W -reps, nilp orbits, orbit method David Vogan Representation theory Weyl group representations, nilpotent irr reps ! nilp orbits, and the orbit method orbits irr reps ! W reps nilp orbits $ W reps David Vogan Explaining the arrows Remembrance of Department of Mathematics things past Massachusetts Institute of Technology Lie groups: structure, actions and representations In honor of Joe Wolf, on his 75th birthday January 11–14, 2012, Bochum W -reps, nilp orbits, Outline orbit method David Vogan Representation What is representation theory about? theory irr reps ! nilp orbits Nilpotent orbits from G reps irr reps ! W reps nilp orbits $ W reps Explaining the W reps from G reps arrows Remembrance of things past W reps and nilpotent orbits What it all says about representation theory The old good fan-fold days W -reps, nilp orbits, Gelfand’s “abstract harmonic analysis” orbit method David Vogan Representation Say Lie group G acts on manifold M. Can ask about theory irr reps ! nilp I topology of M orbits I solutions of G-invariant differential equations irr reps ! W reps I special functions on M (automorphic forms, etc.) nilp orbits $ W reps Method step 1: LINEARIZE. Replace M by Hilbert Explaining the space L2(M). Now G acts by unitary operators. arrows 2 Remembrance of Method step 2: DIAGONALIZE. Decompose L (M) things past into minimal G-invariant subspaces. Method step 3: REPRESENTATION THEORY. Study minimal pieces: irreducible unitary repns of G. What repn theory is about is2 and3. Today: what do irr unitary reps look like? W -reps, nilp orbits, Short version of the talk orbit method David Vogan ∗ Irr (unitary) rep of G (coadjoint) orbit of G on g0 . Representation ! theory fifty years of shattered dreams, broken promises !! irr reps ! nilp but I’m fine now, and not bitter orbits irr reps ! W reps G reductive: coadjt orbit conj class of matrices ! nilp orbits $ W reps Questions re matrices nilpotent matrices Explaining the nilp matrices ! combinatorics: partitions (or. ) arrows Remembrance of partitions (or. ) ! Weyl grp reps things past Conclusion: irr unitary reps ! Weyl grp reps &% nilp orbits Plan of talk: explain the arrows. W -reps, nilp orbits, When everything is easy ’cause of p orbit method David Vogan Corresp G rep nilp coadjt orbit hard=R, easier=Qp. Representation G ⊂ GL(n; Qp) reductive alg, g0 ⊂ gl(n; Qp). theory ∗ ∗ irr reps ! nilp Put NG = nilp elts of g0 , the nilpotent cone. orbits irr reps ! W reps Orbit O has natural G-invt msre µO, homog deg dim O=2, hence tempered distribution; Fourier trans µ = temp gen fn on g . nilp orbits $ W bO 0 reps Theorem (Howe, Harish-Chandra local char expansion) Explaining the arrows π 2 Gb, Θπ character (generalized function on G). Remembrance of things past θπ = lift by exp to neighborhood of 0 2 g0. Then there are unique constants cO so that X WF θπ = cOµbO π −! fOjcO 6= 0g: O on some conj-invt nbhd of 0 2 g0. Depends only on π restr to any small (compact) subgp. π restr to compact open WF(π) singularity of Θπ at e W -reps, nilp orbits, Nilp orbits from G reps by analysis: WF(π) orbit method David Vogan HC Representation (π; Hπ) irr rep −! Θπ = char of π. theory Morally Θπ(g) = tr π(g); unitary op π(g) never trace irr reps ! nilp orbits class, so get not function but “generalized function”: irr reps ! W reps nilp orbits $ W Z reps Θπ(δ) = tr δ(g)π(g) (δ test density on G) Explaining the G arrows Remembrance of Singularity of Θπ (at origin) measures things past infinite-dimensionality of π. Howe definition: ∗ ∗ WF(π) = wavefront set of Θπ at e ⊂ Te (G) = g0 passage Hπ ! WF(π) is analytic classical limit. ∗ WF(π) is G-invt closed cone of nilp elts in g0, so finite union of nilp coadjt orbit closures. W -reps, nilp orbits, Nilp orbits from G reps by comm alg: AV(π) orbit method David Vogan Representation G real reductive Lie, K maximal compact subgp. theory HC K irr reps ! nilp (π; Hπ) irr rep −! Hπ Harish-Chandra module of orbits K -finite vectors; fin gen over U(g) (with rep of K ). irr reps ! W reps HK nilp orbits $ W Choose (arbitrary) fin diml gen subspace π;0, define reps 1 Explaining the K K K X K K arrows Hπ;n =def Un(g) ·Hπ;0; gr Hπ =def Hπ;n=Hπ;n−1: n=0 