Department of Mathematics Aarhus University Denmark PhD Dissertation Harish-Chandra characters of some minimal representations Adam Ehlers Nyholm Thomsen July 31, 2015 Supervisor Prof. Bent Ørsted Abstract Let G be a simple Lie group of Hermitian type. Let K ⊂ G be a maximal compact subgroup and let H ⊂ K be a Cartan subgroup. Then H is also a Cartan subgroup of G. We denote their Lie algebras by fraktur script i.e. g; k and h. Let Λ be a weight of h then Λ can be decomposed into two parameters Λ0 and λ such that Λ0 is a weight of [k; k] \ h + and λ 2 R. Let ∆ (g; h) be a positive system such that any positive non-compact root is + greater than any compact root, let ∆n be the set of positive non-compact roots. Then set n = ⊕ + g where g is the root space of α in g . α2∆n α α C g Let γ be a dominant integral weight and let L C (γ) be the unique simple quotient of k k U(gC) ⊗k +n F C (γ). Where F C (γ) is the finite dimensional highest-weight representation C g of kC with highest-weight γ. The set of (Λ0; λ) such that L C ((Λ0; λ)) is unitarizable g has been classified, take λ minimal such that L C ((0; λ)) is unitarizable. Then we call g L C ((0; λ)) the minimal holomorphic representation of G and denote it by πMin. This representation integrates to a representation of G. There exists α; β such that 1 K πMinjK = ⊕^ n=0F (α + nβ): + Let Z 2 Z(k) such that α(Z) < 0 for all α 2 ∆n . Then CMin = Cone(Ad(G)Z) is a non- trivial, Ad(G)-invariant, proper convex cone in g such that CMin 6= f0g; g. Furthermore o CMin =6 ;. Then there exists a semigroup ГG(CMin) such that G is a subgroup and o ГG(CMin) is homeomorphic to G × CMin. Inside this semigroup there is an ideal ГG(CMin) o o which is homeomorphic to G×CMin. The ideal ГG(CMin) has a complex manifold structure. o There exists a representation of ГG(CMin) which we will also denote by πMin such that: o 1. For s 2 ГG(CMin) the operator πMin(s) is trace-class and the function s 7! tr πMin(s) is holomorphic. o 2. There exists a sequence sn 2 ГG(CMin) such that sn converges to the identity in ГG(CMin). Furthermore the functions G 3 g 7! tr πMin(gsn) converge to the Harish-Chandra character of πMin in the sense of distributions. o In this thesis we calculate tr πMin(h Exp(iX)) for h 2 H and X 2 h \ CMin. Then for ∗ g = su(p; q); sp(n; R); so (2n) we use this to calculate the Harish-Chandra character of πMin. Furthermore we also calculate the Harish-Chandra character of the odd part of the metaplectic representation in this way. For g = so(2; n) we get a reduced expression for tr πMin(h Exp(iX)) which we use to conjecture a character formula for πMin in this case. Dansk resumé Lad G være en simpel Lie gruppe af Hermitisk type. Lad K ⊂ G være en maksiml kompakt undergruppe og lad H ⊂ K være en Cartanundergruppe. Så er H også en i Cartanundergruppe af G. Vi skriver Liealgebraer med frakturskrift(også kaldet gotiske bogstaver) så g = Lie(G), k = Lie(K), etc. Lad Λ være en vægt for h ⊂ k så kan Λ dekomponeres i to parametre Λ0 og λ så Λ0 er en vægt for [k; k] \ h ⊂ [k; k] og λ 2 R. Lad ∆+(g; h) være et positivt system som opfylder at enhver positiv ikke-kompakt rod er + større end enhver kompakt rod. Lad ∆n være mængden af positive ikke-kompakte rødder. Så lader vi n = ⊕ + g hvor g er rodrummet i g tilhørende α. α2∆n α α C g Lad γ være en dominant heltallig vægt og lad L C (γ) være den entydige simple kvo- k k tient af modulet U(gC) ⊗k +n F C (γ). Hvor F C (γ) er den endeligt dimensionelle hø- C g jestevægtrepræsentation af kC med højeste vægt γ. Mængden af par (Λ0; λ) så L C ((Λ0; λ)) kan gøres til en unitær repræsentation er tidligere blevet bestemt. Lad λ være mini- g g mal således at L C ((0; λ)) kan gøres unitær. Vi betegner L C ((0; λ)) med dette λ som den minimale holomorfe repræsentation og skriver πMin. Så giver πMin anledning til en repræsentation af G. Der findes α og β således at 1 K πMinjK = ⊕^ n=0F (α + nβ): + Lad Z 2 Z(k) og vælg Z så det opfylder α(Z) < 0 for alle α 2 ∆n . Sæt CMin := Cone(Ad(G)Z) så er CMin en Ad(G)-invariant konveks kegle i g, som opfylder at CMin 6= o f0g; g. Ydermore da g er simpel er CMin 6= ;. Så eksisterer der en semigruppe ГG(CMin) så G er en undergruppe og ГG(CMin) er homøomorf til G × CMin. I denne