − β ) cos π ( y + β ) , which corresponds to the SL -invariant short star-product. There are twosubcases: β ≥ , which corresponds to the spherical unitary principal series for SL ( C ) , and − < β < , which corresponds to the spherical unitarycomplementary series for the same group (namely, the trace form is exactlythe positive inner product on the underlying Harish-Chandra bimodule).Note that together with the trivial representation (corresponding to β = − ), these representations are well known to exhaust irreducible sphericalunitary representations of SL ( C ) ([V]). Example 4.11. Let n = 3 and P ( x ) = x + β x = x ( x − iβ )( x + iβ ) , where β ∈ R . This gives the algebra defined by formulas (6.17),(6.18) of [BPR],with ζ = 1 ; namely, the generators ˆ X, ˆ Y , ˆ Z of [BPR] are v, u, z , respectively,and the parameter κ of [BPR] is κ = − β − . This is an even quantizationof A = C [ X ] . Thus even short star-products are parametrized by a singleparameter α ; namely, the corresponding σ -invariant s -twisted trace such that T (1) = 1 is determined by the condition that T ( z ) = − α (using the notationof [BPR]).Assume that β > − (i.e., κ < ), so that all the roots of P are in thestrip | Re x | < . We have P ( X ) = ( X + 1)( X + e πβ )( X + e − πβ ) . In this case c = so the trace T , up to scaling, is given by T ( R ( z )) = Z i R R ( x ) w ( x ) | dx | , where w ( x ) = e πix G ( e πix )( e πix + 1)( e πi ( x − iβ ) + 1)( e πi ( x + iβ ) + 1) , with deg( G ) ≤ . Moreover, because of evenness we must have w ( x ) = w ( − x ) , so G ( X ) = X G ( X − ) . Up to scaling, such polynomials G form a1-parameter family, parametrized by α .Following [BPR], Subsection 6.3, let us equip the corresponding quantumalgebra A = A P with the quaternionic structure ρ − given by ρ − ( v ) = − u, ρ − ( u ) = v, ρ − ( z ) = − z, and let us determine which traces are unitary for this quaternionic structure.According to Theorem 4.6, there is a unique such trace (which is automati-cally σ -stable), corresponding to G ( X ) = X . Thus this trace is given by theweight function w ( x ) = 1cos πx cos π ( x − iβ ) cos π ( x + iβ ) , Hence, T ( z k ) = Z i R x k | dx | cos πx cos π ( x − iβ ) cos π ( x + iβ ) , in particular, T ( z k ) = 0 if k is odd.For even k this integral can be computed using the residue formula.Namely, assume β = 0 and let us first compute T (1) . Replacing the contour i R by i R and subtracting, we find using the residue formula: T (1) = 2 π (Res w + Res − iβ w + Res + iβ w ) . Now, Res w = 1 π sinh πβ , while Res − iβ w = Res + iβ w = − π sinh πβ sinh 2 πβ . Thus T (1) = 1sinh πβ − πβ sinh 2 πβ = Note that our ρ is ρ − in [BPR], so we use ρ − while [BPR] use ρ + = ρ − − . WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 27 πβ (cid:18) − πβ (cid:19) = 12 cosh ( πβ ) cosh πβ . Note that this function has a finite value at β = 0 , which is the answer inthat case.Now let us compute T ( z ) . Again replacing the contour i R with i R and subtracting, we get T (1)+2 T ( z ) = T ( z )+ T (( z +1) ) = 2 π (Res x w +Res − iβ x w +Res + iβ x w ) . Now, Res x w = 14 π sinh πβ , while Res − iβ x w + Res + iβ x w = 2 β − π sinh πβ sinh 2 πβ . So T ( z ) = − 14 sinh πβ + 2 β + sinh πβ sinh 2 πβ =1sinh πβ (cid:18) − 14 + β + cosh πβ (cid:19) . Thus, α = − T ( z ) T (1) = 14 + β − cosh πβ = 14 − κ + − cos π q κ + . This gives the equation of the curve in Fig. 2 in [BPR]. We also note thatfor κ = − (i.e., β = 0 ) we get α = − π . the value found in [BPR]. Example 4.12. Let n = 4 and P ( x ) = ( x + β )( x + γ ) = ( x − iβ )( x + iβ )( x − iγ )( x + iγ ) , where β , γ ∈ R . This is an even quantization of A = C [ X ] discussed in[BPR], Subsection 6.4. Thus even short star-products are still parametrizedby a single parameter α ; namely, the corresponding σ -invariant s -twistedtrace