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In this preliminary chapter we will outline an introduction to basic structure and representation theory of Lie groups and algebras. For those who are not already acquainted with this material, the hope is that the little we will say, perhaps supplementing a little from the quoted literature, could be enough to proceed without plunging into a long and serious study of the many things involved in this theory. For those who are already familiar with the matter, this chapter can either be skipped, or can serve as a quick reminder of some of the main points. To keep it as simple as possible, we will mostly explain things in the case of matrix groups, which in any case contains the main examples.

In this chapter we study real and complex Clifford algebras and their representations—the spin modules. This setting is essential for the definition of Dirac operators. We will also discuss the construction of Spin groups, which are certain subgroups of the groups of units in Clifford algebras.

Dirac operators were introduced into representation theory by Parthasarathy [Par] as a tool to construct the discrete series representations. The final results, which applied to all discrete series, were obtained by Atiyah and Schmid in [AS]. In this chapter we study an algebraic version of Parthasarathy’s Dirac operator, due to Vogan. In particular, we explain the notion of Dirac cohomology of Harish-Chandra modules, and Vogan’s conjecture which predicts the infinitesimal character of modules with nonzero Dirac cohomology [V3]. We present a proof of this conjecture following [HP1].

The Borel-Weil Theorem gives a geometric realization of each irreducible representation of a compact connected semisimple Lie group . Equivalently, this is a realization of each irreducible holomorphic representation of the complexification of . The realization is in the space of holomorphic sections of a holomorphic line bundle over the flag variety of .

In this chapter we review the basic constructions involved in cohomological induction, most notably the Zuckerman and Bernstein functors. Our definitions are slightly different from the ones available in the literature. For example, we do not use Hecke algebras which are basic ingredients in the definitions in [KV]. Also, we use a direct description of derived functors, including the g-action; this approach has its roots in [B], [W] and [DV], and it was fully developed in the setting of equivariant derived categories by D. Miličić and the second author, [MP1], [MP2], [MP3], [Pan1], [Pan2]. In particular, this will provide for a very simple treatment of the duality results.

In this chapter we review the basic properties of the (g, )-modules obtained by cohomological induction. These properties are roughly as follows: let be an (g, ⋂ )-module with infinitesimal character λ. Then the cohomologically induced modules have g-infinitesimal character λ + (u), where (u) is the half sum of roots corresponding to u. Under appropriate dominance conditions, they are:

One of the greatest achievements of mathematics in the 20th century is Harish-Chandra’s classification of discrete series representations of semisimple Lie groups. Let be a noncompact semisimple Lie group with a maximal compact subgroup . Discrete series representations are those irreducible unitary representations of which occur as subrepresentations in the Plancherel decomposition of (). Harish-Chandra proved that a necessary and sufficient condition for to have a discrete series is to have a compact Cartan subgroup. He constructed the characters of all discrete series representations. Speaking of Harish-Chandra’s work on discrete series, we quote Varadarajan in his article “Harish-Chandra, His Work, and its Legacy” [Va]: “In my opinion the character problem and the problem of constructing the discrete series were the ones that defined him, by stretching his formidable powers to their limit. The Harish-Chandra formula for the characters of discrete series is the single most beautiful formula in the theory of infinite-dimensional unitary representations.” Harish-Chandra “actually wrote down all the proofs in an extraordinary sequence of 8 papers [1964a]–[1966b], totaling 461 journal pages constituting one of the most remarkable series of papers in the annals of scientific research in our times—remarkable because of how long it took him to reach his goal, remarkable for how difficult the journey was and how it was punctuated by illness, remarkable for how unaided his achievement was, and finally, remarkable for the beauty and inevitability of his theorems.”

In this chapter we prove a formula for dimensions of spaces of automorphic forms which sharpens the result of Langlands and Hotta-Parthasarathy [L], [HoP2]. Let be a connected semisimple noncompact Lie group with finite center. Let ⊂ be a maximal compact subgroup of , and let Γ ⊂ be a discrete subgroup. Assume that Γ\ is compact and that Γ acts freely on . Then = Γ\ is a compact smooth manifold. Furthermore, the action of by right translation on the Hilbert space (Γ\) is decomposed discretely with finite multiplicities: Assume that rank is equal to rank . We calculate the multiplicity (Γ, ) for a discrete series representation .

Let g be a complex reductive Lie algebra with an invariant symmetric bilinear form , equal to the Killing form on the semisimple part of g. In this chapter we consider a parabolic subalgebra q = l ⊕ u of g, with unipotent radical u and a Levi subalgebra l. We will denote by the opposite parabolic subalgebra. Here the bar notation does not mean complex conjugation in general, but it will be a conjugation in the cases we will study the most, so the notation is convenient.

The Dirac operators discussed so far were all associated to nondegenerate symmetric bilinear forms on subspaces of reductive Lie algebras and the Clifford algebras corresponding to these symmetric forms. The Dirac operator to be defined in this chapter is associated to a symplectic form on the odd part of a Lie superalgebra and the corresponding Weyl algebra. In [HP3] we obtain an analog of Vogan’s conjecture for this Dirac operator. Our results build upon the results of [Ko6].