Unitarity of non-spherical principal series Alessandra Pantano July 2005 1
Minimal Principal Series G: a real split semisimple Lie group θ: Cartan involution; g = k p: Cartan decomposition of g a: maximal abelian subspace of p, A = exp G (a), M = Z K (a) = (g, a): the set of restricted roots (δ, V δ ): an irreducible tempered unitary representation of M P = MAN: a minimal parabolic in G with Levi factor MA ν: a real character of A, strictly dominant for N X P (δ, ν) = Ind G P (δ ν) a minimal principal series for G 2
Langlands quotient representation If P is the opposite parabolic, there is an intertwining operator A = A( P : P : δ : ν): X P (δ, ν) X P (δ, ν). Define the Langlands quotient representation to be the closure of the image of this operator: X(δ, ν) = Im(A( P : P : δ : ν)). X(δ, ν) is the unique irreducible quotient of X P (δ, ν). 3
Unitarity of X P (δ, ν) X P (δ ν) has a non-degenerate invariant Hermitian form if and only if there exists ω K satisfying ωp ω 1 = P ω δ δ ω ν = ν. Every non-degenerate invariant Hermitian form on X P (δ ν) is a real multiple of the form induced by the Hermitian operator B = δ(ω) R(ω) A( P : P : δ : ν) from X P (δ ν) to X P (δ ν). X P (δ ν) is unitary if and only if B is semidefinite 4
Computing the signature of B This is a very hard problem. Two reductions are possible: 1 st reduction: a K-type by K-type calculation For all µ in the principal series, we get an operator R µ (ω, ν): Hom K (E µ, X P (δ ν)) Hom K (E µ, X P (δ ν)) which, by Frobenius reciprocity and the minimality of P, becomes R µ (ω, ν): Hom M (E µ M, V δ ) Hom M (E µ M, V δ ). 5
Computing the signature of B 2 nd reduction: a rank-one reduction Decompose ω into a product of simple reflections: w = l i=1 s α i By Gindikin-Karpelevic, R µ (ω, ν) decomposes accordingly: R µ (ω, ν) = l R µ (s αi, γ i ) i=1 A factor R µ (s αi, γ i ) is induced from the corresponding intertwining operator for the SL(2, R) associated to α i The intertwining operator for SL(2, R) is known, so we get explicit formulas for the factors R µ (s αi, γ i ) 6
SL(2, R): the spherical case The intertwining operator X P (triv γ) X P (triv γ) acts by:...... φ 4 φ 2 φ 0 φ +2 φ +4............ c 4 φ 4 c 2 φ 2 c 0 φ 0 c 2 c 4...... φ +2 φ +4 We can normalize the operator so that c 0 = 1 and c 2n = Πn j=1 ((2j 1) γ, α ) Π n j=1 ((2j 1) + γ, α ) n 1 7
SL(2, R): the non spherical case The intertwining operator X P (sign γ) X P (sign γ) acts by: φ 3...... φ 1 φ +1 φ +3...... c 3...... φ 3 c 1 φ 1 +c 1 +c 3...... φ +1 φ +3 We can normalize the operator so that c 1 = 1 and c 2n+1 = (2 γ)(4 γ) (2n γ) (2 + γ)(4 + γ) (2n + γ) n 1 8
G split: the spherical case The operator R µ (ω, ν): = (E µ) M (E µ) M decomposes in operators corresponding to simple reflections Every factor R µ (s α, γ) is again an operator on (E µ) M If µ K α= j Z φ j is the decomposition of µ w.r.t. the SO(2)-subgroup K α attached to α, then (E µ) M = n N Hom M (φ 2n + φ 2n, V triv ) is the decomposition of (E µ) M in MK α -invariant subspaces R µ (s α, γ) preserves this decomposition, and acts on Hom M (φ 2n + φ 2n, V triv ) by the scalar c 2n 9
The operator R µ (s α, γ), for δ trivial Hom M (φ 0, V tr ) Hom M (φ 2 + φ 2, V tr ) Hom(φ 4 + φ 4, V tr )...... R µ (s α, γ) : 1 c 2 c 4 Hom M (φ 0, V tr ) Hom M (φ 2 + φ 2, V tr ) Hom M (φ 4 + φ 4, V tr )...... R µ (s α, γ) depends on the decomposition of µ w.r.t K α 10
