Weyl Group Representations, Nilpotent Orbits, and the Orbit Method

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 ∈ Gb, Θπ character (generalized function on G). Remembrance of things past θπ = lift by exp to neighborhood of 0 ∈ g0.

Then there are unique constants cO so that X WF θπ = cOµbO π −→ {O|cO 6= 0}. O on some conj-invt nbhd of 0 ∈ 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 ∞ 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) = {w ∈ W | wλ − λ = integer comb of roots} 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 ∈ 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 ,→ {(O, S)},(S loc sys on O). c irr reps → nilp orbits Write σ 7→ (O(σ), S(σ)) (σ ∈ 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 ∃! σ(O) ∈ 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 ∈ Wc close to trivial ⇔ a(σ) small ⇔ O(σ) large. σ ∈ Wc triv ⇔ a(σ) = 0 ⇔ O(σ) = princ nilp. σ ∈ Wc sgn ⇔ a(σ) = |∆+| ⇔ O(σ) = zero nilp. W -reps, nilp orbits, W (λ) reps and nilpotent orbits orbit method David Vogan ∗ λ ∈ h ∆(λ) integral roots endoscopic gp G(λ)(C) ∗ Representation Weyl group W (λ), nilp cone N (λ)). theory

∗ irr reps → nilp σ(λ) ∈ 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 ∃! σ ∈ Fb, σL ⊂ σ ⊂ X. reps a(σ(λ)) Explaining the W (λ) ⊂ W , σ(λ) ⊂ S (h) σ ∈ Wc, a(σ) = a(σ(λ)). arrows Remembrance of Theorem Representation σ ∈ Wc constructed from special things past

σ(λ) ∈ 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) = {0} 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 ∈ h endoscopic gp G(λ)(C) theory irr reps → nilp Block B of reps of G(R) of inﬂ char λ orbits conn comp D of W (λ)-graph Π(G(R))(λ) irr reps → W reps nilp orbits ↔ W real form G(λ)(R); D ' D(λ) ⊂ Π(G(λ)(R)) reps Explaining the Conclusion: for each double cell of reps of inﬂ char λ arrows Remembrance of ∗ things past C ⊂ Π(G(R))(λ) σ ∈ Wcnilp ↔ O⊂N l'&↑↑ ∗ C(λ) ⊂ Π(G(λ)(R)) ↔ σ(λ) ∈ W\(λ)spec ↔O (λ) ⊂ N (λ)

Further conclusion: to understand G reps nilpotent orbs, must understand W graphs special W reps. Real nilp orb(s)WF (C) (Howe) refine this correspondence. W -reps, nilp orbits, Special reps and W -graphs orbit method David Vogan Want to understand W graphs special W reps. Representation Silently fix reg int infl char ξ (perhaps for endoscopic gp). theory

x ∈ W -graph = HC/Langlands param for irr of G(R) irr reps → nilp + orbits = (H(R)x , ∆x , Λx ) mod G(R) conj irr reps → W reps = (H , ∆+, θ , ( /2 stuff) ) mod G conj x x x Z Z x nilp orbits ↔ W + reps =( Hp, ∆p , θx , (Z/2Z stuff)x ) Explaining the Last step: move to (Cartan, pos roots) from pinning. arrows /2 stuff grades θ-ﬁxed roots as cpt/noncpt (real form) Remembrance of Z Z things past and −θ-ﬁxed roots as nonparity/parity (block). Rep of W on graph is sum (over real Cartans) of induced from stabilizer of (θx , Z/2Z stuff)...

. . . so maps to sum of induced from stabilizer of θx . W -graph for G(R) quotient with basis {involutions}. Kottwitz calculated RHS ten years ago: for classical G it’s sum of all special reps of W , each with mult given by Lusztig’s canonical quotient (of π1(O)). W -reps, nilp orbits, What is to be done? orbit method David Vogan

Representation theory W -rep with basis {Langlands params} quotient irr reps → nilp with basis {involutions}. orbits irr reps → W reps

Kottwitz calculation of RHS explains( G(R) reps) → nilp orbits ↔ W (complex special orbits). reps Explaining the Lusztig-V (arxiv 2011): there’s a W -graph with vertex arrows set {involutions}. Remembrance of things past PROBLEM: Relate LV W -graph structure on involutions to classical one on Langlands params. PROBLEM: reﬁne Kottwitz calculation to include Z/2Z stuff, to explain/calculate Howe’s wavefront set map( G(R) reps) → (real special orbits). W -reps, nilp orbits, Trying to outwit an old friend orbit method David Vogan

Representation theory

irr reps → nilp orbits

irr reps → W reps

nilp orbits ↔ W reps

Explaining the arrows

Remembrance of things past W -reps, nilp orbits, Being outwitted by an old friend. . . orbit method David Vogan

Representation theory

irr reps → nilp orbits

irr reps → W reps

nilp orbits ↔ W reps

Explaining the arrows

Remembrance of things past W -reps, nilp orbits, . . . who has old friends of his own. . . orbit method David Vogan

Representation theory

irr reps → nilp orbits

irr reps → W reps

nilp orbits ↔ W reps

Explaining the arrows

Remembrance of things past W -reps, nilp orbits, . . . doesn’t know when to quit. orbit method David Vogan

Representation theory

irr reps → nilp orbits

irr reps → W reps

nilp orbits ↔ W reps

Explaining the arrows

Remembrance of things past