Let $G$ is a connected reductive group and $X$ a smooth irreducible projective curve, both defined over a field $F$. The goal of the global Langlands program over function fields is to do harmonic analysis on the moduli space (stack) ${\rm Bun}_G(X)$ of principal $G$-bundles on $X$, for various fields $F$. This means that we want to find the spectral decomposition for a family of commuting Hecke operators (or functors) acting on a suitable space of functions (or category of sheaves) on ${\rm Bun}_G(X)$. More precisely, the Langlands program aims to parametrize this spectrum by data related to the Langlands dual group $G^\vee$, namely appropriate $G^\vee$-local systems on $X$. The corresponding eigenvectors (or eigensheaves) are called automorphic functions (sheaves). There is also a ramified generalization of this theory, in which one considers $G$-bundles with level structure at a finite set of points of $X$.
In the most classical setup, developed by Langlands himself around 1970 (the arithmetic Langlands program), $F$ is a finite field $\Bbb F_q$, so ${\rm Bun}_G(X)(F)$ is an infinite countable set, and one considers Hecke operators $H_{x,\lambda}$ acting on $L^2({\rm Bun}_G(X)(\Bbb F_q))$ (a family of bounded, commuting normal operators). In this case, the Langlands conjecture (a version of which was recently proved by D. Gaitsgory and S. Raskin) states that the spectrum of Hecke operators is parametrized by local systems on $X$ with values in $G^\vee$ (or representations of the absolute Galois group of the field $\Bbb F_q(X)$), understood in a suitable sense.
In 1990s Beilinson and Drinfeld proposed a geometric generalization of this theory (the geometric Langlands program). In this generalization, $F=\Bbb C$, and the role of the space $L^2({\rm Bun}_G(X)(\Bbb F_q))$ is played by the category of (critically twisted) D-modules (de Rham version) or constructible sheaves of $\Bbb C$-vector spaces (Betti version) on the stack ${\rm Bun}_G(X)$; the two versions are related by the Riemann-Hilbert correspondence. On this category there is an action of a family of commuting Hecke functors, and the goal of the geometric Langlands program is to “diagonalize” these functors by finding and parametrizing their eigensheaves. The geometric Langlands conjecture of Beilinson and Drinfeld was that such eigensheaves are parametrized by $G^\vee$-local systems on the Riemann surface $X(\Bbb C)$, i.e., in the de Rham version, holomorphic $G^\vee$-bundles on $X$ with a connection, and in the Betti version, finite dimensional complex representations of its fundamental group. This conjecture has recently been proved in general by D. Gaitsgory and his collaborators. Also, Kapustin and Witten showed that the geometric Langlands program is related to topologically twisted 4-dimensional $N=4$ supersymmetric gauge theory.
There is also a version of this theory in which $F$ is any field and the constructible sheaves are $\ell$-adic (i.e., of $\overline{\Bbb Q}_\ell$-vector spaces). This version, with $F=\Bbb F_q$ where $(\ell,q)=1$, connects to the arithmetic version via the sheaf-function correspondence (i.e., taking the trace of Frobenius in the étale cohomology of the sheaf to get a function). This gives $\ell$-adic, rather than complex, functions, but it does not matter too much since cuspidal automorphic functions (i.e., ones from the discrete spectrum of Hecke operators) take values in $\overline{\Bbb Q}$.
In the de Rham version, the automorphic D-module $M_L$ attached to a $G^\vee$-local system $L$ is easiest to describe when $L$ is an oper, i.e., a connection on a special oper $G^\vee$-bundle $E$, induced from the oper bundle for the principal $SL_2$ of $G^\vee$, which is (for ${\rm genus}(X)\ne 1$) the non-trivial extension of $K_X^{-1/2}$ by $K_X^{1/2}$. The space of opers, ${\rm Op}_{G^\vee}(X)$, is an affine space whose underlying vector space is the Hitchin base $\mathcal B=\oplus_{i=1}^r H^0(X,K_X^{d_i})$, where $d_i$ are the degrees of basic invariants for $G$ and $r$ is its rank. Beilinson and Drinfeld showed that the classical Hitchin system $\pi: T^*{\rm Bun}_G(X)\to \mathcal B$ admits a quantization, which gives an inclusion (in fact, an isomorphism)
$D: \Bbb C[{\rm Op}_{G^\vee}(X)]\to \mathcal D({\rm Bun}_G(X),K^{1/2})$ (differential operators on half-forms, i.e., critically twisted ones). This quantization is constructed using the Casimirs of the affine Lie algebra at the critical level, constructed around 1990 by B. Feigin and E. Frenkel. The twisted D-module $M_L$ is then defined by the system of quantum Hitchin differential equations
$\qquad D(f)\psi=f(L)\psi,\ f\in \Bbb C[{\rm Op}_{G^\vee}(X)].$
Finally, in the last 7 years, jointly with E. Frenkel and D. Kazhdan, we developed an analytic version of the Langlands program. This program develops the earlier ideas and results of Braverman-Kazhdan, Kontsevich, Langlands and Teschner. In this version, $F$ is a local field, so there are two cases: archimedean ($F=\Bbb R,\Bbb C$) and non-archimedean ($F$ is a finite extension of $\Bbb Q_p$ or $F=\Bbb F_q((t))$). In both cases, we are interested in the spectrum of Hecke operators $H_{x,\lambda}$ acting on the Hilbert space $L^2({\rm Bun}_G(X)(F))$, where ${\rm Bun}_G(X)(F)$ is now generically an analytic $F$-variety, and $L^2$ stands for the space of square integrable half-densities. The operators $H_{x,\lambda}$ are densely defined on the Hilbert space, and conjecturally extend to bounded (in fact, compact) commuting normal operators; this conjecture has been proved in the ramified setting for genus $0$ and $1$ for $G=PGL_2$.
