Langlands parameters and finite-dimensional representations

Transcription

1 Langlands parameters and finite-dimensional representations Department of Mathematics Massachusetts Institute of Technology March 21, 2016

2 Outline What Langlands can do for you Representations of compact Lie groups Representations of finite Chevalley groups Representations of p-adic maximal compacts

3 Old reasons for listening to Langlands GL n everybody s favorite reductive group/local F. Want to understand GL n (F) = set of irr repns (hard). Classical approach (Harish-Chandra et alia 1950s): 1. find big compact subgp K GL n (F); 2. understand K (supposed to be easy?) 3. understand reps of GL n (F) in terms of restriction to K. Langlands (1960s) studies GL n (F) (global reasons). Global suggests: Better: GLn (F) n-diml reps of Gal(F/F). GLn (F) n-diml reps of Weil group of F. bij Harris/Taylor: GLn (F). n-diml of Weil-Deligne(F). Meanwhile (Howe et alia 1970s... ) continue GL n (F) K. One difficulty (of many): K not so easy after all. First question: what s Langlands tell us about K?

4 Representations of compact Lie groups This is introduction number two. Suppose K is a compact Lie group. Famous false fact: we understand K. Proof we don t: O(n) = maximal compact in GL n (R). Fix irreducible τ Ô(n). How do you write down τ? ( Highest weight?? ) How do you calculate mult of τ in principal series? Second question is branching O(n) O(1) n. Today: Ô(n) temp irr of GL n(r)/unram twist certain Langlands parameters... Second question: what s Langlands tell us about K?

5 Representations of finite Chevalley groups This is introduction number three. Suppose G is a reductive group defined over F q. Deligne-Lusztig and Lusztig described irr reps of G(F q ). Can their results be formulated in spirit of Langlands? Deligne-Lusztig start with ratl max torus T G, char θ : T(F q ) C. Lusztig: (T, θ) semisimple conj class x G(F q ). This is a step in the right direction, but not quite a Langlands classification. Third question: what s Langlands tell us about G(Fq )?

6 Structure of compact conn Lie grps K compact connected Lie T maximal torus. X (T) = def lattice of chars λ: T S 1 C X (T) = def lattice of cochars ξ : S 1 T. Adjoint rep of T on cplx Lie algebra decomposes k C = t C k C,α ; α X (T)\{0} defines finite set R = R(K, T) of roots of T in K. Each root α gives rise to root TDS φ α : SU(2) K, defined up to conjugation by T. im dφ α t + k C,α + k C, α φ α diagonal α : S 1 K coroot for α. Get R = R (K, T) X (T), (finite set in bijection with R) coroots of T in K.

7 We do understand compact conn Lie grps Saw: cpt conn Lie K T max torus (X, R, X, R ): dual lattices (X, X ), finite subsets (R, R ) in bijection. Pair (α, α ) s α : X X, s α (λ) = λ λ, α α, s α = t s α : X X. PROPERTIES: for all α R 1. RD1: α, α = 2 (so s 2 α = Id) 2. RD2: s α R = R, s α R = R, (s α β) = s α (β ) 3. RDreduced: 2α R, 2α R. Axioms root datum; W = s α α R = Weyl group. Root datum is based if we fix (R +, R,+ ) (pos roots). Axioms symm in X, X : (X, R, X, R) = dual root datum. Theorem (Grothendieck) 1. Each root datum unique cpt conn Lie grp. 2. k = k: root datum unique conn reductive alg grp /k. 3. k k: red alg grp /k Gal(k/k) based root datum.

8 Representations of compact conn Lie grps Recall cpt conn Lie K T max torus (X, R, X, R ). (X, R, X, R ) K C complex conn reductive alg = Spec(K-finite functions on K) K = max compact subgp of K C. irr reps of K = irr alg reps of K C = X /W. K = def cplx alg group (X, R, X, R) cplx dual gp. Theorem (Cartan-Weyl) 1. K ( homs φc : S 1 K ) / ( K-conj), E(φ c ) φ c. 2. Each side is X /W. Theorem (Zhelobenko) Write KC = cont irr reps of K C 1. KC (homs φ: C K) / ( K-conj), X(φ) φ. 2. X(φ) K Ind K T (C φ S 1 ). 3. E(φ S 1) = lowest K-type of X(φ).

9 Langlands classification for real groups G complex reductive alg group, Γ = Gal(C/R). Fix inner class of real forms σ = action Γ (based root datum). Definition Cartan involution for σ is inv alg aut θ of G such that σθ = θσ is compact real form of G. inner class of real forms σ = inner class of alg invs θ. Definition L-group for (G, {σ}) is L G = def G Γ. Definition Weil grp W R = C, j, 1 C W R Γ 1. Definition Langlands param = ( ) φ: W R L G / conj by } {{ } G. ss image, respect Γ Theorem (Langlands, Knapp-Zuckerman) 1. Param φ L-packet Π(φ) of reps π j G(R, σj ). 2. L-packets disjoint; cover all reps of all real forms. 3. Π(φ) indexed by ( G φ / G0). φ 4. (3) is (correctably) false. See Adams-Barbasch-Vogan.

