Robert Langlands: On the Geometric Theory 

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I'm not sure. I think he means going from a function (or section) on the double-covering of a space to a "function" on the covered space itself. I wrote "function" since at every point you get two values, i.e. you get a "double-valued function", not a proper function. It seems he refers to putting the two values as the two diagonal entries of a 2x2 matrix, so you get a proper function from the covered space into the diagonal 2x2 matrices. Note that this resulting function will have some funny properties. If you go around the space once along a curve for which the double-covering is non-trivial, the function will not go back to the original value (i.e. 2x2 diagonal matrix) but to this matrix with the two diagonal entries swapped. P.S.: He said "function with two variables" which does not make sense to me. I think he misspoke and wanted to say "function with two values."

BTW, this seems to be his paper that he is attempting to summarize here: publications.ias.edu/sites/default/files/iztvestiya-english_7.pdf

@@EdwinSteiner Thanks! Sounds like some sort of "twist" is involved (it's funny that he should mention Atiyah & Bott afterwards, who later involved themselves with twisted K-theory). But I am not familiar with this sort of thing at all.

As one tries to see why the Atiyah-Bott story for the YM connection on a GL(2)-bundle over the elliptic curve and the Hecke eigenfunction story for the elliptic curve with group GL(2) are equivalent, one defines a 1-form on the double cover ( of the 1 dim'l part of Bun_GL(2)(E) where E is the ell curve) and considers the pushforward of the 1-form by the projection map into the space. When one does this, one gets a 2-form on the space, so one gets the same 2*2 diagonal matrices over the space whose conjugacy class was given by the eigenvalues of Hecke operators at every point of E. From here, one can understand this equivalence conceptually. As for the YM connection, one chooses such a metric on the total space of the bundle over E such that the YM connection is constant and one integrates them to get the YM equations which one realises as the Hecke eigenfunctions. Hence by "pushing down" he just meant the pushforward of the 1-form on the double cover by the projection map into the space, which gives a 2-form on the space.

Hey ... it was complicated.

Well done...thank you ...enjoyed listening

ok boomer