Local Langlands and character sheaves

Seminar/Forum

Russell Love Theatre
Peter Hall
Let $$G$$ be a connected reductive group and $$G^{\vee}$$ its dual group. For example, $$G=SO{2n+1}$$ and $$G^{\vee}=Sp{2n}$$ the symplectic group on a $$2n$$-dimensional space. The Langlands correspondence can be interpreted as attaching monodromy to representations (and vice versa); it has at least three incarnations: (1) In number theory, for certain automorphic representations generalizing modular forms, it predicts a matching Galois representation (étale monodromy over the field $$\mathbb{Q}$$ of rational numbers). (2) In geometry, it relates the objects on the moduli space of principal $$G$$-bundles (representation) on a Riemann surface $$X$$ to objects on the space of $$G^{\vee}$$-valued local systems (monodromy) on $$X$$. (3) In pure algebra, there is the Deligne-Lusztig theory and Lusztig’s theory of character sheaves; to each irreducible representation of $$G(F_q)$$ or character sheaf on G a monodromy as a semisimple element in $$G^{\vee}$$. While the theory for (3) is completed by Lusztig, (1) and (2) remain wildly open. In this talk, we survey three connections between (3) and local version of (1), raising new questions and proposing new conjectures. There will be no theorem in this talk.