R µ (s α, γ), for δ trivial and µ petite If µ is petite, (E µ) M = Hom M (φ 0, V tr ) Hom M (φ 2 + φ +2, V tr ) (E µ) M = Hom M (E µ, V tr ) carries a representation Ψ µ of W, Hom M (φ 0, V δ ) (+1)-eigenspace of Ψ µ (s α ) Hom M (φ 2 + φ +2, V δ ) ( 1)-eigenspace of Ψ µ (s α ) R µ (s α, γ) acts by c 0 on the (+1)-eigenspace of Ψ µ (s α ), and by c 2 on the ( 1)-eigenspace of Ψ µ (s α ). This gives: ( ) ( ) c0 + c 2 c0 c 2 R µ (s α, γ) = Id + Ψ µ (s α ) 2 2 11
R µ (s α, γ), for δ trivial and µ petite Hom M (φ 0, F δ ) 1 Hom M (φ 0, F δ ) Hom M (φ 2 + φ 2, F δ Hom ) M (φ 4 + φ 4, F δ )...... 1 γ, α 1+ γ, c α 4 c 6...... Hom M (φ 2 + φ 2, F δ ) Hom M (φ 4 + φ 4, F δ ) +1 eigenspace of Ψ µ (s α ) R µ (s α, γ)= 1 eigenspace of Ψ µ (s α ) +1 on the (+1)-eigenspace of Ψ µ (s α ) 1 γ, ˇα 1+ γ, ˇα on the ( 1)-eigenspace of Ψ µ (s α ) 12
Relevant W -types In the p-adic case, the spherical representation X(ν) is unitary if and only if the operator R τ (ω, ν) is positive semidefinite, for every representation τ of W Dan and Dan have determined a subset of W -representations (the relevant set) that detects unitarity: X(ν) is unitary R τ (ω, ν) is pos. semidef. for all τ relevant If µ is petite and W acts on (E µ) M by τ, then the real operator R µ (ω, ν) = the p-adic operator R τ (ω, ν) 13
Relevant W -types A n 1 (n-k, k), j n/2 B n, C n (n-j, j) (0), j n/2, (n-k) (k), k=0... n D n (n-j, j) (0), j n/2, (n-k) (k), k < n/2 (n/2) (n/2) ± if n is even F 4 1 1, 2 3, 8 1, 4 2, 9 1, E 6 1 p, 6 p, 20 p, 30 p, 15 q, E 7 1 a, 7 a, 27 a, 56 a, 21 b, 35 b, 105 b, E 8 1 x, 8 z, 35 x, 50 x, 84 x, 112 z, 400 z, 300 x, 210 x. 14
Relevant K-types Barbasch has proved that for every G real and split, there is a set of petite K-types such that the (E µ) M s realize all the relevant W-representations (relevant K-types). These relevant petite K-types provide non-unitarity certificates for spherical Langlands quotients: X(triv. ν) is unitary only if R µ (ω, ν) is positive semidefinite for every relevant K-type µ For relevant K-types, the operator R µ (ω, ν) can easily be constructed by means of Weil groups computations. 15
Classical groups The spherical unitary dual for a classical split reductive group is independent of whether the field is real or p-adic. Therefore X(triv ν) is unitary in the real case X(ν) is unitary in the p-adic case R τ (ω, ν) is positive semidefinite, for all relevant W -types τ R µ (ω, ν) is positive semidefinite, for all relevant K-types µ 16
Spherical - Not Spherical S: R µ (ω, ν) is an endomorphism of Hom M (E µ, V triv ), and this space has a W -representation N-S: R µ (ω, ν) is an endomorphism of Hom M (E µ, V δ ), and this space does not have a W -representation S: Every factor R µ (s α, γ) acts on Hom M (E µ, V triv ) N-S: R µ (s α, γ) carries Hom M (E µ, V δ ) into Hom M (E µ, V s α δ ) S: The action of R µ (s α, γ) is easy N-S: the action of R µ (s α, γ) is more complicated 17
The action of R µ (s α, γ), for α good If α is a good root, R µ (s α, γ) behaves just like in the spherical case: R µ (s α, γ) is an endomorphism of Hom M (E µ M, V δ ) Hom M (E µ M, V δ ) = n N Hom M (φ 2n + φ 2n, V δ ) R µ (s α, γ) acts on Hom M (φ 0, V δ ) by 1. For all n 1, it acts on Hom M (φ 2n + φ 2n, V δ ) by the scalar: c 2n = Πn j=1 ((2j 1) γ, α ) Π n j=1 ((2j 1) + γ, α ) 18
The action of R µ (s α, γ), for α good Hom M (φ 0, V δ ) Hom M (φ 2 + φ 2, V δ ) Hom(φ 4 + φ 4, V δ )...... R µ (s α, γ) : c 0 = 1 Hom M (φ 0, V δ ) c 2 = 1 γ, ˇα 1+ γ, ˇα Hom M (φ 2 + φ 2, V δ ) c 4 Hom M (φ 4 + φ 4, V δ ) c 6...... When µ is petite (of level 3): +1 on the (+1)-eigenspace of Ψ µ (s α ) R µ (s α, γ)= 1 γ, ˇα on the ( 1)-eigenspace of Ψ 1+ γ, ˇα µ (s α ) Ψ µ is the representation of Wδ 0 on Hom M (E µ, V δ ) 19
