Hecke operators and analytic Langlands correspondence for curves over local fields. (English) Zbl 1539.11146

Etingof, Pavel; Frenkel, Edward; Kazhdan, David

Duke Math. J. 172, No. 11, 2015-2071 (2023).

The paper under review is the third in a series of works [P. Etingof et al., Proc. Symp. Pure Math. 103, 137–202 (2021; Zbl 1475.14021); Geom. Funct. Anal. 32, No. 4, 725–831 (2022; Zbl 1495.14021)] by the authors on the foundations of an analytic Langlands correspondence for curves over (primarily Archimedean) local fields. This paradigm builds on earlier ideas of [A. Braverman and D. Kazhdan, Nagoya Math. J. 184, 57–84 (2006; Zbl 1124.22006); M. Kontsevich, Prog. Math. 270, 213–247 (2009; Zbl 1279.11065); R. P. Langlands, “On the analytic form of the geometric theory of automorphic forms”, Preprint, http://publications.ias.edu/rpl/section/2659; J. Teschner, “Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence”, Preprint, arXiv:1707.07873]. By analogy, the Langlands correspondence for a smooth projective curve \(X\) over \(\mathbb F_q\) relates the joint spectrum of Hecke operators acting on \(L^2\)-functions on the \(\mathbb F_q\)-points of the stack \(\mathrm{Bun}_G\) of \(G\)-bundles on \(X\), to Galois data associated to \(X\) and the Langlands dual group \(^LG\). The current paper studies the same problem, replacing \(\mathbb F_q\) with a local field \(F\). The authors define analogues for Hecke operators on a dense subspace of the Hilbert space of half-densities on \(\mathrm{Bun}_G\), where \(G\) is an \(F\)-split connected reductive group, and conjecture that they extend to the full space.
In particular, let \(\mathrm{Bun}_G(X,S)\) be the moduli stack of pairs \((\mathscr F,r_S)\) where \(\mathscr F\) is a \(G\)-bundle on \(X\) and \(r_S\) a reduction to the Borel \(B\) of the restriction of \(\mathscr F\) to a reduced divisor \(S\) of \(X\) over \(F\). The authors introduce a open and dense substack of regularly stable pairs, or rather its corresponding coarse moduli space \(\mathrm{Bun}^\circ_G(X,S)\). The Hilbert space \(\mathscr H_G(X,S)\) is then defined as the completion of the space of smooth compactly supported sections of \(\Omega^{1/2}_\mathrm{Bun} = ||K^{1/2}_\mathrm{Bun}||\), a complex line bundle obtained from the square root of the canonical line bundle on \(\mathrm{Bun}^\circ(X,S)\). For a dominant integral coweight \(\lambda\) of \(G\), define a Hecke correspondence \[ q: Z(\lambda) \to \mathrm{Bun}_G(X,S) \times \mathrm{Bun}_G(X,S)\times (X\backslash S), \] where \(Z(\lambda)\) classifies certain tuples \((\mathscr F,\mathscr P, x,t)\) where \(\mathscr F,\mathscr P\) are principle \(G\)-bundles on \(X\), \(x\) a closed point in \(X\backslash S\), and \(t:\mathscr F|_{X\backslash x}\to \mathscr P|_{X\backslash x}\) an isomorphism. Further, denote by \(p_i\) the projection of the left hand side to the \(i\)-th factor, and \(q_i = p_i\circ q\). Let \(K_2\) be the canonical line bundle of \(q_2\times q_3\). Then the main Theorem 1.1 of the paper is that there exists a canonical isomorphism \[ q_1^*(K_\mathrm{Bun}^{1/2})\stackrel{\sim}{\to} q^*_2(K_\mathrm{Bun}^{1/2})\otimes K_2 \otimes q_3^*(K_X^{-\langle\lambda,\rho\rangle} \] where \(\rho\) is the half-sum of positive roots.
For \(\mathscr P \in \mathrm{Bun}_G^\circ(X,S)(F)\) and \(x\in (X\backslash S)(F)\), let \(Z_{\mathscr P,x} = (q_2\times q_3)^{-1}(\mathscr P\times x)\). Then for certain compactly supported half densities \(f\) on \( \mathrm{Bun}_G^\circ(X,S)(F)\), define the Hecke operator \[ \hat H_\lambda(x): f(\mathscr P) = \int_{Z_{\mathscr P,x}} q_1^*(f) \in \mathscr H_G(X,S) \otimes (\Omega_X^{-\langle\lambda,\rho\rangle})_x, \] and varying \(x\in X\backslash S\), we obtain \(\hat H_\lambda\) as an operator-valued section of \(\Omega_X^{-\langle\lambda,\rho\rangle}\) over \(X\backslash S\). The authors conjecture several properties of this Hecke operator, including that they extend to a family of commuting compact normal operators on \(\mathscr H_G = \mathscr H_G(X,S)\), denoted by \(H_\lambda(x)\), and that there exists an orthogonal decomposition \[ \mathscr H_G = \widehat\bigoplus_{s}\mathscr H_G(s), \] where \(s\) runs over the the joint spectrum of the commutative algebra generated by \(H_\lambda(x)\) as \(\lambda\) and \(x\) vary, and \(\mathscr H_G(s)\) are finite-dimensional joint eigenspaces.
In the case \(F=\mathbb C\), the authors furthermore conjecture that the joint spectrum is bijective with the set of \(^LG\)-opers on \(X\) with real monodromy. They also conjecture an explicit formula connecting the eigenvalues of Hecke operators with global differential operators, and prove it for \(G = \mathrm{GL}_n\) assuming a compactness conjecture.

