Notes on the Orbit Metho d and Quantization Reyer Sjamaar Les Diablerets March 1 Contents Orbit metho d and quantization Index theorem in symplectic geometry 2 Orbit metho d and quantization A A Kirillov G Lie group innitedimensional group quan tum group Category of unitary representations of G Objects continuous homomorphisms T G UH H a Hilb ert space Morphism intertwining op erator from T to T continuous linear A H H A H H T g T g 2 1 A H H 3 Example X Gmanifold with Ginvariant mea sure Unitary representation on L X T g f x f g x Map F X X induces intertwining map F L X L X if is absolutely continuous wrt F T is indecomp osable if T T T for nonzero T and T T is irreducible if do es not have nontrivial invariant subspaces For unitary representation irreducible in decomp osable Unirrep unitary irreducible representation 4 Main problems of representation theory Describ e unitary dual G funirreps of Ggequivalence Decomp ose any T into unirreps Z T g T g dy y Y Sp ecial cases for H G closed littl e group G T a for T G decomp ose restriction Res H G S b for S H decomp ose induction Ind H Compute character of T G 5 Ad b let S H UH Supp ose GH has G G invariant measure Ind S L sections of H H G H Obtained by taking space of functions f G H satisfying f g h S hf g and completing wrt inner product Z hf f i hf x f xi dx H GH Ad let C G Put Z T g T g dg G With luck T H H is of trace class and Tr T is a distribution on G the charac ter of T 6 Solutions proposed by orbit metho d Let g Lie algebra of G Coadjoint repre sentation nonunitary representation of G on g G g G the space of coadjoint orbits Let T b e unirrep corresponding to O O g G For H G have projection pr g h Then Z G Res for O g G T mO O T 0 O O H 0 O h H 0 pr O O Z G Ind mO O T for O h H T 0 O O H 0 O g G 0 pr O O Same mO O Frobenius reciprocity 7 For O g G let character of T O O Kirill ov character formula for g Z q ihf i j exp e df O O Fourier transform of df canonical mea O p j where j derivative of sure on O j j r l l r left resp right Haar measure wrt Leb esgue measure Theorem Kirillov Ab ove is exactly right for connected simply connected nilp otent groups where j 8 Examples n n G R Then g G g R Unirrep n corresponding to R is ihxi T x e H C Fourier analysis Heisenb erg group G group of matrices g g C B g g A Typical element of Lie algebra g is B C A Basis C B C B C B z q p A A A 9 Note p q z z generates centre of g Complete list of unirreps Stonevon Neumann For T G L R is generated by d q ix z i p dx tp tq ie T e f x f x t T e f x itx tz it e f x T e e Note T p T q T z uncertainty principle G C is generated by For R S p i q i z 10 Description of gG Adjoint action g g C B g g g A Adjoint orbits O 3 2 1 11 Description of g G Identify g with lower triangular matrices Typ ical element is B C f f A f f Pairing hf i Tr f f f f Coadjoint action g f lower triangular part of g f g B C f g f A f f g f 12 Coadjoint orbits O f 3 f 2 f 1 Twodimensional orbits correspond to T zero dimensional orbits to S 13 Explanation for orbit metho d Classical Quantum Symplectic manifold M Hilb ert space H QM or PH Observable function f skewadjoint op erator Qf on H Poisson bracket ff g g commutator Qf Qg Hamiltonian ow of f PS in UH Diracs rules Qc ic c constant f Qf is linear Qf Qf Qff f g Ie f Qf is a Lie algebra homomor phism C M uH So Lie algebra homomorphism g C M gives rise to unitary representation of G on H Last rule if G acts transitively QM is a unirrep 14 Hamiltonian actions M symplectic manifold on which G acts Action is Hamiltonian if there exists Gequivariant map M g called moment map or Hamil tonian such that dh i M where vector eld on M induced by g M If G connected equivariance of is equivalent to transp ose map g C M dened by m m is homomorphism of Lie algebras Triple M is a Hamiltonian Gmanifold Notation comp osite map M R comp onent of g 15 Examples Q any manifold w Gaction G Di Q M T Q with lifted action g q p g q g p where q Q p T Q d where q v p v projection M Q q p Moment map q p p Q Poisson structure on g for C g f g f gf hf d d i f f Here d d g g f f Leaves orbits for coadjoint action For coad joint orbit O moment map is inclusion O g 16 Theorem KirillovKostantSouriau Let M b e homogeneous Hamiltonian Gmanifold Then M g is lo cal symplectomorphism onto its image Hence if G compact is global symplectomorphism