The action of R µ (s α, γ), for α bad The reflection s α may fail to stabilize δ, so R µ (s α, γ): Hom M (E µ, V δ ) Hom M (E µ, V s α δ ) may fail to be an endomorphism The operator R µ (s α, γ) carries Hom M (φ 2n+1 + φ 2n 1, V δ ) Hom M (φ 2n+1 + φ 2n 1, V s α δ ) For T in Hom M (φ 2n+1 + φ 2n 1, V δ ), R µ (s α, γ)t is the map φ 2n+1 + φ 2n 1 V s α δ, (v + + v ) c 2n+1 T (v + v ) 20
The operator τ α µ For every K-type µ and every root α, we have an operator τ α µ : Hom M ( Eµ, V δ) Hom M ( Eµ, V s α δ ), S S µ(σ 1 α ) For any integer k 0, set U k = Hom M (φ k + φ k, V δ ). Then If α is good, Hom M (E µ, V δ )=Hom M (E µ, V sα δ )= n N U 2n. The operator τµ α acts on U 2n by ( 1) n If α is bad, Hom M (E µ, V δ ) = n N U 2n+1. The image of an element T in U 2n+1, is the map φ 2n+1 + φ 2n 1 V s α δ, (v + + v ) ( 1) n+1 i T (v + v ) 21
The action of R µ (s α, γ), for α bad Hom M (φ 1 + φ 1, V δ ) Hom M (φ 3 + φ 3, V δ ) Hom(φ 5 + φ 5, V δ )...... R µ (s α, γ) : +ic 1 τ α µ ic 3 τ α µ ic 5 τ α µ Hom M (φ 1 + φ 1, V s α δ ) Hom M (φ 3 + φ 3, V s α δ )...... Hom M (φ 5 + φ 5, V s α δ ) If µ is petite of level 2, then R µ (s α, γ) = ic 1 τ α µ 22
A very special case Assume that there is a minimal decomposition of ω in simple roots that involves only good roots. If µ is petite, every factor R µ (s α, γ) behaves like in the spherical case, and depends only on the representation ψ µ of the Weyl group of the good co-roots on Hom M (δ, ν). The full intertwining operator R µ (ω, ν) can be constructed in terms of ψ µ, and coincides with the p-adic operator R ψµ. This gives a way to compare the non-spherical principal series for our real split group with a spherical principal series for the p-adic split group associated to the root system of the good co-roots. 23
An example Let G be the double cover of the real split E 8, and let δ be the genuine representation. Because Wδ 0 = W, we are in the very special case. A direct computation shows that not every relevant W (E 8 )-type can be realized as the representation of W on Hom M (E µ, V δ ). Therefore it might happen that the non-spherical genuine principal series X(δ ν) for the real E 8 is unitary, even if the spherical principal series X(ν) for the p-adic E 8 is not unitary. The unitarity of X(ν) could be ruled out exactly by the W -type that we are unable to match. 24
Another kind of matching... Most often, a minimal decomposition of ω involves both good and bad roots. We would still like to construct the intertwining operator in terms of the representation ψ µ of Wδ 0 on Hom M (E µ, V δ ). When is this possible? Claim: If w belongs to Wδ 0 and µ is a petite K-type of level at most two ( ), then the intertwining operator R µ (ω, ν) for G coincides with the p-adic operator R ψµ for the split group associated to the root system of the good co-roots. The hypothesis can be weakened: the characters ±3 of K α are allowed for the simple good roots α that appear in a minimal decomposition of ω as an element of W 0 δ. 25
Conclusions Suppose that The split group corresponding to W 0 δ is a classical group Every relevant W δ 0 type appears in some Hom M (E µ, V δ ) The matching of the intertwining operators is possible (in particular, ω is in W 0 δ ). Then you obtain non-unitarity certificates for X P (δ, ν). In the cases of (E 8, δ 135 ), (E 6, δ 27 ), (E 6, δ 36 ), (F 4, δ 3 ), (F 4, δ 12 ), there is a complete matching of Wδ 0-types, w 0 always belongs to Wδ 0 and the good roots determine a classical split group, so we expect to be able to get non-unitarity certificates. 26