"Hecke transformations are one of the most important ingredients in geometric Langlands. What they mean in terms of physics had bothered me for a long time and eventually had been the last major stumbling block in interpreting geometric Langlands in terms of physics and gauge theory. Finally, while on an airplane flying home from Seattle, it struck me that a Hecke transformation in the context of geometric Langlands is simply an algebraic geometerās way to describe the effects of a āāt Hooft operatorā of quantum gauge theory. I had never worked with āt Hooft operators, but they were familiar to me, as they had been introduced in the late 1970s as a tool in understanding quantum gauge theory. The basics of how to work with āt Hooft operators and what happens to them under electric-magnetic duality were well known, so once I could reinterpret Hecke transformations in terms of āt Hooft operators, many things were clearer to me."
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Edward Witten, as quoted in (quote from p. 505)
https://en.wikiquote.org/wiki/Geometric_Langlands_correspondence
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