such that T (1) = 1 is determined by the condition that T ( z ) = − α . Assume that β , γ > − , so that all the roots of P are in the strip | Re x | < . We have P ( X ) = ( X + e πβ )( X + e − πβ )( X + e πγ )( X + e − πγ ) . In this case c = 0 so the trace T , up to scaling, is given by T ( R ( z )) = Z i R R ( x ) w ( x ) | dx | , where w ( x ) = G ( e πix )( e πi ( x − iβ ) + 1)( e πi ( x + iβ ) + 1)( e πi ( x − iγ ) + 1)( e πi ( x + iγ ) + 1) , with deg( G ) ≤ and G (0) = 0 . Moreover, because of evenness we must have w ( x ) = w ( − x ) , so G ( X ) = X G ( X − ) . Up to scaling, such polynomials G form a 1-parameter family, parametrized by α .Let us equip the corresponding quantum algebra A = A P with the quater-nionic structure ρ + given by ρ + ( v ) = u, ρ + ( u ) = v , ρ + ( z ) = − z , and let usdetermine which traces are unitary for this quaternionic structure. Accord-ing to Theorem 4.6, there is a unique such trace (which is automatically σ -stable), corresponding to G ( X ) = X . Thus this trace is given by theweight function w ( x ) = 1cos π ( x − iβ ) cos π ( x + iβ ) cos π ( x − iγ ) cos π ( x + iγ ) . Thus, T ( z k ) = Z i R x k | dx | cos π ( x − iβ ) cos π ( x + iβ ) cos π ( x − iγ ) cos π ( x + iγ ) , in particular, T ( z k ) = 0 if k is odd.As before, for even k this integral can be computed using the residueformula. Namely, assume β = 0 , γ = 0 , β = ± γ , and let us first compute T (1) . Replacing the contour i R by i R and subtracting, we find using theresidue formula: T (( z + 1) ) − T ( z ) = T (1) = − π (Res − iβ x w + Res + iβ x w + Res − iγ x w + Res + iγ x w ) . WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 29 Now, Res − iβ x w + Res + iβ x w = 2 βπ sinh π ( β + γ ) sinh π ( γ − β ) sinh 2 πβ . Thus T (1) = 1sinh π ( β + γ ) sinh π ( γ − β ) (cid:18) β sinh 2 πβ − γ sinh 2 πγ (cid:19) . Note that this function is regular when βγ ( β − γ )( β + γ ) = 0 , and thecorresponding limit is the answer in that case.We similarly have T (( z + 1) ) − T ( z ) = 6 T ( z ) + T (1) = − π (Res − iβ x w + Res + iβ x w + Res − iγ x w + Res + iγ x w ) , and Res − iβ x w + Res + iβ x w = β − β π sinh π ( β + γ ) sinh π ( γ − β ) sinh 2 πβ . Thus T ( z ) + T (1) = 2sinh π ( β + γ ) sinh π ( γ − β ) (cid:18) β − β sinh 2 πβ − γ − γ sinh 2 πγ (cid:19) . Hence T ( z ) = 1sinh π ( β + γ ) sinh π ( γ − β ) (cid:18) γ + 4 γ πγ − β + 4 β πβ (cid:19) . Thus α = − T ( z ) T (1) = 112 + 13 β sinh 2 πγ − γ sinh 2 πββ sinh 2 πγ − γ sinh 2 πβ . This is the equation (in appropriate coordinates) of the surface computednumerically in [BPR] and shown in Fig. 4 of that paper. In particular, for β = γ = 0 , we get α = 112 − π . Thus τ = 128 α = π − π = 4 . ... is the number given by the compli-cated expression (B.16) of [BPR] (as was pointed out in [DPY]). Remark 4.13. Similar calculations can be found in [DPY], Subsection 8.1. The case of a closed strip. Suppose now that all roots α of P satisfy | Re α | ≤ . Recall that we have P ( x ) = P ∗ ( x ) Q ( x + ) Q ( x − ) where allroots of P ∗ ( x ) satisfy | Re x | < and all roots of Q ( x ) belong to i R . For any R ∈ C [ x ] write R = R Q + R , where deg R < deg Q .By Proposition 3.10 each g t -twisted trace can be obtained as T ( R ( z )) = Z i R R ( x ) Q ( x ) w ( x ) | dx | + φ ( R ) , where w ( x ) = e πicx G ( e πix ) P ( e πix ) and φ is any linear functional. Proposition 4.14. Suppose that T is a trace as above and w ( x ) has poleson i R . Then T does not give a positive definite form.Proof. Let Q ∗ ( x ) = Q ( x ) Q ( − x ) ; note that Q ∗ ( x ) ≥ for x ∈ i R . Thenthere exists a linear functional ψ such that for any R = R Q ∗ + R with deg R < deg Q ∗ we have T ( R ( z )) = Z i R R ( x ) Q ∗ ( x ) w ( x ) | dx | + ψ ( R ) . Suppose that T gives a positive definite form. Then T ( S ( z ) S ( − z )) > for