Hecke operators and analytic Langlands correspondence for curves over local fields (Q6177379)

The paper under review is the third in a series of works [\textit{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 [\textit{A. Braverman} and \textit{D. Kazhdan}, Nagoya Math. J. 184, 57--84 (2006; Zbl 1124.22006); \textit{M. Kontsevich}, Prog. Math. 270, 213--247 (2009; Zbl 1279.11065); \textit{R. P. Langlands}, ``On the analytic form of the geometric theory of automorphic forms'', Preprint, \url{http://publications.ias.edu/rpl/section/2659}; \textit{J. Teschner}, ``Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence'', Preprint, \url{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.