semigruppe findes o et semigruppeideal ГG(CMin), som under den samme homøomorfi som før bliver sendt til o o G × CMin. Idealet ГG(CMin) er en kompleksmangfoldighed. Der findes en repræsentation o af ГG(CMin) som vi også kalder πMin som opfylder o 1. Lad s 2 ГG(CMin) så er operatoren πMin(s) sporklasse og funktionen s 7! tr πMin(s) er holomorf. o o 2. Der findes en følge sn 2 ГG(CMin), så sn konvergerer til identiteten i ГG(CMin). Ydermere konvergerer funktionerne G 3 g 7! tr πMin(gsn) til Harish-Chandra karakteren af πMin. Denne konvergens er i distributionsforstand. o I denne afhandling beregner vi tr πMin(h Exp(iX)) når h 2 H og X 2 h \ CMin.I ∗ tilfældene g = su(p; q); sp(n; R) og so (2n) bruger vi dette til at beregne en formel for Harish-Chandra karakteren af πMin. Vi beregner også Harish-Chandra karakteren af den ulige metaplektiske repræsentation med disse metoder. I tilfældet so(2; 2n) beregner vi et reduceret udtryk for tr πMin(h Exp(iX)) som vi bruger til at opstille en formodning om Harish-Chandra karakteren. ii Acknowledgments First of all I would like to thank my supervisor Professor Bent Ørsted for showing me a beautiful and complex area of mathematics and for suggesting a problem and an approach that is imminently suitable for a graduate thesis. I would like to thank him for his unending patience and tireless support even when I have taxed it beyond what any reasonable person could expect. I owe a great debt of gratitude to Professor Jorge Vargas for patiently and repeatedly explaining many fundamental things to me, and for supporting my progress from half-way across the globe. If there are any insights in this thesis they can ultimately be traced to one them. I want to thank Professor Toshiyuki Kobayashi for letting me visit and experience the mathematical community at Tokyo University and in Japan. I also want to thank Dr. Jan Möllers for always answering my e-mails and questions promptly. I would like to thank my family, friends and the community of students and young mathematicians at Aarhus University. Without you I would have given up much earlier in the process, your unwavering and kind support has been invaluable. iii Contents 1 Introduction 1 1.1 Harish-Chandra characters . 1 1.2 Minimal representations . 2 1.3 Ol’shanski˘ısemigroups . 3 1.4 Results . 4 1.5 Structure of this thesis . 6 1.6 Deficiencies and outlook . 7 1.7 Notation . 8 2 Characters on Ol’shanski˘ı semigroups 9 2.1 Setup . 9 2.2 Complex Lie groups . 11 2.3 Ad(G)-invariant cones . 12 2.4 Ol’shanski˘ısemigroups . 13 2.5 Representation theory of Ol’shanski˘ısemigroups . 19 2.6 Pencil of K-types . 21 2.7 SL(2; R) considerations . 24 o 2.7.1 A curve in ГSL(2;R)(CMin) ....................... 25 2.7.2 Lifts to SL(2f ; R) ............................ 26 2.8 Characters on the Ol’shanski˘ısemigroups . 28 2.8.1 Cayley transforms . 29 2.8.2 Uniformity of characters . 35 3 Minimal Representations 39 3.1 Minimal holomorphic representations . 39 3.2 Technicalities . 41 3.3 sp(n; R) ..................................... 45 3.3.1 Some structure theory . 45 3.3.2 The metaplectic representation . 51 3.3.3 The character on the compact Cartan subgroup . 52 3.3.4 The character function . 58 3.4 su(p; q) ...................................... 62 3.4.1 Structure and realization . 62 3.4.2 The minimal representation . 68 3.5 so∗(2n) ..................................... 70 3.5.1 Structure and realization . 71 3.5.2 Minimal representation . 76 v Contents 3.6 so(2; 2n) ..................................... 79 3.6.1 Structure and realization . 80 3.6.2 Minimal representations . 83 3.6.3 Combinatorics . 84 3.6.4 The character . 87 3.7 so(2; 2n + 1) ................................... 92 3.7.1 Structure and realization . 92 3.7.2 Combinatorics . 95 Bibliography 99 vi 1 Introduction The aim of this thesis is to calculate the Harish-Chandra characters for minimal represen- tations of simple Lie groups of Hermitian type. In the thesis we introduce some general results that we hope should be applicable to the calculation of characters of highest-weight representations of Lie groups of Hermitian type. However they rely on an assumption of continuous extension of the character on the Ol’shanski˘ısemigroup to the group.