
SIGMA 17 (2021), 029, 31 pages arXiv:2009.09437
https://doi.org/10.3842/SIGMA.2021.029
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday
Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A
Pavel Etingof ^{a}, Daniil Klyuev ^{a}, Eric Rains ^{b} and Douglas Stryker ^{a}
^{a)} Department of Mathematics, Massachusetts Institute of Technology, USA
^{b)} Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA
Received September 22, 2020, in final form March 08, 2021; Published online March 25, 2021
Abstract
Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345392] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type $A_{n1}$. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short starproducts for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3dimensional superconformal field theory. In particular, we confirm their conjecture that for $n\le 4$ a unitary short starproduct is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If $n=2$, this, in particular, recovers the theory of unitary spherical HarishChandra bimodules for ${\mathfrak{sl}}_2$. Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary HarishChandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short starproducts corresponding to twisted traces, which are generalizations of discrete Painlevé systems.
Key words: starproduct; orthogonal polynomial; quantization; trace.
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References
 Bavula V.V., Generalized Weyl algebras and their representations, St. Petersburg Math. J. 4 (1993), 7192.
 Beem C., Peelaers W., Rastelli L., Deformation quantization and superconformal symmetry in three dimensions, Comm. Math. Phys. 354 (2017), 345392, arXiv:1601.05378.
 Bleher P., Its A., Semiclassical asymptotics of orthogonal polynomials, RiemannHilbert problem, and universality in the matrix model, Ann. of Math. 150 (1999), 185266, arXiv:mathph/9907025.
 Dedushenko M., Fan Y., Pufu S.S., Yacoby R., Coulomb branch operators and mirror symmetry in three dimensions, J. High Energy Phys. 2018 (2018), no. 4, 037, 111 pages, arXiv:1712.09384.
 Dedushenko M., Pufu S.S., Yacoby R., A onedimensional theory for Higgs branch operators, J. High Energy Phys. 2018 (2018), no. 3, 138, 83 pages, arXiv:1610.00740.
 Etingof P., Stryker D., Short starproducts for filtered quantizations, I, SIGMA 16 (2020), 014, 28 pages, arXiv:1909.13588.
 Fokas A.S., Its A.R., Kitaev A.V., The isomonodromy approach to matrix models in $2$D quantum gravity, Comm. Math. Phys. 147 (1992), 395430.
 Klyuev D., On unitarizable HarishChandra bimodules for deformations of Kleinian singularities, arXiv:2003.11508.
 Klyuev D., Twisted traces and positive forms on generalized $q$Weyl algebras, in preparation.
 Koekoek R., Swarttouw R.F., The Askey scheme of hypergeometric orthogonal polynomials and its $q$analogue, arXiv:math.CA/9602214.
 Losev I., Finitedimensional representations of $W$algebras, Duke Math. J. 159 (2011), 99143, arXiv:0807.1023.
 Magnus A.P., Associated AskeyWilson polynomials as LaguerreHahn orthogonal polynomials, in Orthogonal Polynomials and their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, Springer, Berlin, 1988, 261278.
 Rains E.M., Generalized Hitchin systems on rational surfaces, arXiv:1307.4033.
 Szegő G., Orthogonal polynomials, 4th ed., American Mathematical Society, Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, R.I., 1975.
 Vogan Jr. D.A., Unitary representations of reductive Lie groups, Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987.