Reviewer: Tian An Wong (Dearborn)

MSC:

11R39

Langlands-Weil conjectures, nonabelian class field theory

Keywords:

algebraic curve; differential operators; Hecke operators; Hilbert space; Langlands program; local field; normal operators; opers; principal \(G\)-bundle

Citations:

Zbl 1475.14021; Zbl 1495.14021; Zbl 1124.22006; Zbl 1279.11065

References:

[1]	D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, and Y. VARSHAVSKY, The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support, preprint, arXiv:2010.01906v2 [math.AG].
[2]	A. BEILINSON and J. BERNSTEIN, “A proof of Jantzen conjectures” in I. M. Gelfand Seminar, Adv. Soviet Math. 16, Part 1, Amer. Math. Soc., Providence, 1993, 1-50. · Zbl 0790.22007
[3]	A. BEILINSON and V. DRINFELD, Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint, 1991, http://www.math.uchicago.edu/ drinfeld/langlands/QuantizationHitchin.pdf.
[4]	A. BEILINSON and V. DRINFELD, Opers, preprint, arXiv:math/0501398v1 [math.AG].
[5]	A. BRAVERMAN and D. KAZHDAN, Some examples of Hecke algebras for two-dimensional local fields, Nagoya Math. J. 184 (2006), 57-84. · Zbl 1124.22006 · doi:10.1017/S0027763000009314
[6]	A. D’AGNOLO and P. SCHAPIRA, Radon-Penrose transform for \( \mathcal{D} -modules \), J. Funct. Anal. 139 (1996), no. 2, 349-382. · Zbl 0879.32012 · doi:10.1006/jfan.1996.0089
[7]	V. G. DRINFELD and V. V. SOKOLOV, Equations of Korteweg-deVries type and simple Lie algebras (in Russian), Dokl. Akad. Nauk 258 (1981), no. 1, 11-16; English translation in Math. Dokl. 23 (1981), 457-462. · Zbl 0513.35073
[8]	P. ETINGOF, E. FRENKEL, and D. KAZHDAN, “An analytic version of the Langlands correspondence for complex curves” in Integrability, Quantization, and Geometry, II: Quantum Theories and Algebraic Geometry, Proc. Sympos. Pure Math. 103, Amer. Math. Soc., Providence, 2021, 137-202. · Zbl 1475.14021
[9]	P. ETINGOF, E. FRENKEL, and D. KAZHDAN, Analytic Langlands correspondence for (( \operatorname{PGL}_2 on \mathbb{P}^1\)with parabolic structures over local fields, Geom. Funct. Anal. 32 (2022), no. 4, 725-831. · Zbl 1495.14021 · doi:10.1007/s00039-022-00603-w
[10]	J. FÆRGEMAN and S. RASKIN, Non-vanishing of geometric Whittaker coefficients for reductive groups, preprint, arXiv:2207.02955 [math.RT].
[11]	G. FALTINGS, Real projective structures on Riemann surfaces, Compos. Math. 48 (1983), no. 2, 223-269. · Zbl 1514.53095 · doi:10.1007/s00229-022-01377-z
[12]	B. FEIGIN and E. FRENKEL, “Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras” in Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys. 16, World Sci. Publ., River Edge, 1992, 197-215. · Zbl 0925.17022 · doi:10.1142/s0217751x92003781
[13]	E. FRENKEL, Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005), no. 2, 297-404. · Zbl 1129.17014 · doi:10.1016/j.aim.2004.08.002
