Dirac Cohomology And Cohomology Of Locally Symmetric Spaces

Suppose G is a connected real reductive linear group with a maximal com-pact subgroup K, and Γ is a torsion-free discrete subgroup of G. Vogan [Vog97] showed that the de Rham cohomology of locally symmetric space Γ\G/K is iso-morphic to the (g, K)-cohomology of C∞(Γ\G)K.According to the Matsushi-ma Theorem [BW00], to study the cohomology of locally symmetric spaces, it is necessary to get all the irreducible submodule of C∞(Γ\G)K, or more gen-erally, all the discrete series representations, with nonzero (g, K)-cohomology, and then to calculate their cohomology. Also an irreducible unitary module has nonzero (g, K)-cohomology if and only if it has the same infinitesimal charac-ter with the trivial module, which is Xp ([Vog97]). In Chapter 6 of this thesis we deduce the above connection between those cohomologies. In Chapter 5 the cases SU(1,1) and SU(2,1) are discussed. We list all the discrete series with nonzero (g,K)-cohomology by cohomological induction, and calculate their K-types using the Blattner’s Formula.Besides, the Dirac cohomology of a (g, K)-module can be defined using the Dirac operators. For a discrete series X with nonzero (g, K)-cohomology, its Dirac cohomology can determine its (g,K)-cohomology ([HP06]): H*(g,K;X)=HomK(HD(C),HD(X)). In the last section of Chapter 6, cohomologically induced modules are used to prove the above equivalence for G = SU(2,1).This thesis mainly consists of six chapters.In Chapter 1, we introduce the background of the classification of discrete series in the representation theory of Lie groups.In Chapter 2, preliminaries of the infinite-dimensional representation the-ory and an explicitly construction of all the SL(2, R)-discrete series are given.In Chapter 3, spin modules, Dirac operators and Dirac cohomology are defined. For SU(n,1), we decompose the spin module of C(p) as an t-odule.In Chapter 4, we recall some concepts in homological algebra, and then introduce the cohomological induction.In Chapter 5, we exhaust all the discrete series of SU(1,1) and SU(2,1) with infinitesimal character χρ, and calculate their K-types.In Chapter 6, we deduce the connection between cohomology of locally symmetric spaces and (g,K)-cohomology, and then prove the connection be-tween (g, K)-cohomology and Dirac cohomology of SU(2,1)-discrete series, using the results in the previous chapters.