Introduction to Representations of Real Semisimple Lie Groups Matvei Libine September 13, 2021 Abstract These are lecture notes for a one semester introductory course I gave at Indiana University. The goal was to make this exposition as clear and elementary as possible. A particular emphasis is given on examples involving SU(1, 1). These notes are in part based on lectures given by my graduate advisor Wilfried Schmid at Harvard University and PQR2003 Euroschool in Brussels [17] as well as other sources, such as [10, 13, 14, 23, 25, 27]. The text is formated for convenient viewing from iPad and other tablets. Contents 1 Introduction 4 2 Some Geometry and Examples of Representa- tions 7 2.1 ComplexProjectiveLine. 7 arXiv:1212.2578v3 [math.RT] 9 Sep 2021 2.2 ExamplesofRepresentations . 10 3 The Universal Enveloping Algebra 14 3.1 TheDefinition .................. 14 3.2 Some Properties of the Universal Enveloping Al- gebra....................... 16 3.3 Geometric Realization of U(g).......... 20 1 4 Irreducible Representations of sl(2, R) 21 4.1 Preliminaries . 21 4.2 Classification of Irreducible sl(2, C)-Modules . 27 5 The Complexification, the Cartan Decomposi- tion and Maximal Compact Subgroups 36 5.1 AssumptionsonGroups . 36 5.2 ComplexificationsandRealForms. 38 5.3 TheCartanDecomposition . 40 5.4 MaximalCompactSubgroups . 44 6 Definition of a Representation 48 6.1 A Few Words on Topological Vector Spaces . 48 6.2 TheDefinition .................. 52 6.3 AdmissibleRepresentations . 56 7 Harish-ChandraModules 57 ∞ 7.1 KR-finite and C Vectors ............ 57 7.2 ProofofTheorem61. 59 7.3 Harish-ChandraModules . 67 7.4 InfinitesimalEquivalence. 73 7.5 AFewWordsaboutGlobalization. 76 7.6 Unitary Representations and Infinitesimal Equiv- alence....................... 78 8 Construction of Representations of SU(1, 1) 80 8.1 Construction of Representations of SL(2, R).. 80 8.2 Construction of Representations of SU(1, 1) .. 82 8.3 ActionoftheLieAlgebra . 85 ∞ 1 8.4 The Harish-Chandra Module of (πχ, C (S )).. 87 2 ∞ 1 8.5 Decomposition of (πχ, C (S )) into Irreducible Components ................... 89 8.6 Unitary Representations of SU(1, 1) ...... 93 9 Notes on Schur’s Lemma 99 9.1 ClassicalVersion . 99 9.2 Schur’s Lemma for Unitary Representations . 100 9.3 Dixmier’sLemma . 100 9.4 InfinitesimalCharacters . 103 10 Iwasawa and KAK Decompositions 104 10.1 IwasawaDecomposition . 104 10.2 KAK Decomposition . 110 11 Principal Series Representations 111 11.1 GR-equivariantVectorBundles . 111 11.2 Construction of the Principal Series Representa- tions ....................... 113 11.3 Admissibility of the Principal Series Representa- tions ....................... 116 11.4 SubrepresentationTheorem . 118 11.5 Unitary Principal Series Representations . 119 12 Group Characters 121 12.1 Hilbert-SchmidtOperators. 121 12.2 TraceClassOperators . 124 12.3 TheDefinitionofaCharacter . 125 12.4 PropertiesofCharacters . 129 12.5 Harish-Chandra Regularity Theorem . 131 12.6 Characters of Representations of SU(1, 1) ... 134 3 1 Introduction Let G be a real semisimple or reductive Lie group (intuitively this means that its Lie algebra g is semisimple or reductive). Examples: SL(n, R), O(p,q), SU(p,q), ...are real semisim- ple Lie groups and GL(n, R), U(p,q), ...are reductive real Lie groups. Gˆ = set of isomorphism classes of irreducible unitary represen- tations. Thus, for each i ∈ Gˆ, we get an irreducible unitary representation (πi, Vi). G – compact: dim Vi < ∞, unitarity is automatic (every ir- reducible representation is unitary), πi : G × Vi → Vi is real analytic, so get a representation of g on Vi. Non-compact: Typically dim Vi = ∞, there are many interest- ing non-unitary representations, analyticity fails miserably. To get around this problem and construct representation of g, look at the space of C∞ or Cω vectors; solution: Harish-Chandra modules. G – compact: We have Peter-Weyl Theorem: 2 ∗ L (G) ≃ Vi ⊗ V ≃ End(Vi) i∈Gˆ i i∈Gˆ G×G equivariant isomorphismMd of vectorMd spaces, the hat denotes the Hilbert space direct sum. We also have a Fourier Inversion Theorem: Define ∞ C (G) ∋ f 7→ fˆ(i)= f(g) · πi(g) dg ∈ End(Vi), ZG then f(e)= Tr fˆ(i) · dim Vi. Xi∈Gˆ 4 Compare this with Fourier series: f(x)= fˆ(n) · einx, f(0) = fˆ(n), Xn∈Z Xn∈Z 1 π fˆ(n)= f(x) · e−inx dx, f ∈C∞(R/2πZ). 