Unitary Representations With Nonzero Dirac Cohomology For Complex Lie Groups

The representation theory of Lie groups has important applications in anal-ysis,differential geometry,topology and quantum mechanics.An important open problem in this domain is the classification of the unitary dual.More precisely,let G be a real reductive Lie group.Then the unitary dual of G,denoted by G,is the set of all the equivalence classes of irreducible unitary representations of G.To understand G better,and to sharpen the Dirac inequality,Vogan formu-lated the notion of Dirac cohomology in 1997 in three MIT Lie group seminars.He also raised several conjectures.In 2002,Huang-Pandzic confirmed the Vogan conjecture.Since then,Dirac cohomology became a new invariant of Lie group rep-resentations,and the classification of Gd-the modules in G with non-vanishing Dirac cohomology-became an interesting open problem.In this paper,we will use Dirac inequality,cohomological induction,Vogan pencil and the atlas software to study the classification of Gd for complex con-nected simple Lie groups.The structure of this thesis is as follows:In Chapter 1,we firstly introduce the research background and the current situation of the study of irreducible unitary representations with non-zero Dirac cohomology.Secondly,we give some basic theory,which is necessary for later discussions.Finally,we summarize the main results which are obtained in this thesis.From now on,we specialize G to be a complex connected simple Lie group.In Chapter 2,we recall from Barbasch and Pandzic's work that for an irreducible representation to be unitary and to have nonzero Dirac cohomology,it must be of the form J(?,-s?),where s?W is an involution,and 2? is dominant integral and regular.Then we partition the candidate representations into s-families,and obtain a finiteness theorem for the classification of Gd by utilizing Dirac inequality and cohomological induction.In Chapter 3,we firstly investigate the Dirac cohomology for tempered repre-sentations,minimal representations and model representations.Then we study the spherical unitary dual.Indeed,we give necessary conditions for a spherical unitary representation to have nonzero Dirac cohomology.Then we develop an algorithm which explicitly pins down the irreducible spherical unitary representations with non-zero Dirac cohomology for any fixed G.By using the software atlas,we have carried out the classification for complex classical Lie groups with rank no more than six,and for the exceptional Lie groups G2,F4,E6.In Chapter 4,by using Dirac inequality,Vogan pencil and the atlas software,we propose an algorithm for the classification of Gd for any fixed G.More precisely,we firstly give a necessary condition for an irreducible unitary representation to have nonzero Dirac cohomology.Then we reduce the candidate representations to finitely many ones.Finally,by utilizing the atlas software,we can check whether these candidates are members of G one by one.The complete classification of F4d is obtained by this algorithm.In Chapter 5,we firstly introduce an improved algorithm which is due to Dong.Then the complete classification of Gd for some classical Lie groups with small ranks is given by this improved algorithm.