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I'm been wondering about this for a while and hope someone can enlighten me.

In this interview of Robert Langlands's from 2010, on pg 21 (Question 8) he states "At one point, when fairly young, I read the book of Coddington and Levinson on ordinary differential equations and was taken with the topic of differential equations with irregular singular points. There are traces of this topic in the long letter to Weil about the Hecke theory. The same topic reappears in the “geometric Langlands programme” over C, where it has quite a different flavour than in Coddington-Levinson, whose references will have been Poincare and G. D. Birkhoff. In its current guise it is little explored. I hope to find time to learn more about this before I am done."

(1) From having seen Edward Frenkel speak on dualities in physics and geometric Langlands, and from speaking to people far more familiar with all of the above topics than myself, I can guess that the link with geometric Langlands may be as follows:

ODEs with irregular singular points $\longleftrightarrow$ asymptotic series/resummation methods $\longleftrightarrow$ getting non-perturbative physical theories from perturbative ones $\longleftrightarrow$ the link with geometric Langlands, with the last correspondence as described say here https://arxiv.org/abs/0906.2747. Is this accurate? Are there any insightful comments one would like to add?

(2) Even if the above has some semblance of truth, I am not sure about what the link is to the arithmetic Langlands program: "There are traces of this topic in the long letter to Weil about Hecke theory."

The second last sentence is fascinating - I have talked to other people who also think "In its current guise it is little explored" (which came as a big surprise to me - ODEs with singularities?!), but to think understanding this better might lead to insights into both versions of the Langlands program ((3) could it?) sounds very interesting.

P.S. That I shared the arxiv paper above should not be taken as an indication that I understand its contents with any amount of depth at all, though if you feel like assuming familiarity with technicalities in the geometric Langlands program will lead to a full answer, please go ahead (and provide references if you can!)

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  • $\begingroup$ At the very least, the Casselman (-Milicic?) subrepresentation theorem for real reductive groups, improving Harish-Chandra's subquotient theorem, using an elaboration of Deligne's work on PDE, is an essential "local" result for the modern representation theory of real reductive groups, which, yes, is essential for a high-end viewpoint on automorphic forms, Langlands' programme, and such... Is this the sort of thing you're wanting to hear? $\endgroup$
    – paul garrett
    Commented Oct 13, 2020 at 22:33
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    $\begingroup$ My guess is the comment has rather concrete meaning. The study of irregular singular points of ODE is more or less the study of $\mathcal{D}$-modules on the formal disk. The space of local Langlands parameters in geometric Langlands is a moduli space of $\mathcal{D}$-modules on the formal disk. So the study of singularities of ODE is the geometric counterpart of the study of ramification in the arithmetic setting. $\endgroup$
    – dhy
    Commented Oct 13, 2020 at 23:23
  • $\begingroup$ I apologize for the delay in response. @dhy I think I gathered something similar from talking to people about the geometric Langlands side, but is there a more direct link to the arithmetic side? Langlands says "there are traces of this in the long letter to Weil about the Hecke theory" and I can't imagine this is what he was thinking of. Even with geometric Langlands, does it not seem Langlands is referring to something more concrete, especially considering his views on the geometric theory after the Abel paper (please excuse my naivety if this is not so)? $\endgroup$
    – Waleed Qaisar
    Commented Oct 27, 2020 at 19:53
  • $\begingroup$ @paulgarrett I'm afraid I'm too ignorant to say "what I want to hear", but again I think it's unlikely that this is what Langlands is referring to in the letter to Weil? $\endgroup$
    – Waleed Qaisar
    Commented Oct 27, 2020 at 19:53
  • $\begingroup$ @WaleedQaisar, I cannot claim to truly know, but in those years Harish-Chandra had already been using various ideas about higher-rank/dimension analogues of "ODE with regular singular points", in his work going back to the 1950s. Apparently he left a large, unfinished manuscript on this (dunno whether it reached his collected works... I've tried to buy a copy, but they're out of print?), which in fact did not quite succeed in proving the more-general things he needed for his repn theory of semi-simple real Lie groups... [cont'] $\endgroup$
    – paul garrett
    Commented Oct 27, 2020 at 20:11

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