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