all nonzero S ∈ C [ x ] . Taking S ( x ) = Q ∗ ( x ) S ( x ) and using Lemma 4.2,we deduce that Q ∗ ( x ) w ( x ) , hence w ( x ) , is nonnegative on i R . In particular,all poles of w ( x ) have order at least .Without loss of generality assume that w ( x ) has a pole at zero. Let R n ( x ) := ( F n Q ∗ + b )( F n Q ∗ + b ) , where b ∈ R . Suppose that F n is a sequence of polynomials that tendsto the function f := χ ( − ε,ε ) (the characteristic function of the interval) inthe space L ( i R , ( Q ∗ + Q ∗ ) w ) . In particular, F n tends to f in the spaces L ( i R , Q ∗ w ) and L ( i R , Q ∗ w ) . Then we deduce from the Cauchy-Schwartzinequality that F n F n tends to f in the space L ( i R , Q ∗ w ) , and F n and F n tend to f in L ( i R , Q ∗ w ) .We have T ( R n ( z )) = T (( F n F n Q ∗ + F n b + F n b )( z ) Q ∗ ( z ) + b ) = Z i R ( F n F n Q ∗ + F n bQ ∗ + F n bQ ∗ ) w | dx | + φ ( b ) . WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 31 Therefore, when n tends to infinity, T ( R n ( z )) → Z i R ( f Q ∗ + 2 f bQ ∗ ) w | dx | + φ ( b ) . We have φ ( b ) = Cb for some C ≥ . Suppose that w has a pole of order M ≥ at and Q ∗ has a zero of order N > at . Then Q ∗ w has a zeroof order N − M at zero and Q ∗ w has a zero of order N − M at zero. Wededuce that Z i R F n Q ∗ w | dx | → c ε N − M +1 , Z i R F n Q ∗ w | dx | → c ε N − M +1 , n → ∞ , where c = c ( ε ) , c = c ( ε ) are functions having strictly positive limits at ε = 0 . Therefore lim n →∞ T ( R n ( z )) = Cb + 2 c ε N − M +1 b + c ε N − M +1 . This is a quadratic polynomial of b with discriminant D = 4 ε N − M +2 ( c − Cc ε M − ) . Since M ≥ , for small ε this discriminant is positive. In particular, forsome b , lim n →∞ T ( R n ( z )) < , so for this b and some n , T ( R n ( z )) < , acontradiction. (cid:3) Now we are left with the case when w ( x ) has no poles on i R . In this case T ( R ( z )) = R i R R ( x ) w ( x ) | dx | + η ( R ) , where η is some linear functional. Proposition 4.15. T gives a positive definite form only when η ( R ) = P j c j R ( z j ) , where c j ≥ and z j ∈ i R are the roots of Q .Proof. Suppose that this is not the case. Then it is easy to find a polynomial S such that η (( SS ) ) < . Then using Lemma 4.2(2) for M = Q , we find F n such that F n Q + S tends to zero in L ( i R , w ) . We deduce that T (( F n Q + S )( z )( F n Q + S )( z )) → η (( SS ) ) < , which gives a contradiction. (cid:3) In the proof of Proposition 4.14 we got that w is nonnegative on i R . Wealso note that Q ( z ) divides P ( z − )) , hence T ( R ( z − ) R ( − z ) P ( z − )) = Z i R R ( z − ) R ( − z ) P ( z − ) w ( z ) | dz | . Using the proof of Proposition 4.4 we see that λP ( z ) w ( z + ) is nonnegativeon i R . Assume that Q ( z − ) Q ( z + ) is positive on R . Then this is equivalentto λP ∗ ( z ) w ( z + ) being nonnegative on i R . So we have proved the followingtheorem. Theorem 4.16. Suppose that P ( x ) = P ∗ ( x ) Q ( x − ) Q ( x + ) , where all rootsof P ∗ belong to the set | Re x | < and all roots of Q belong to i R . Supposethat α , . . . , α k are all the different roots of Q . Then positive traces T are inone-to-one correspondence with e T , c , . . . , c k ≥ , where e T is a positive tracefor P ∗ ; namely, T ( R ( z )) = e T ( R ( z )) + X c i R ( α i ) . The general case. Let A is be a filtered quantization of A with con-jugation ρ given by formula (2.2). Let P ( x ) be its parameter. Let e P ( x ) bethe polynomial defined in Subsection 3.5: it has the same degree as P ( x ) and its roots are obtained from the roots of P ( x ) by minimal integer shiftinto the strip | Re x | ≤ . Also recall from Subsection 3.5 that P ◦ denotesthe following polynomial: all roots of P ◦ belong to strip | Re x | ≤ and themultiplicity of α, | Re α | ≤ in P ◦ equals to multiplicity of α in P . Let n ◦ := deg( P ◦ ) .Proposition 3.11 says that any trace T can be represented as T = Φ + e T ,where Φ is a linear functional such that Φ( R ) = m X j =1 X k c jk R ( k ) ( z j ) ,z j / ∈ i R , and e T is a trace for e P . Furthermore, if Φ = 0 then T is just a tracefor P ◦ . Proposition 4.17. Let T be a trace such that Φ = 0 . Then T does not givea positive definite form.Proof. For big enough k we have Φ(( x − z ) k · · · ( x − z m ) k C [ x ]) = 0 . Recallthat there exists polynomial Q ∗ ( x ) nonnegative on i R such that for R = WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 33 R Q ∗ + R , deg R < deg Q ∗ , we have e T ( R ) = R i R R Q ∗ w | dx | + ψ ( R ) , where ψ is some linear functional. Let U ( x ) be a polynomial divisible by Q ∗ suchthat Φ( U ( x ) C [ x ]) = 0 .Let L be any polynomial. Using Lemma 4.2 for M = U , we find asequence G n = U S n that tends to L in the space L ( i R , Q ∗ w | dx | ) . Wededuce that H n ( x ) := ( G n ( x ) − L ( x ))( G n ( − x ) − L ( − x )) tends to zero in L ( i R , Q ∗ w ) . We have e T ( H n ( z ) Q ∗ ( z )) = R i R H n ( x ) Q ∗ w | dx | . We concludethat e T ( H n ( z ) Q ∗ ( z )) = k H n k L ( i R ,Q ∗ w ) tends to zero when n tends to infinity.It follows that T ( H n ( z )) tends to Φ( Q ∗ ( x ) H n ( x )) = Φ( Q ∗ ( x ) L ( x ) L ( − x )) .Since H n is nonnegative on i R , we have T ( H n ( z )) > . Now we get acontradiction with Lemma 4.18. There exists F ( x ) ∈ C [ x ] such that Φ( Q ∗ ( x ) F ( x ) F ( − x )) < .Proof. Let r be the biggest number such that there exists j with c jr = 0 . Let F ( x ) := G ( x )( x − z ) r +1 · · · ( x − z j ) r · · · ( x − z m ) r +1 . Here we omit x − z j ∗ in the product, where j ∗ = j is such that z j ∗ = − z j .We note that c ik ( Q ∗ ( x ) F ( x ) F ( − x )) ( k ) ( z i ) = 0 for all i, k except k = r and i = j or i = j ∗ . It follows that Φ( Q ∗ ( x ) F ( x ) F ( − x )) = c jr ( Q ∗ ( x ) F ( x ) F ( − x )) ( r ) ( z j ) + c j ∗ r ( Q ∗ ( x ) F ( x ) F ( − x )) ( r ) ( z j ∗ ) = c jr Q ∗ ( z j ) F ( r ) ( z j ) F ( − z j ) + ( − r c j ∗ r Q ∗ ( z j ∗ ) F ( z j ∗ ) F ( r ) ( − z j ∗ ) = c jr a + c j ∗ r a, where a := Q ∗ ( z j ) F ( r ) ( z j ) F ( − z j ) . Pick a ∈ C so that c jr a + c j ∗ r a = 2Re( c jr a ) < , and choose G ∈ C [ x ] which gives this value of a (e.g., we can choose G to belinear). Then Φ( Q ∗ ( x ) F ( x ) F ( − x )) < , as desired. (cid:3)(cid:3) If A P ◦ is the quantization with parameter P ◦ then there is a conjugation ρ ◦ on A P ◦ given by the formulas ρ ◦ ( v ) = λ ◦ u, ρ ◦ ( u ) = ( − n λ − ◦ v, ρ ◦ ( z ) = − z, where λ ◦ := ( − n − n ◦ λ . Therefore we can consider the cone of positivedefinite forms for A P ◦ with respect to ρ ◦ . Corollary 4.19. The cone of positive definite forms on A P with respect to ρ coincides with the cone of positive definite forms on A P ◦ with respect to ρ ◦ .Namely, a trace T : C [ x ] → C for A gives a positive definite form if and onlyif T is a trace for A P ◦ that gives a positive definite form on A P ◦ .Proof. We deduce from Proposition 4.17 that each trace T that gives a pos-itive definite form should have Φ = 0 . By Proposition 3.11, in this case T is a trace for the polynomial P ◦ ( x ) . So there exists polynomial Q ∗ such thatfor R = R Q ∗ + R , deg R < deg Q ∗ , and T ( R ( z )) = Z i R Q ∗ ( x ) R ( x ) w ( x ) | dx | + φ ( R ) . Using Proposition 4.14 and its proof, we deduce that w has no poles andthat w ( x ) and λw ( x + ) P ( x ) are nonnegative on i R . Therefore T ( R ( z )) = Z i R R ( x ) w ( x ) | dx | + ψ ( R ) , where ψ is some linear functional. Using Proposition 4.15, we deduce thatthis trace is positive if and only if ψ ( R ) = P j c j R ( z j ) , where c j ≥ and z j ∈ i R .Since ( − n − n ◦ P ( x ) P ◦ ( x ) is positive on i R , we see that λw ( x + ) P ( x ) is non-negative on i R if