Remembrance of K things past gr Hπ is fin gen over poly ring S(g) (with rep of K ). K AV(π) =def support of gr Hπ = K -invariant alg cone of nilp elts in (g=k)∗ ⊂ g∗ = Spec S(g): passage Hπ ! AV(π) is algebraic classical limit. W -reps, nilp orbits, Interlude: real nilpotent cone orbit method David Vogan Representation G real reductive Lie, K max compact, θ Cartan inv. theory irr reps ! nilp g0 real Lie alg, g = g0 ⊗R C, G(C) cplx alg gp. orbits K (C) = complexification of K : cplx reductive alg. irr reps ! W reps ∗ ∗ nilp orbits $ W N ⊂ g = nilp cone; finite union of G(C) orbits. reps N ∗ = N ∗ \ g∗; finite union of G orbits. Explaining the R def 0 arrows ∗ ∗ ∗ N =def N \ (g=k) ; finite union of K (C) orbits. Remembrance of θ things past Theorem (Kostant-Sekiguchi, Schmid-Vilonen). There’s natural bijection N ∗=G $ N ∗=K ( ); WF(π) $ AV(π): R θ C Conclusion: two paths reps nilp coadj orbits, π 7! WF(π) and π 7! AV(π) are same! W -reps, nilp orbits, And now for something completely different. orbit method David Vogan . to distinguish representations: Z(g) =def center of U(g). π irr rep infinitesimal character ξ(π): Z(g) ! is Representation C theory homomorphism giving action in π. irr reps ! nilp Fix Cartan subalgebra h ⊂ g, W = Weyl group. orbits irr reps ! W reps ∗ HC: there’s bijection [infl chars ξλ : Z(g)] ! C $ h =W . nilp orbits $ W reps Π(G)(λ) = (finite) set of irr reps of infl char ξλ Explaining the W (λ) = fw 2 W j wλ − λ = integer comb of rootsg arrows Remembrance of Theorem (Lusztig-V). If λ is regular, then Π(G)(λ) has things past natural structure of W (λ)-graph. In particular, 1. W (λ) acts on free Z-module with basis Π(G(R))(λ). 2. There’s preorder ≤LR on Π(G)(λ): y ≤LR x , x appears in w · y (w 2 W (λ)) , π(x) subquo of π(y) ⊗ F (F fin diml of Ad(G)) 3. Each double cell( ∼LR class) in Π(G)(λ) is W (λ)-graph, so carries W (λ)-repn. 4. WF(π(x)) constant for x in double cell. W -reps, nilp orbits, W reps and nilpotent orbits orbit method David Vogan ∗ Nilp cone N = fin union of G(C) orbits O. Representation theory Springer corr W ,! f(O; S)g,(S loc sys on O). c irr reps ! nilp orbits Write σ 7! (O(σ); S(σ)) (σ 2 Wc). irr reps ! W reps a(σ) Write a(σ) =lowest degree with σ ⊂ S (h) nilp orbits $ W 1. a(σ) ≥ [dim(N ∗) − dim(O)]=2. Equality iff S(σ) trivial. reps Explaining the 2. For each O 9! σ(O) 2 Wc corr to (O; trivial). arrows a(σ(O)) 3. σ(O) has mult one in S (h). Remembrance of 4. Every special rep of W (Lusztig) is of form σ(O). things past Wc ⊃ Wcnilpotent ⊃ Wcspecial Type An−1: all size p(n). E8: 112 Wc ⊃ 70 Wcnilp ⊃ 46 Wcspecial. σ 2 Wc close to trivial , a(σ) small ,O(σ) large. σ 2 Wc triv , a(σ) = 0 ,O(σ) = princ nilp. σ 2 Wc sgn , a(σ) = j∆+j , O(σ) = zero nilp. W -reps, nilp orbits, W (λ) reps and nilpotent orbits orbit method David Vogan ∗ λ 2 h ∆(λ) integral roots endoscopic gp G(λ)(C) ∗ Representation Weyl group W (λ), nilp cone N (λ)). theory ∗ irr reps ! nilp σ(λ) 2 W\(λ)special $ O(λ) ⊂ N (λ); codim = 2a(σ(λ)). orbits irr reps ! W reps Proposition L ⊂ F fin gps, L 3 σ ⊂ X (reducible) rep of F. b L nilp orbits $ W If σL has mult one in X then 9! σ 2 Fb, σL ⊂ σ ⊂ X. reps a(σ(λ)) Explaining the W (λ) ⊂ W , σ(λ) ⊂ S (h) σ 2 Wc; a(σ) = a(σ(λ)). arrows Remembrance of Theorem Representation σ 2 Wc constructed from special things past σ(λ) 2 W\(λ)special belongs to Wcnilp. Get endoscopic induction special nilps for G(λ)(C) nilps for G(C); preserves codimension in nilpotent cone. O(λ) principal O principal O(λ) = f0g O orbit for maxl prim ideal of infl char λ G(λ)(C) Levi )O(λ) O Lusztig-Spaltenstein induction W -reps, nilp orbits, Our story so far. orbit method David Vogan ∗ Representation λ 2 h endoscopic gp G(λ)(C) theory irr reps ! nilp Block B of reps of G(R) of infl char λ orbits irr reps ! W reps conn comp D of W (λ)-graph Π(G(R))(λ) nilp orbits $ W reps real form G(λ)(R); D ' D(λ) ⊂ Π(G(λ)(R)) Explaining the Conclusion: for each double cell of reps of infl char λ arrows Remembrance of ∗ things past C ⊂ Π(G(R))(λ) σ 2 Wcnilp $ O⊂N l'&"" ∗ C(λ) ⊂ Π(G(λ)(R)) $ σ(λ) 2 W\(λ)spec $O (λ) ⊂ N (λ) Further conclusion: to understand G reps nilpotent orbs, must understand W graphs special W reps.