[14]	E. FRENKEL, Is there an analytic theory of automorphic functions for complex algebraic curves?, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), no. 042. · Zbl 1446.14005 · doi:10.3842/SIGMA.2020.042
[15]	E. FRENKEL and D. BEN-ZVI, Vertex Algebras and Algebraic Curves, 2nd ed., Math. Surveys Monogr. 88, Amer. Math. Soc., Providence, 2004. · Zbl 1106.17035 · doi:10.1090/surv/088
[16]	E. FRENKEL and C. TELEMAN, Geometric Langlands correspondence near opers, J. Ramanujan Math. Soc. 28A (2013), 123-147. · Zbl 1325.14023
[17]	D. GAITSGORY, Outline of the proof of the geometric Langlands conjecture for \( \operatorname{GL}_2\), Astérisque 370 (2015), 1-112. · Zbl 1406.14008
[18]	D. GALLO, M. KAPOVICH, and A. MARDEN, The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. (2) 151 (2000), no. 2, 625-704. · Zbl 0977.30028 · doi:10.2307/121044
[19]	I. M. GELFAND and D. KAZHDAN, Certain questions of differential geometry and the computation of the cohomologies of the Lie algebras of vector fields (in Russian), Dokl. Akad. Nauk 200 (1971), 269-272; English translation in Math. Dokl. 12 (1971), 1367-1370. · Zbl 0238.58001
[20]	W. M. GOLDMAN, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), no. 3, 297-326. · Zbl 0595.57012 · doi:10.4310/jdg/1214440978
[21]	A. B. GONCHAROV, Differential equations and integral geometry, Adv. Math. 131 (1997), no. 2, 279-343. · Zbl 0931.53038 · doi:10.1006/aima.1997.1669
[22]	M. KASHIWARA, D-Modules and Microlocal Calculus, Trans. Math. Monogr. 217, Amer. Math. Soc., Providence, 2003. · Zbl 1017.32012 · doi:10.1090/mmono/217
[23]	M. KONTSEVICH, “Notes on motives in finite characteristic” in Algebra, Arithmetic, and Geometry, in Honor of Y. I. Manin, Vol. II, Progr. Math. 270, Birkhäuser Boston, Boston, 2010, 213-247. · Zbl 1279.11065 · doi:10.1007/978-0-8176-4747-6_7
[24]	R. P. LANGLANDS, On the analytic form of the geometric theory of automorphic forms, preprint, http://publications.ias.edu/rpl/section/2659 (accessed 7 April 2023).
[25]	Y. LASZLO and C. SORGER, The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. Éc. Norm. Supér. (4) 30 (1997), no. 4, 499-525. · Zbl 0918.14004 · doi:10.1016/S0012-9593(97)89929-6
[26]	G. LAUMON, Transformation de Fourier généralisée, preprint, arXiv:alg-geom/9603004v1.
[27]	M. S. NARASIMHAN and S. RAMANAN, “Geometry of Hecke cycles, I” in C. P. Ramanujam: A Tribute, Tata Inst. Fund. Res. Stud. Math. 8, Springer, Berlin, 1978, 291-345. · Zbl 0427.14002
[28]	M. ROTHSTEIN, Connections on the total Picard sheaf and the KP hierarchy, Acta Appl. Math. 42 (1996), no. 3, 297-308. · Zbl 0848.58032 · doi:10.1007/BF01064170
[29]	J. TESCHNER, Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence, preprint, arXiv:1707.07873v2 [math-ph]. · Zbl 1439.14051
[30]	A. WEIL, “Adèles et groupes algébriques” in Séminaire Bourbaki, Vol. 5, no. 186, Soc. Math. France, Paris, 1959, 249-257. · Zbl 0118.15801