LECTURE 1
The Classical Theory: Part I
The first two lectures will be largely elementary and expository. They will
deal with the upper-half-plane H and Riemann sphere P1 from the points of view
of Hodge theory, representation theory and complex geometry. The topics to be
covered will be
(i) compact Riemann surfaces of genus one (= 1-dimensional complex tori)
and polarized Hodge structures (PHS) of weight one;
(ii) the space H of PHS’s of weight one and its compact dual
P1
as homoge-
neous complex manifolds;
(iii) the geometry and representation theory associated to H;
(iv) equivalence classes of PHS’s of weight one, as parametrized by Γ\H, and
automorphic forms;
(v) the geometric representation theory associated to
P1,
including the real-
ization of higher cohomology by global, holomorphic data;
(vi) Penrose transforms in genus g = 1 and g 2.
Assumptions.
• basic knowledge of complex manifolds (in this lecture mainly Riemann
surfaces);
• elementary topology and manifolds, including de Rham’s theorem;
• some familiarity with classical modular forms will be helpful but not
essential;1
• some familiarity with the basic theory of Lie groups and Lie algebras.
Complex tori of dimension one. We let X = compact, connected complex
manifold of dimension one and genus one. Then X is a complex torus C/Λ where
Λ = {n1π1 + n2π2}n1,n2∈Z ⊂ C
1The
classical theory will be covered in the article [Ke1] by Matt Kerr in the Contemporary
Mathematics volume, published by the AMS and that is associated to the CBMS conference.
5
http://dx.doi.org/10.1090/cbms/118/02
