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N-point spherical functions and asymptotic boundary KZB equations

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Abstract

Let G be a split real connected Lie group with finite center. In the first part of the paper we define and study formal elementary spherical functions. They are formal power series analogues of elementary spherical functions on G in which the role of the quasi-simple admissible G-representations is replaced by Verma modules. For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular part of G. We show that they produce eigenstates for spin versions of quantum hyperbolic Calogero–Moser systems. In the second part of the paper we define and study special subclasses of global and formal elementary spherical functions, which we call global and formal N-point spherical functions. Formal N-point spherical functions arise as limits of correlation functions for boundary Wess–Zumino–Witten conformal field theory on the cylinder when the position variables tend to infinity. We construct global N-point spherical functions in terms of compositions of equivariant differential intertwiners associated with principal series representations, and express them in terms of Eisenstein integrals. We show that the eigenstates of the quantum spin Calogero–Moser system associated to N-point spherical functions are also common eigenfunctions of a commuting family of first-order differential operators, which we call asymptotic boundary Knizhnik–Zamolodchikov–Bernard operators. These operators are explicitly given in terms of folded classical dynamical r-matrices and associated dynamical k-matrices.

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Notes

  1. Throughout this paper we use “spin” in the sense how this term is used in physics as the description of internal degrees of freedom of one-dimensional quantum particles.

  2. In our follow-up paper [58], we consider the space \(C_{\sigma _\ell ,{\underline{\tau }},\sigma _r}^\infty (G^{\times (N+1)})\) of \(V_\ell \otimes {\mathbf {U}}\otimes V_r^*\)-valued functions \({\widetilde{f}}\) on \(G^{\times (N+1)}\) satisfying the transformation behaviour

    $$\begin{aligned} {\widetilde{f}}(k_\ell g_0h_1^{-1},h_1g_1h_2^{-1},\ldots ,h_Ng_Nk_r^{-1}) =(\sigma _\ell (k_\ell )\otimes \tau _1(h_1)\otimes \cdots \otimes \tau _N(h_N)\otimes \sigma _r^*(k_r)) {\widetilde{f}}(g_0,\ldots ,g_N) \end{aligned}$$

    for \((k_\ell ,h_1,\ldots ,h_N,k_r)\in K\times G^{\times N}\times K\). This space is preserved by the action of the commutative algebra of biinvariant differential operators on \(G^{\times (N+1)}\), and N-point \(\sigma ^{(N)}\)-spherical functions f produce simultaneous eigenfunctions \({\widetilde{f}}\in C^{\infty }_{\sigma ,{\underline{\tau }},\sigma _r}(G^{\times (N+1)})\) of the biinvariant differential operators on \(G^{\times (N+1)}\) via the formula

    $$\begin{aligned} {\widetilde{f}}(g_0,\ldots ,g_N):=\bigl (id _{V_\ell } \otimes \tau _1(g_0^{-1})\otimes \tau _2(g_1^{-1}g_0^{-1})\otimes \cdots \otimes \tau _N(g_{N-1}^{-1}\cdots g_1^{-1}g_0^{-1})\otimes id _{V_r^*}\bigr ) f(g_0\cdots g_N). \end{aligned}$$

    In this paper we do not use to full extent the \(G^{\times N}\)-action on \({\mathbf {U}}\). This will be done in the followup paper [58], where we will focus on superintegrability.

  3. For more background on Eisenstein integrals and their role in harmonic analysis see, e.g., [28,29,30, 63].

  4. From the perspective of footnote 2, the eigenvalue equations with respect to the asymptotic boundary KZB operators arise from the action of the biinvariant differential operator \(\Omega _i-\Omega _{i-1}\) on \({\widetilde{f}}\), where \(\Omega \) is the quadratic Casimir of G and \(\Omega _i\) is its interpretation as biinvariant differential operator acting on the ith-coordinate of \(G^{\times (N+1)}\).

  5. See (3.16) for details.

  6. Note here the remarkable fact, well known to specialists in harmonic analysis, that for generic \(z\in Z({\mathfrak {g}})\) and \(\lambda \in {\mathfrak {h}}^*\) the requirement that the formal \(End (V_\ell \otimes V_r^*)\)-valued power series \(f=\sum _{\mu \le \lambda }f_{\lambda -\mu }\xi _\mu \) is an eigenfunction of the radial component of z with eigenvalue \(\zeta _\lambda (z)\) will uniquely define the coefficients \(f_\gamma \in End (V_\ell \otimes V_r^*)\) in terms of \(f_0\in End (V_\ell \otimes V_r^*)\). This in particular holds true for \(z=\Omega \). The quadratic Casimir \(\Omega \) is a natural choice since its radial component is an explicit second-order differential operator that produces the Hamiltonian of the \(\sigma \)-spin quantum Calogero–Moser system, solvable by asymptotic Bethe ansatz, see Sect. 1.3.

  7. This factorisation can be used to derive the asymptotic KZB equations for Etingof’s and Schiffmann’s [15] generalised weighted trace functions in a manner similar to the one as described above for N-point spherical functions, see [62] (weighted traces are naturally associated to the symmetric space \(G\times G/diag (G)\), with \(diag (G)\) the group G diagonally embedded into \(G\times G\)).

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Acknowledgements

We thank Ivan Cherednik, Pavel Etingof, Giovanni Felder, Gert Heckman, Erik Koelink, Christian Korff, Tom Koornwinder, Eric Opdam, Maarten van Pruijssen, Taras Skrypnyk and Bart Vlaar for discussions and comments. We thank Sam van den Brink for carefully reading the first part of the paper and pointing out a number of typos. The work of J.S. and N.R. was supported by NWO 613.009.126. In addition the work of N.R. was partially supported by NSF DMS-1601947 and by RSF 21-11-00141. He also would like to thank ETH-ITS for the hospitality during the final stages of the work. The work on this paper was completed before N.R. retired from the University of California at Berkeley. He would like to thank the Department of Mathematics at UC Berkeley and all colleagues there for many happy and productive years.

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  1. KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, Amsterdam, 1098 XG, The Netherlands

    J. V. Stokman & N. Reshetikhin

  2. Yau Center for Mathematical Sciences, Tsinghua University, Beijing, China

    N. Reshetikhin

  3. St. Petersburg University, Saint Petersburg, Russia

    N. Reshetikhin

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Stokman, J.V., Reshetikhin, N. N-point spherical functions and asymptotic boundary KZB equations. Invent. math. 229, 1–86 (2022). https://doi.org/10.1007/s00222-022-01102-3

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