2π Z−π Non-compact: Not every irreducible unitary representation oc- curs in L2(G). In fact, it is still an open problem to describe Gˆ for an arbitrary real semisimple Lie group. We have an abstract Fourier Inversion Theorem: 2 ∗ L (G) ≃ Vi⊗ˆ Vi dµ(i) ≃ EndHS(Vi) dµ(i), ˆ ˆ Zi∈G Zi∈G where µ(i) denotes a measure on Gˆ, called Plancherel measure. For compact G, µ(i) = dim Vi and supp(µ) = Gˆ. For non- compact G, supp(µ) ( Gˆ. ˆ ∞ f(e)= Tr f(i) dµ(i), f ∈Cc (G). ˆ ZG G – compact: We have Borel-Weil-Bott Theorem for i H (GC/BC, Lλ), irreducible representations are uniquely deter- mined by their characters,Weyl and Kirillov’s character formu- las. Non-compact: Na¨ıve notion of character fails. Harish-Chandra: get character as a conjugation-invariant distribution on G, does determine representation up to equivalence. Regularity theo- rem: characters are real-analytic functions with singularities. No totally explicit formula for characters. Kashiwara-Schmid i [11] constructed representations of G in H (GC/BC, F), where 5 F is a G-equivariant sheaf on GC/BC. This can be regarded as an analogue of Borel-Weil-Bott theorem. Then Schmid-Vilonen [18] gave analogues of Weil and Kirillov’s character formulas. G – compact: If H ⊂ G is a closed subgroup, .(L2(G/H) ֒→ L2(G Non-compact: If H ⊂ G is a closed non-compact subgroup, .(L2(G/H) 6֒→ L2(G For example, if G = SL(2, R), Γ ⊂ G a discrete subgroup, even then L2(G/Γ) is not known in general. Strategy for understanding representations: In order to under- stand representations of compact groups K, we restricted them to a maximal torus T , and the representation theory for T is trivial. Equivalently, in order to understand finite-dimensional representations of a semisimple Lie algebra k, we restrict them to a Cartan subalgebra h. Now, to understand representations of a real semisimple Lie group G we restrict them to a maximal compact subgroup K, and the representation theory for K is assumed to be understood. (A particular choice of a maximal compact subgroup K ⊂ G does not matter, just like a partic- ular choice of a maximal torus T ⊂ K or a Cartan subalgebra h ⊂ k is not essential.) 6 2 Some Geometry and Examples of Representations 2.1 Complex Projective Line The complex projective line, usually denoted by CP 1 or P1(C), is C2 \ {0} modulo equivalence, where z λz 1 ∼ 1 , ∀λ ∈ C×. z λz 2 2 Note that z z /z z 1 1 ∼ 1 2 if z 6=0, 1 ∼ if z 6=0. z 1 2 z z /z 2 2 2 2 1 z 1 We identify points 1 ∈ CP with z ∈ C, then the comple- ment of C in CP 1 is 1 , and we think of this point as ∞. 0 Thus CP 1 = C ∪ {∞} ≈ S2 (as C∞ manifolds). The group SL(2, C) acts on CP 1 by fractional linear trans- formations: a b z az + bz 1 = 1 2 , c d z cz + dz 2 1 2 a b z az + b az+b = ∼ cz+d . c d 1 cz + d 1 az+b The map z 7→ cz+d from C ∪ {∞} to itself is also called a M¨obius or fractional linear transformation. Note that the center {±Id} ⊂ SL(2, C) acts on CP 1 trivially, so we actually have an action on CP 1 of PGL(2, C)= GL(2, C)/{center} = SL(2, C)/{±Id}. 7 Recall that an automorphism of CP 1 or any complex mani- fold M is a complex analytic map M → M which has an inverse that is also complex analytic. Theorem 1. All automorphisms of CP 1 are of the form az + b a b z 7→ , ∈ SL(2, C). cz + d c d Moreover, two matrices M1, M2 ∈ SL(2, C) produce the same 1 automorphism on CP if and only if M2 = ±M1. Next we turn our attention to the upper half-plane H = {z ∈ C; Im z > 0}. Note that SL(2, R) preserves H: If a,b,c,d ∈ R and Im z > 0, then az + b 1 az + b az¯ + b Im = − cz + d 2i cz + d cz¯ + d 1 (az + b)(cz¯ + d) − (az¯ + b)(cz + d) = 2i |cz + d|2 1 (ad − bc)(z − z¯) Im z = = > 0. 2i |cz + d|2 |cz + d|2 Thus the group SL(2, R) acts on H by automorphisms. Theorem 2. All automorphisms of the upper half-plane H are of the form az + b a b z 7→ , ∈ SL(2, R). cz + d c d Moreover, two matrices M1, M2 ∈ SL(2, R) produce the same automorphism on H if and only if M2 = ±M1. 8 The group PGL(2, R) = SL(2, R)/{±Id} is also the auto- morphism group of the lower half-plane H¯ = {z ∈ C; Im z < 0} and preserves R ∪ {∞}. Note that the Cayley transform z − i z 7→ , z + i which is the fractional linear transformation associated to the 1 −i matrix C = 1 i , takes H to the unit disk D = {z ∈ C; |z| < 1}. Of course, the group of automorphisms of D is obtained from the group of automorphisms of H by conjugating by C: C · PGL(2, R) · C−1 ⊂ PGL(2, C).