Start: Wednesday, October 13, 2010
Lectures: Wednesday: 09:00 - 11:00, Thursday: 11:00 - 13:00
Seminar: tba
ESI, Schrödinger lecture hall

 
Abstract:
The deRham cohomology groups of a locally symmetric space are closely related to Lie algebra cohomology groups. In this way questions about the former groups can sometimes be translated into questions about cohomological properties of infinite-dimensional representations of the underlying real Lie group G. This leads to the basic problem of determining (up to infinitesimal equivalence) the irreducible unitary representations of G with non-vanishing relative Lie algebra cohomology. This problem was solved by Vogan-Zuckerman, based on previous work of Kumaresan and Parthasarathy. To understand their solution, we have to know what it means to describe a unitary representation. Since an explicit realization for the representations with non-zero cohomologyis not at hand they have to be described by specifying some invariants which any unitary representation has. The invariants needed are the eigenvalue of the Casimir operator of the representation, and the restriction ofthe representation of G to a maximal compact subgroup K⊂G.

In detail: We aim to cover the following topics in this course:

(1) We first consider the cohomology of a compact locally symmetric space X=Γ | G/K (with Γ a discrete subgroup in G) and prove the Matsushima formula relating the Hodge Laplacian of forms of degree p on X with the action of the Casimir on functions on Γ | G. This is then easily translated into a problem of classifying representations π of the group G on which the Casimir of G vanishes and which share a K-type with the exterior algebra of p (the orthocomplement of k in the Lie algebra g of G).

(2) Next we analyse the structure of the spinor module S associated to p as a representation of K, and relate it to the exterior algebra of p.We prove Parthasarathy's Dirac Operator Inequality and use this to analyse the representations common with S and S&otmes;π.

(3) We prove Kumaresan's result on which K-types can occur in common with the exterior algebra of p and a cohomological representation π. This is fairly involved and we spend three or four lectures on this.

(4) We prove the result of Vogan-Zuckerman that the only representations with cohomology which can be unitary are the A_q representations.

(5) Then we construct (following Vogan-Zuckerman) the representations A_q.

(6) Using the Weil representation, we then show in some special cases that the A_q representations indeed contribute to cohomology of arithmetic groups.