Character, Character Cycle, Fixed Point Theorem and Group Representations

Abstract

This chapter explores that among many methods to derive Weyl's character formula, there is an application of the fixed point theorem to a line bundle on the flag variety. Namely, any finite-dimensional irreducible representation of a reductive group G is obtained as the cohomology group of an equivariant line bundle on the flag variety. Therefore, the trace of the action of an element g of G is obtained as the sum of the contributions at each fixed point. On the other hand, Harish-Chandra defined the character of an infinite-dimensional representation of a real semisimple group GR as an invariant eigendistribution. The chapter presents a character formula in terms of the geometry of flag manifold as a conjecture and presents a proof of it for discrete series. The correspondence of Harish-Chandra modules and K-equivariant sheaves is completed by adding representations of GR and GR-equivariant sheaves. Thereafter, the character is calculated from GR-equivariant sheaves.