SL2 Seminar

In this talk I hope to explain how the representation theory of $GL_n(F_q)$ (with q being a power of p) in char. p, and also in mixed characteristic $(0,p)$, plays an important role in the theory of Galois representations and the Langlands program. The case of $GL_2(F_q)$ is already very interesting, and I will focus on that case(but will also try to indicate what is general to $GL_n$, and what is special to the $GL_2$ case).There is a key method in number-theory, called patching (due to Taylor--Wiles and Kisin), which is at the basis of the connection. In my talk, I *won't* really try to explain where this comes from (that would take us away from rep. theory and into the thick of the arithmetic theory of automorphic forms), but *will* instead focus on the axiomatic structures that come out of the method (``patching functors''). A priori there are lots of patching functors, but we can hope that actually there is only one of them (i.e. that all patching functors are naturally isomorphic). For reps. of $GL_2(F_q),$ this is more-or-less proved (as I will try to explain), via results of Le--Morra--Schraen/Hu--Wang/Le/Dotto--Le; nevertheless, in some sense we don't really understand *why* it's true (as is the case for many results in the Langlands program!), and there are lots of surprising, seemingly coincidental, phenomena for which we would like to have conceptual explanations. The study of patching functors uses a mixture of rep'n theory and number-theoretic methods. I will explain some of the key rep'n theoretic input, and also explain what else we would like to know about the rep'n theory, which might help shed more light on the structure and meaning of patching functors.