and only if λ ◦ w ( x + ) P ◦ ( x ) is nonnegative on i R . UsingTheorem 4.16 we then deduce that T is positive for P ( x ) if and only if it ispositive for P ◦ ( x ) . (cid:3) So we have proved the following theorem. Theorem 4.20. Let A = A P be a filtered quantization of A with parameter P equipped with a conjugation ρ such that ρ = g t . Let ℓ be the numberof roots α of P such that | Re α | < counted with multiplicities and r bethe number of distinct roots α of P with Re α = − . Then the cone C + of ρ -equivariant positive definite traces on A is isomorphic to C × C , where C = R r ≥ , and C is the cone of nonzero polynomials G such that(1) G has degree less than ℓ .(2) G (0) = 0 if t = 1 .(3) G ( X ) ≥ when X > . WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 35 (4) G ( X ) is either nonnegative or nonpositive when X < depending onwhether ρ = ρ + or ρ − .The conditions are the same as in Theorem 4.6. Explicit computation of the coefficients a k , b k of the3-term recurrence for orthogonal polynomials anddiscrete Painlev´e systems As noted in Subsection 3.2, to compute the short star-product associ-ated to a trace T , one needs to compute the coefficients a k , b k of the 3-termrecurrence for the corresponding orthogonal polynomials: p k +1 ( x ) = ( x − b k ) p k ( x ) − a k p k − ( x ) . Also recall ([Sz]) that a k = ν k ν k − , where ν k := ( P k , P k ) . Finally, recall that ν k = D k D k − , where D k is the Gram determinant for , x, ..., x k − , i.e., D k = det ≤ i,j ≤ k − ( x i , x j ) = det ≤ i,j ≤ k − ( M i + j ) , where M r is the r -th moment of the weight function w ( x ) , i.e., M r = Z i R x r w ( x ) | dx | . In the even case w ( − x ) = w ( x ) we have b k = 0 , so p k +1 ( x ) = xp k ( x ) − a k p k − ( x ) , and p k can be easily computed recursively from the sequence a k . If the poly-nomials p k are q -hypergeometric (i.e., obtained by a limiting procedure fromAskey-Wilson polynomials), then D k , ν k , a k admit explicit product formulas,but in general they do not admit any closed expression and do not enjoy anynice algebraic properties beyond the above.In our case, the hypergeometric case only arises for n = 1 or, in specialcases, n = 2 , but the fact that the weight function for general n is essentiallya higher complexity version of the weight function for n = 1 suggests thatthere is still a weaker algebraic structure in the picture.In fact, by [M] it follows immediately from the fact that the formal Stielt-jes transform satisfies an inhomogeneous first-order difference equation with rational coefficients that the corresponding orthogonal polynomials p m ( x ) inthe x -variable satisfy a family of difference equations (cid:18) p m ( x + ) p m − ( x + ) (cid:19) = A m ( x ) (cid:18) p m ( x − ) p m − ( x − ) (cid:19) such that the matrix A m ( x ) has rational function coefficients of degree boundedby a linear function of n alone. (Here we work with the “ x ” version of thepolynomials, to avoid unnecessary appearances of i .)Since the results of [M] are stated in significantly more generality thanwe need, we sketch how they apply in our special case. Let Y be the matrix Y ( x ) = (cid:18) F ( x )0 1 (cid:19) , where F is the formal Stieltjes transform of the given trace. Moreover, foreach n , let q n ( x ) p n ( x ) be the n -th Pad´e approximant to F ( x ) (with monic denom-inator), so that q n ( x ) p n ( x ) − F ( x ) = O ( x − n − ) . If we define Y n ( x ) := (cid:18) p n ( x ) − q n ( x ) p n − ( x ) − q n − ( x ) (cid:19) Y ( x ) for n > , then Y n = (cid:18) x n + o ( x n ) O ( x − n − ) x n − + o ( x n − ) O ( x − n ) . (cid:19) Lemma 5.1. The denominator p n of the n -th Pad´e approximant to F ( x ) isthe degree n monic orthogonal polynomial for the associated linear functional T .Proof. If F = F T , then we find p n ( x ) F ( x ) = T (cid:18) p n ( x ) x − z (cid:19) = T (cid:18) p n ( x ) − p n ( z ) x − z (cid:19) + T (cid:18) p n ( z ) x − z (cid:19) (where we evaluate T on functions of z , and x is a parameter). The twoterms correspond to the splitting of p n ( x ) F ( x ) into its polynomial part andits part vanishing at x = ∞ , so that q n ( x ) = T (cid:18) p n ( x ) − p n ( z ) x − z (cid:19) WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 37 and T (cid:18) p n ( z ) x − z (cid:19) = p n ( x ) F ( x ) − q n ( x ) = O ( x − n − ) . Comparing coefficients of x − m − for ≤ m < n implies that T ( z m p n ( z )) = 0 as required. (cid:3) Remark 5.2. 1. It also follows that Y n ( x ) = N n x − n − + O ( x − n − ) , Y n ( x ) = N n − x − n + O ( x − n − ) . 2. Note that this is an algebraic/asymptotic version of the explicit so-lution of [BI] to the Riemann-Hilbert problem for orthogonal polynomialsintroduced in [FIK]. Lemma 5.3. We have det( Y n ) = N n − for all n > .Proof. The definition of Y n implies that det( Y n ) ∈ C [ x ] , while the (formal)asymptotic behavior implies that det( Y n ) = N n − + O ( x ) . (cid:3) The inhomogeneous difference equation satisfied by F trivially inducesan inhomogeneous difference equation satisfied by Y : Y ( x + ) = t − L ( x ) P ( x ) t − ! Y ( x − ) (cid:18) t (cid:19) where L ( x ) = P ( x )( F ( x + ) − tF ( x − )) ∈ C [ x ] . It follows immediately that Y n satisfies an analogous equation Y n ( x + ) = A n ( x ) Y n ( x − ) (cid:18) t (cid:19) , where A n ( x ) = (cid:16) p n ( x + ) − q n ( x + ) p n − ( x + ) − q n − ( x + ) (cid:17) (cid:16) t − L ( x ) P ( x ) t − (cid:17) (cid:16) p n ( x − ) − q n ( x − ) p n − ( x − ) − q n − ( x − ) (cid:17) − . Since det( Y n ) = N n − , det( Y ) = 1 , we can use the standard formula for theinverse of a × matrix to rewrite this as A n ( x ) = N − n − (cid:16) p n ( x + ) − q n ( x + ) p n − ( x + ) − q n − ( x + ) (cid:17) (cid:16) t − L ( x ) P ( x ) t − (cid:17) (cid:16) − q n − ( x − ) q n ( x − ) − p n − ( x − ) p n ( x − ) (cid:17) . It follows immediately that P ( x ) A n ( x ) has polynomial coefficients. We canalso compute the asymptotic behavior of A n ( x ) using the expression A n ( x ) = Y n ( x + ) (cid:18) t − (cid:19) Y n ( x − ) − to conclude that A n ( x ) = 1 + nx + O ( x ) A n ( x ) = − (1 − t − ) a n x + O ( x ) A n ( x ) = − t − x + O ( x ) A n ( x ) = t − (1 − nx ) + O ( x ) , which when t = 1 refines to A n ( x ) = 1 + nx + O ( x ) A n ( x ) = − (2 n +1) a n x + O ( x ) A n ( x ) = n − x + O ( x ) A n ( x ) = 1 − nx + O ( x ) . Restricting to the first column of Y n ( x ) gives the following. Proposition 5.4. The orthogonal polynomials satisfy the difference equation (cid:18) p n ( x + ) p n − ( x + ) (cid:19) = A n ( x ) (cid:18) p n ( x − ) p n − ( x − ) (cid:19) . Note that it is not the mere existence of a difference equation with ratio-nal coefficients that is significant (indeed, any pair of polynomials satisfiessuch an equation!), rather it is the fact that (a) the poles are bounded inde-pendently of n , and (b) so is the asymptotic behavior at infinity.If we consider (for t = 1 ) the family of matrices satisfying the aboveconditions; that is, P A n is polynomial, det( A n ) = t − , and A n ( x ) = 1 + nx + O (cid:0) x (cid:1) (5.1) A n ( x ) = O (cid:0) x (cid:1) (5.2) A n ( x ) = − t − x + O (cid:0) x (cid:1) (5.3) A n ( x ) = t − (cid:0) − nx (cid:1) + O (cid:0) x (cid:1) , (5.4) WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 39 we find that the family is classified by a rational moduli space. To be precise,let f ( x ) := (1 − t − ) − P ( x ) A n ( x ) , and let g ( x ) ∈ C [ x ] / ( f ( x )) be the reduc-tion of P ( x ) A n ( x ) modulo f ( x ) . Then f and g both vary over affine spacesof dimension deg( q ) − , and generically determine A n . Indeed, A