10 Langlands classification for real max cpts G cplx reductive endowed with inner class of real forms σ inner class of alg invs θ; L G = L-group. K = G θ = cplxified max cpt of G(R, σ). Defn Compact Weil grp W R,c = S 1, j, 1 S 1 W R,c Γ 1. Defn Compact param = ( ) φ c : W R,c L G / conj by G. } {{ } respect Γ Theorem. 1. Param φ c L c -pkt Π c (φ c ) of irr reps µ j of K j = G θ j. 2. L c -packets disjoint; cover all reps of all K = G θ. 3. Π c (φ c ) indexed by ( G φ c / G φ c ) {lowest K-types of all π Π(φ)} = Π c (φ WR,c ). 5. (3) is (correctably) false...

11 Example of O 2n G = GL 2n (R), L G = GL 2n (C) Γ. Cartan involution is θg = t g 1, K = O 2n (C). Recall W R,c = S 1, j, je iθ j 1 = e iθ, j 2 = 1 S 1. Theorem says Ô 2n 2n-diml reps of W R,c. Irr reps of W R,c are 1. 1-diml trivial rep δ + (e iθ ) = 1, δ + (j) = diml sign rep δ (e iθ ) = 1, δ (j) = For m > 0 integer, 2-dimensional representation τ m (e iθ ) = ( e imθ 0 0 e imθ ), τ m (j) = ( ) 0 1 ( 1) m 0. n-dimensional rep pos ints m 1 > > m r > 0, non-neg ints (a 1,..., a r, p, q) so 2n = 2a a r + p + q. Rep is a 1 τ m1 + + a r τ mr + pδ + qδ. Highest weight for O 2n rep is (m 1 + 1,..., m 1 + 1,..., m } {{ } r + 1,..., m r + 1, 1,..., 1, 0,..., 0). } {{ }} {{ }} {{ } a 1 times a r times min(p,q) q p /2

12 groups k = F q finite field; Γ = Gal(k/k) = lim m Z/mZ. Generator is arithmetic Frobenius F = qth power map. k-ratl form of conn reductive alg G = action of Γ on based root datum = fin order aut. Definition L-group for G/k is L G = def G Γ. Here G taken over C, or Q l, or... : field for repns. Definition Weil grp W k = lim m F q m; W k Γ trivial. Definition Langlands param = ( ) ρ: W k G / conj by } {{ } G. respect Γ ρ(w k ) G (not L G) since W k 1 Γ. Respect Γ = exists f L G mapping to F, Ad(f)φ(γ) = ρ(f γ). keep coset f G ρc 0 as part of ρ c. Deligne-Langlands param φ = ( (ρ φ, N φ ) (N g ρ φ, Ad(f)N = qn) ).

13 Langlands parameters for F q G B T conn red alg /F q, F : G G Frobenius. Get Γ action on W permuting gens Γ W = W Γ w = wf (another) Frobenius morphism T T. Deligne-Lusztig built chars of G(F q ) from virt chars R T θ : T ratl maxl torus, θ char of T (F q ). Proposition. For any rational = F-stable max torus T G,! W-conj class of w so (T, F) (T, w). Prop (Macdonald) T w { ρ: W k T wf φ(γ) = φ(f γ) }. Conclusion: L-params ρ for G = DL-pairs (T, θ ). R T θ and R T θ overlap ρ, ρ G-conjugate. G(F q ) partitioned by Langlands parameters. So far this is Deligne-Lusztig 1976: (relatively) easy. Using Deligne-Langlands params to shrink L-pkts harder...

14 Lusztig s big orange book G B T conn red alg /F q, L G L-group. Def φ = (ρ, N) special if N g ρ is special nilp. Recall that φ remembers coset f G ρ,n 0. Theorem (Lusztig). Irreducible reps of G(F q ) are partitioned into packets Π(φ) by special DL parameters φ. The packet Π(φ) is indexed by irr chars of Lusztig quotient of G φ / G φ 0. Missing params (non-special N, comp rep not factoring) reps of smaller reductive groups.

15 Lifting finite to p-adic G B T conn red alg /k = F q. Fix p-adic F O P, O/P k. Γ F = Gal F/F; 1 I F Γ F Γ k 1. Weil group of F is preimage of Z = F, so 1 I F W F F 1. Set P F = wild ramif grp I F ; then I F /P F W k. Fix p-adic G based root datum of G/k, Γ F acts via Γ k. G/k and G/F have same L-group L G. Prop L-params for G/k = (tamely ramif params for G/F) IF. Def cpt Weil grp W F,c = inertia subgroup I F. Def cpt param is ρ c : I F L G s.t. extn to L-param. Extension to cpt Deligne-Langlands params φ c = (ρ c, N) easy.

16 Wild conjectures G/F conn reduc alg, inner class of F-forms σ. {K j (σ)} maxl cpt subgps of G(F, σ). L G = G Γ F L-group for (G, {σ}). Conjecture 1. Cpt DL param φ c L c -pkt of irr reps µ j (σ) of K j (σ). 2. L c packets are disjoint. 3. φ any ext of φ c Π c (φ c ) = {LKTs of all π Π(φ)}. 4. Π c (φ c ) indexed by ( G φ c / G φ c ) Πc (φ c ) = all irrs Bushnell-Kutzko type. NOTE: some K j G j (F q ), G j smaller than G. Corr reps should correspond to non-special N, etc. Chance that this is formulated properly is near zero. I know this because I m teaching Bayesian inference this semester. Hope that it s wrong in interesting ways.