Sketch proof M homogeneous image of is single orbit in g and therefore a symplectic manifold equivariant is Poisson map Conclu sion preserves symplectic form If G compact all coadjoint orbits are simply connected 17 Prequantization First attempt QM L M where n n Liouville volume element on M For f function on M put Qf f skewsymmetric op erator on L Hamil f tonian vector eld of f Wrong Qc Second try Qf if f But then Qf Qf if f 3 Qf where f ff f g Sign convention ff g g g g f f 18 Third attempt supp ose d Put Qf i f f f Works But dep ends on and what if not exact Note rst two terms are covariant dierentation wrt connection oneform Denition KostantSouriau M is prequan tizable if there exists a Hermitian line bundle L prequantum bundle with connection r such that curvature is Prequantum Hilb ert space is L sections of L and op erator asso ciated to f C M is Qf r if f 19 Example P n n dx dy L R C M R k k k P x dy Inner product k k k Z h i x y x y dx dy 2n R y and x so x y k k k k Qx k y k iy Qy k k x k Snag prequantization is to o big For n get L R R is homogeneous space under Heisenb erg group but L R is not unirrep for this group 20 Polarizations Polarization on M integrable Lagrangian sub C C bundle of T M ie subbundle P T M st C P is Lagrangian in T M for all m and vec m m tor elds tangent to P are closed under Lie bracket P is totally real if P P P is complex if P P Frobenius real p olarization Lagrangian fo liation of M NewlanderNirenb erg complex p olarization complex structure J on M st P is spanned by z in holomorphic co ordinates z k k P is Kahler if it is complex and J is a Riemannian metric 21 Section s of L is p olarized if r s for all v v tangent to P Denition QM L p olarized sections of L Problems Existence of p olarizations Qf acts on QM only if preserves P f Polarized sections are constant along real leaves of P Squareintegrability M compact P complex but not Kahler there are no p olarized sections QM indep endent of P 22 Coadjoint orbits O coadjoint orbit through f g Assume G simply connected O prequantizable G action on O lifts to L Innitesimally lift of L O L where g generator of scalar S action L on L Ginvariant p olarization P of O is determined C C by p g inverse image of P under g f f C T O f P integrable p subalgebra P Lagrangian f j ie f j is in p pp nitesimal character and dim p dim G C R dim G f R 23 C P real p p for p g Let P group generated by exp p Assume f p R exp onentiates to character S P S then f G S QM Ind f P 0 If P complex QM is holomorphically induced representation Example G compact and simply connected Let T maximal torus t p ositive Weyl chamb er f t Then O Gf integral f in integral lattice All invariant p olarizations are complex and are C determined by parabolic subalgebras p g In f C fact O GG G P where P exp p f C P QO holomorphic sections of G S f unirrep with highest weight f 24 Character formula Z q ihf i j exp e df O O where q h i h i Y e e j h i Y h f i dim QO volO h i where sum of p ositive ro ots Compare Weyl dimension formula Y h f i dim QO h i shift 25 Index theorem in symplectic geometry Recall table Classical Quantum Symplectic manifold M Hilb ert space H QM or PH Observable function f skewadjoint op erator Qf on H Poisson bracket ff g g commutator Qf Qg Hamiltonian ow of f PS in UH Continuation Hamiltonian Gaction on M unitary representation on QM Moment p olytop e M highest weights of irreducible comp onents Symplectic crosssection highestweight spaces t Symplectic quotients isotypical comp onents G QO QM O G Hom 26 Lemma ker d T Gm where Gm m m Gorbit through m im d g where g f g m m m M m Hence if f g is regular value of G acts f lo cally freely on f Theorem Meyer MarsdenWeinstein If f is regular value of nullfoliation of j 1 f is equal to Gorbits of G action Hence the f quotient M f G O G is a f f f symplectic orbifold Conjecture GuilleminSternb erg Q R G QM QM G This implies QM HomQO QM O 27 Proved by Guillemi nS ternb erg in Kahler case using geometric invariant theory In compact case can make life easier by chang ing denition of QM regard prequantum bundle L as element of K M Let M G b e map to a p oint Dene QM L regarded as element of K RepG rep G resentation ring Disadvantages works only for compact M and G dimension can b e negative no natural inner product Advantages by and large satises Diracs