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On the Representation Theory of Semisimple Lie Groups On the Representation Theory of Semisimple Lie Groups by Faisal Al-Faisal A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Pure Mathematics Waterloo, Ontario, Canada, 2010 c Faisal Al-Faisal 2010 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract This thesis is an expository account of three central theorems in the representation theory of semisimple Lie groups, namely the theorems of Borel{Weil{Bott, Casselman{ Osborne and Kostant. The first of these realizes all the irreducible holomorphic repre- sentations of a complex semisimple Lie group G in the cohomology of certain sheaves of equivariant line bundles over the flag variety of G. The latter two theorems describe the Lie algebra cohomology of a maximal nilpotent subalgebra of g with coefficients in an irreducible g-module. Applications to geometry and representation theory are given. Also included is a brief overview of Schmid's far-reaching generalization of the Borel{ Weil{Bott theorem to the setting of unitary representations of real semisimple Lie groups on (possibly infinite-dimensional) Hilbert spaces. iii Acknowledgements I would like to express my sincere thanks to my family and friends for their support, to my advisor Professor Wentang Kuo for his patience and insight, to my readers Professors Spiro Karigiannis and Kevin Purbhoo for their helpful comments, and to the Department of Pure Mathematics at the University of Waterloo for being my home away from home. Special thanks are also due to Professors Brian Forrest, Kathryn Hare and Nico Spronk for their guidance, help and inspiration all through my time at Waterloo; it has been truly appreciated. Lastly, I dedicate this thesis to my parents, whose love and encouragement got me to where I am today. iv Contents Introduction 1 1 Preliminaries 5 1.1 Complex Semisimple Lie Groups . 5 1.2 The Levi Decomposition . 6 1.3 Maximal Tori and Borel Subgroups . 7 1.4 The Jordan{Chevalley Decomposition . 8 1.5 The Weyl Group . 8 1.6 Representations . 9 1.6.1 Main Definitions . 9 1.6.2 Lie Algebra Representations . 10 1.6.3 Representations of Complex Semisimple Lie Groups . 10 1.6.4 Weight Space Decompositions . 11 1.7 The Root Space Decomposition . 12 1.7.1 The Setup . 12 1.7.2 Positive and Negative Roots . 14 1.7.3 The Killing Form . 15 1.7.4 Weyl Chambers . 15 v 1.7.5 The Weyl Group Revisited . 16 1.8 Cartan{Weyl Theory . 17 1.8.1 Integrality and Dominance . 17 1.8.2 The Theorem of the Highest Weight . 18 1.9 Homogeneous Spaces . 19 1.9.1 The Case of a Complex Lie Group . 19 1.9.2 The Case of a Complex Linear Algebraic Group . 20 1.10 The (Generalized) Flag Variety . 20 1.11 Parabolic Subgroups and Parabolic Subalgebras . 22 1.12 The Bruhat Decomposition . 23 1.13 Compact Real Forms . 24 1.14 The Infinitesimal Character . 25 2 The Borel{Weil{Bott Theorem 27 2.1 Introduction . 27 2.2 Equivariant Vector Bundles . 28 2.3 The Borel{Weil Theorem . 32 2.4 The Borel{Weil{Bott Theorem . 42 2.5 Applications . 51 2.5.1 Cohomology of Certain Homogeneous Spaces . 51 2.5.2 The Complex Structure on G=P .................... 55 2.5.3 Weyl's Dimension Formula . 58 2.6 Generalizing to Algebraic Groups over Arbitrary Fields . 60 vi 3 Lie Algebra Cohomology and the Theorems of Casselman{Osborne and Kostant 63 3.1 Introduction . 63 3.2 Lie Algebra Cohomology . 64 3.2.1 The Definition of Hp(g;V )....................... 64 3.2.2 Functoriality . 68 3.2.3 n-cohomology . 69 3.2.4 The Long Exact Sequence Revisited . 70 3.2.5 Cohomology of Injective Modules . 71 3.3 The Casselman{Osborne Theorem . 71 3.4 Kostant's Theorem . 73 3.5 Applications . 78 3.5.1 The Borel{Weil{Bott Theorem . 78 3.5.2 Weyl's Character Formula . 82 4 Discrete Series Representations and Schmid's Theorem 86 4.1 Introduction . 86 4.2 Preliminaries . 87 4.3 Harish-Chandra's Parametrization of the Discrete Series . 91 4.4 Schmid's Theorem . 95 4.5 On the Proof of Schmid's Theorem . 