Representations of semisimple lie groups utah math department. Ill by harishchandra the main object of this paper is to define the character of an irreducible quasisimple1 representation ir of a connected semisimple lie group g on a hubert space this will be done as follows. The goal of this paper is to study the irreducible representations of semisimple lie algebras. Let 7r be a quasisimple irreducible representation 6c of a connected semisimple lie group g. On the degrees and rationality of certain characters of finite chevalley groups, trans. On the other hand, the grouphas two components, so there are two one dimensional. We denote by c g the space of all complexvalued functions. Classification of irreducible tempered representations of semisimple lie groups article pdf available in proceedings of the national academy of sciences 737. The equivariance is with respect to a group whose connected component is a torus. Faithful representations of semisimple lie algebras. Proof of the kazhdanlusztig conjectures in the integral case, to appear in inventiones math. In this paper i build up to a useful theorem characterizing the irreducible.
A simple example of a reducible representation is obtained by taking two irreducible representations on the space v1 and v2, say, and forming the space v1. Let 8 be a complex semisimple lie algebra, and b a cartan subalgebra. Irreducible characters of semisimple lie groups iii. Pdf representations of semisimple lie groups researchgate. Roughly speaking, the standard representations are non. For a large part, they summarise relevant material from knapps book 12. Irreducible character an overview sciencedirect topics. In addition, when gis a nonsimplylaced group, we show that there are three. The discrete series of semisimple groups peter hochs september 5, 2019 abstract these notes contain some basic facts about discrete series representations of semisimple lie groups. Lecture speciali es to the representation theory of. In this paper we classify and give kazhdanlusztig type character formulas for equivariantly irreducible representations of lie algebras of reductive algebraic groups over a field of large positive characteristic. To each irreducible admissible representation of g, langlands in 17 has associated a natural induced representation, of a kind we will call standard. Representation theory of compact groups and complex.