This is the last lecture in the series. I will discuss the restriction functor for Harish-Chandra modules.

Abstract: A periodic pencil of flat connections on a smooth algebraic variety X is a linear family of flat connections ∇(s1, …, sn) = d − Xr i=1 Xn j=1 sjBijdxi , where {xi} are local coordinates on X and Bij : X → MatN are matrixvalued regular functions. A pencil is periodic if it is generically invariant under the shifts sj 7→ sj + 1 up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic p, the p-curvature operators {Ci , 1 ≤ i ≤ r} of a periodic pencil ∇ are isospectral to the commuting endomorphisms C ∗ i := Pn j=1(sj − s p j )B (1) ij , where B (1) ij is the Frobenius twist of Bij . This allows us to compute the eigenvalues of the p-curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.