101 Bibliography 103 vii Introduction Let G be a complex semisimple Lie group and fix a Borel subgroup B ⊂ G. Then the irre- ducible holomorphic representations of G lie in one-to-one correspondence with a certain subset|the set of so-called dominant weights|of the character group Bb. This correspon- dence identifies an irreducible representation with its highest weight: If V is an irreducible representation of highest weight λ, then there is a unique B-invariant line in V on which B acts by multiplication by λ. In the reverse direction, one can start with a dominant weight λ and attempt to induce it to a representation of G. Taking our cue from finite groups1, we define G −1 indH (λ) = ff 2 Hol(G; C): f(gb) = λ(b) f(g) for all b 2 B and g 2 Gg and we let G act on this space by the left regular representation, viz. (gf)(g0) = f(g−1g0); 0 G where g; g 2 G and f 2 indH (λ). This construction makes sense for any λ 2 Bb. However, G indH (λ) is nonzero if and only if λ is dominant, in which case it is an irreducible represen- tation of G of highest weight λ. This is the content of the Borel{Weil theorem (Theorem 2.3.1). We can rephrase this induction procedure in more geometric terms. We begin by noting that each λ 2 Bb gives rise to a holomorphic line bundle Lλ = G ×λ C = G × C=f(gb; v) ∼ (g; λ(b)v)g 1 ∼ −1 If B ⊂ G are finite and V is a B-module, then C[G] ⊗C[B] V = ff : G ! V : f(gb) = b f(g)g: 1 over the projective variety G=B. There is a natural G-action on Lλ that lies over the G-action on G=B, i.e., Lλ is a G-equivariant line bundle. It follows that there is a linear G- q action on the sheaf cohomology spaces H (G=B; Lλ). It is then easy to see (cf. Proposition 2.2.5) that we have an isomorphism of representations 0 ∼ G H (G=B; Lλ) = indH (λ): In other words, the space of global holomorphic sections of Lλ can be identified with G indH (λ). In this setting the Borel{Weil theorem becomes the statement that Lλ has global 0 holomorphic sections only when λ is dominant, in which case H (G=B; Lλ) is an irreducible representation of G of highest weight λ. Thus the representation theory of G tells us something about the geometry of G=B. Conversely, we obtain geometric realizations of all the irreducible holomorphic representations of G. 0 In view of the above description of H (G=B; Lλ) it is natural to wonder if similar descriptions exist for the higher cohomology of Lλ. There is indeed a remarkable description ∗ of H (G=B; Lλ). By applying Kodaira's vanishing theorem, Borel and Hirzebruch [10] q found that H (G=B; Lλ) = 0 for all q > 0 if λ is dominant. Combined with the Borel{ ∗ Weil theorem, this says that if λ is dominant, then H (G=B; Lλ) is concentrated in a single degree, namely 0, at which it is an irreducible representation of G of highest weight λ. In a follow-up paper [11] Borel and Hirzebruch take up the matter of nondominant λ 2 Bb. They use Hirzebruch's Riemann{Roch formula to show that the Euler characteristic χ P i (G=B; Lλ) = i≥0 dim H (G=B; Lλ) either vanishes or else is equal to (plus or minus) the dimension of an irreducible representation of G. This led them to conjecture that, in ∗ general, H (G=B; Lλ) either vanishes entirely or else is concentrated in a single degree, at which it is an irreducible representation of G. This conjecture was proved by Bott in [13] and now goes by the name of the Borel{Weil{Bott theorem (Theorem 2.4.1). Bott's q theorem includes concrete descriptions of the degree qλ at which H (G=B; Lλ) is nonzero qλ and of the highest weight of the resulting irreducible representation H (G=B; Lλ). It should be remarked that the geometric content, i.e. the description of the cohomology of Lλ, is what makes this theorem interesting; we have gained no new representation theoretic information: all the irreducible representations already occur in degree 0. Some applications are given in Section 2.5. 2 In [13] one can also find a reformulation of Bott's result in terms of Lie algebra coho- mology. Bott's basic observation was that one can use the Dolbeault complex to compute ∗ H (G=B; Lλ). First one notes that G=B = K=T , where K is a maximal compact subgroup of G and T = K \B is a maximal torus in K. Thus G=B endows the a priori real manifold K=T with a complex structure, and the antiholomorphic tangent bundle of K=T is then seen to be modeled on a maximal nilpotent subalgebra n− of the Lie algebra g of G; here n− can be identified as the nilradical of the Borel subalgebra opposite to the Borel subalgebra b corresponding to B.
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