n ( x ) isclearly determined by f , and since A n ( x ) P ( x ) is specified by the asymp-totics up to an additive polynomial of degree deg( P ) − , it is determined by f and g . For generic f , g , this also determines A n ( x ) , since the determinantcondition implies that for any root α of f , A n ( α ) A n ( α ) = t − . More-over, this constraint forces P ( x ) ( A n ( x ) A n ( x ) − t − ) to be a multiple of f ( x ) , and thus the unique value of A n ( x ) compatible with the determinantcondition gives a matrix satisfying the desired conditions.Moreover, given such a matrix, the three-term recurrence for orthogo-nal polynomials tells us that the corresponding A n +1 is the unique matrixsatisfying its asymptotic conditions and having the form A n +1 ( x ) = (cid:18) x + − b n − a n (cid:19) A n ( x ) (cid:18) x − − b n − a n (cid:19) − . It is straightforward to see that a n , b n are determined by the leading termsin the asymptotics of A n ( x ) , and thus in particular are rational functionsof the parameters. We thus find that the map from the space of matrices A n to the space of matrices A n +1 is a rational map, and by considering theinverse process, is in fact birational, corresponding to a sequence F n of bira-tional automorphisms of A P ) − . Note that the equation A , though notof the standard form, is still enough to determine A , and thus gives (ra-tionally) a P deg( P ) − worth of initial conditions corresponding to orthogonalpolynomials. (There is a deg( P ) -dimensional space of valid functions F , butrescaling F merely rescales the trace, and thus does not affect the orthogonalpolynomials.) Example 5.5. As an example, consider the case P ( x ) = x , corresponding,e.g., to w( y ) = e πcy cosh πy , with c ∈ (0 , . In this case, deg( P ) = 2 , so we get a -dimensional fam-ily of linear equations, and thus a second-order nonlinear recurrence, with a1-parameter family of initial conditions corresponding to orthogonal polyno-mials. Since the monic polynomial f is linear, we may use its root as one parameter f n , and g n = A n ( f n ) as the other parameter. We thus find that A n ( x ) = (5.5) (1 − f n x )(1 + f n + nx ) + f n g n x − a n − t − x (1 − f n +1 x ) − t − x (1 − f n x ) t − (cid:16) (1 − f n x )(1 + f n − nx ) + f n g n x (cid:17)! (5.6)where(5.7) a n = t ( t − n g n − f n ( g n − g n and f n , g n are determined from the recurrence f n +1 = f n ( f n ( g n − − ng n )( f n ( g n − − n ) n g n − f n ( g n − (5.8) g n +1 = ( f n ( g n − − ng n ) tg n ( f n ( g n − − n ) . (5.9)The three-term recurrence for the orthogonal polynomials is then p n +1 ( x ) = ( x − b n ) p n ( x ) − a n p n − ( x ) , where a n is as above and b n = − f n +1 − ( t + 1)( n + ) t − . The initial condition is given by f = b + t + 12( t − , g = 1 . (Note that the resulting A is not actually correct, but this induces thecorrect values for f , g , noting that the recurrence simplifies for n = 0 to f = − f , g = 1 /tg .) It follows from the general theory of isomonodromydeformations [R2] that this recurrence is a discrete Painlev´e equation (Thiswill also be shown by direct computation in forthcoming work by N. Witte.).We also note that the recurrence satisfies a sort of time-reversal symmetry:there is a natural isomorphism between the space of equations for t, n and thespace for t − , − n , coming (up to a diagonal change of basis) from the duality A ( A T ) − , and this symmetry preserves the recurrence. (This follows fromthe fact that if two equations are related by the three-term recurrence, thenso are their duals, albeit in the other order.) WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 41 Remark 5.6. The fact that A n ( x ) has a nice expression in terms of a n and f n +1 follows more generally from the fact (via the three-term recurrence) that A n ( x ) = − a n A n +1 ( x ) . One similarly has A n ( x ) = A n +1 ( x ) − ( x + − b n ) A n +1 ( x ) , so that in general f n +1 ( x ) ∝ P ( x ) A n ( x ) and g n +1 ( x ) = P ( x ) A n ( x ) mod f n +1 ( x ) . In particular, applying this to n = 0 tells us that the orthogonalpolynomial case corresponds to the initial condition f ( x ) ∝ L ( x ) , g ( x ) = t − P ( x ) mod L ( x ) .The above construction fails for t = 1 , because the constraint on theasymptotics of the off-diagonal coefficients of A n is stricter in that case: A n ( x ) = n − x + O (cid:0) x (cid:1) (5.10) A n ( x ) = O (cid:0) x (cid:1) . (5.11)The moduli space is still rational, although the arguments is somewhat sub-tler. We can still parametrize it by f n ( x ) := P ( x ) A n ( x ) and g n ( x ) := P ( x ) A n ( x ) mod f n ( x ) as above, which is certainly enough to determine P ( x ) A n ( x ) modulo f n ( x ) . This still leaves two degrees of freedom in thediagonal coefficients, but det( A n ( x )) + O ( x ) depends only on the diagonalcoefficients and is linear in the remaining degrees of freedom, so we can solvefor those. Once again, having determined the coefficients on and below thediagonal, the coefficient follows from the determinant, and can be seen tohave the correct poles and asymptotics. Note that now the dimension of themoduli space is q ) − ; that the dimension is even in both cases followsfrom the existence of a canonical symplectic structure on such moduli spaces,see [R2].There is a similar reduction in the number of parameters when the traceis even (forcing t = ( − n and P ( x ) = ( − n P ( − x ) ). The key observationin that case is that Y n ( − x ) = ( − n (cid:18) − (cid:19) Y n ( x ) (cid:18) − (cid:19) implying that A n satisfies the symmetry A n ( − x ) = (cid:18) − (cid:19) A n ( x ) − (cid:18) − (cid:19) . Since A n is × and has determinant t − = ( − n , this actually imposeslinear constraints on the coefficients of A n : A n ( − x ) = ( − n A n ( x ) A n ( − x ) = ( − n A n ( x ) A n ( − x ) = ( − n A n ( x ) A n ( − x ) = ( − n A n ( x ) . In particular, A n ( x ) has only about half the degrees of freedom one wouldotherwise expect, and for any root of that polynomial, A n ( α ) A n ( − α ) = 1 ,again halving the degrees of freedom (and preserving rationality). Example 5.7. Consider the case P ( x ) = x + β x with t = − and eventrace (e.g., for β = 0 , the weight function w( y ) = πy ). Then A n ( x ) hasthe form x − f n ) x + β x , and A n ( √ f n ) is of norm 1, which can be parametrized inthe form A n ( p f n ) = g n + √ f n g n − √ f n . Applying this to both square roots gives two linear conditions on A n ( x ) ,which suffices to determine it, with A n ( x ) following by symmetry and A n ( x ) from the remaining determinant conditions. We thus obtain A n ( x ) = n ( x − f n ) x ( x + β ) + f n ( f n + β )( g n + x )( g n − f n ) x ( x + β ) − a n x − f n +1 x ( x + β ) x − f n x ( x + β ) − n ( x − f n ) x ( x + β ) + f n ( f n + β )( g n − x )( g n − f n ) x ( x + β ) ! , where a n = − n f n ( f n + β ) g n − f n and f n , g n are determined by the recurrence g n +1 = − n − g n a n ng n − f n f n +1 = − ( ng n − f n ) g n +1 f n a n , with initial condition f = − β − − a , g = 0 . WISTED TRACES ON QUANTIZED KLEINIAN SINGULARITIES OF TYPE A 43 Remark 5.8. One can perform a similar calculation for the case P ( x ) = x − e x + e with even trace; again, one obtains a second-order nonlinearrecurrence, but the result is significantly more complicated, even for e = e = 0 .In each case, when the moduli space is -dimensional, so that the con-ditions uniquely determine the equation, we get an explicit formula for A n .This, of course, is precisely the case that the orthogonal polynomial is clas-sical. References [B] V. 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