This raises the question what parametrizes the spectrum of the Hecke operators: can we describe it in terms of the Langlands dual group? In the non-archimedian case, we don't even have a conjecture how to do this, even though it has been shown by Braverman, Kazhdan and Polishchuk that cuspidal automorphic functions of arithmetic Langlands (i.e., over $\Bbb F_q$) can be lifted to ones over a local field $F$ with residue field $\Bbb F_q$ if we fix a lift of $X$ over its ring of integers $\mathcal O_F$.
The situation is better in the archimedian case, however. In this case, a prominent role is played by quantum Hitchin operators,
which (conjecturally) act on $L^2({\rm Bun}_G(X)(F))$ as commuting unbounded normal operators and commute with Hecke operators, hence have the same spectral decomposition. This connects the story to geometric Langlands, showing that automorphic functions of analytic Langlands theory should be parametrized by a certain (discrete) family of $G^\vee$-opers. And indeed, for $F=\Bbb C$, these (conjecturally) turn out to be the opers with real monodromy, i.e., ones whose monodromy representation can be conjugated into the split real form $G^\vee(\Bbb R)$. For such opers, the automorphic function $\psi$ on ${\rm Bun}_G(X)(\Bbb C)$ is simply a single valued analytic joint eigenfunction of $D(f)$ and $\overline{D(f)}$, which is unique up to scaling:
$$
\quad D(f)\psi_L=f(L)\psi_L,\ \overline{D(f)}\psi_L=\overline{f(L)}\psi_L.
$$
Gaiotto and Witten showed that this conjecture is compatible to the structure of branes in the Kapustin-Witten theory. Also, this conjecture is proved in genus zero and (almost) in genus $1$ in the ramified case, for $G=PGL_2$; the last steps in genus zero were made in a recent paper of Ambrosino and Teschner using Sklyanin's separation of variables transform. If $F=\Bbb R$, the situation is more complicated (as it depends on the structure of real ovals on $X$, real forms of $G$, etc.)
but still in many cases one can formulate the condition on the oper (always a topological one, i.e., in the Betti realization) which makes it
a point of the spectrum.
In the simplest case of $PGL_2$ and genus zero with tame ramification at $4$ points, ${\rm Bun}_G(X)$ is 1-dimensional and the eigenfunctions $\psi_L$ have the form
$$
\psi_L(z,\overline z)=\eta_1(z)\overline{\eta_2(z)}+\eta_2(z)\overline{\eta_1(z)},
$$
where $\eta_1,\eta_2$ is a basis of solutions of a Heun equation (in the most basic case, Lam\'e equation).
In the mini-course I will review this theory and consider various examples of arising spectral decompositions.
References
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P. Etingof, E. Frenkel, D. Kazhdan, An analytic version of the Langlands correspondence for complex curves, in Integrability, Quantization, and Geometry, Vol. II,
Proc. Sympos. Pure Math. 103.2, pp. 137-202, Amer. Math. Soc., Providence, RI, 2021,
arXiv:1908.09677.
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P. Etingof, E. Frenkel, and D. Kazhdan, Hecke operators and analytic Langlands correspondence for curves over local fields, Duke Math. Journal 172 (2023) 2015-2071,
arXiv:2103.01509.
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P. Etingof, E. Frenkel, D. Kazhdan, Analytic Langlands correspondence for P GL(2)
on P1 with parabolic structures over local fields, Geom. Funct. Anal. Vol. 32 (2022) 725-831, arXiv:2106.05243.
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P. Etingof, E. Frenkel, and D. Kazhdan,
A general framework for the analytic Langlands correspondence, arXiv:2311.03743.
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P. Etingof, H. Liu, Hitchin systems and their quantization, arXiv:2409.09505.
