Sakellaridis-Venkatesh conjectures for real classical symmetric - - PowerPoint PPT Presentation
Sakellaridis-Venkatesh conjectures for real classical symmetric spaces David Renard, joint work with C. Moeglin 27 juin 2019 COLE POLYTECHNIQUE David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical
ÉCOLE POLYTECHNIQUE –
David Renard, joint work with C. Moeglin 27 juin 2019
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 1 / 27
G : connected reductive algebraic group defined over a local field F. X = G/H spherical variety defined over F (a Borel subgroup of G has a dense orbit on X). X = X(F).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 2 / 27
G : connected reductive algebraic group defined over a local field F. X = G/H spherical variety defined over F (a Borel subgroup of G has a dense orbit on X). X = X(F). L2(X ) : unitary rep. of G = G(F) L2
d(X ) : discrete spectrum, sum of irreducible unitary representations of G in L2(X )
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 2 / 27
G : connected reductive algebraic group defined over a local field F. X = G/H spherical variety defined over F (a Borel subgroup of G has a dense orbit on X). X = X(F). L2(X ) : unitary rep. of G = G(F) L2
d(X ) : discrete spectrum, sum of irreducible unitary representations of G in L2(X )
Goal : describe L2
d(X ) in terms of Langlands L-groups, Arthur-Langlands parameters, etc
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 2 / 27
G : connected reductive algebraic group defined over a local field F. X = G/H spherical variety defined over F (a Borel subgroup of G has a dense orbit on X). X = X(F). L2(X ) : unitary rep. of G = G(F) L2
d(X ) : discrete spectrum, sum of irreducible unitary representations of G in L2(X )
Goal : describe L2
d(X ) in terms of Langlands L-groups, Arthur-Langlands parameters, etc
Motivation : periods of automorphic representations, etc.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 2 / 27
More precisely : define an L-group LGX , together with an L-morphism ϕ : LGX × SL(2, C) − → LG
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 3 / 27
More precisely : define an L-group LGX , together with an L-morphism ϕ : LGX × SL(2, C) − → LG Any discrete Langlands parameter φd : WF → LGX extends to φd : WF × SL(2, C) − → LGX × SL(2, C) (identically on SL(2, C)), and composing with ϕ gives
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 3 / 27
More precisely : define an L-group LGX , together with an L-morphism ϕ : LGX × SL(2, C) − → LG Any discrete Langlands parameter φd : WF → LGX extends to φd : WF × SL(2, C) − → LGX × SL(2, C) (identically on SL(2, C)), and composing with ϕ gives ψ = ϕ ◦ φd : WF × SL(2, C) − → LG, an Arthur parameter.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 3 / 27
Π(G, ψ) : Arthur packet attached to ψ
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 4 / 27
Π(G, ψ) : Arthur packet attached to ψ Conjecture : Any irreducible unitary rep. π of G occuring in L2(X ) should be in an Arthur packet Π(G, ψ) with ψ = ϕ ◦ φd as above.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 4 / 27
Π(G, ψ) : Arthur packet attached to ψ Conjecture : Any irreducible unitary rep. π of G occuring in L2(X ) should be in an Arthur packet Π(G, ψ) with ψ = ϕ ◦ φd as above. Given X , ϕ :
LGX × SL(2, C) −
→ LG is fixed, and for each π ⊂ L2(X ), there exists φd = φd(π) with the properties above.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 4 / 27
F = R, G classical (meaning GL(.), SO(.), Sp(.), or U(.)) X = G/H symmetric space, ie. : σ : involution of G over R, H = Gσ, and H(R)e ⊂ H ⊂ H(R)
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 5 / 27
F = R, G classical (meaning GL(.), SO(.), Sp(.), or U(.)) X = G/H symmetric space, ie. : σ : involution of G over R, H = Gσ, and H(R)e ⊂ H ⊂ H(R) We propose a little extension of the setting :
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 5 / 27
F = R, G classical (meaning GL(.), SO(.), Sp(.), or U(.)) X = G/H symmetric space, ie. : σ : involution of G over R, H = Gσ, and H(R)e ⊂ H ⊂ H(R) We propose a little extension of the setting : χ : unitary character of H, gives a line bundle Lχ over X = G/H. L2(X )χ : square integrable sections, with discrete part L2
d(X )χ
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 5 / 27
F = R, G classical (meaning GL(.), SO(.), Sp(.), or U(.)) X = G/H symmetric space, ie. : σ : involution of G over R, H = Gσ, and H(R)e ⊂ H ⊂ H(R) We propose a little extension of the setting : χ : unitary character of H, gives a line bundle Lχ over X = G/H. L2(X )χ : square integrable sections, with discrete part L2
d(X )χ
Same conjecture, with an L-group LGX ,χ which also depends on χ
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 5 / 27
F = R, G classical (meaning GL(.), SO(.), Sp(.), or U(.)) X = G/H symmetric space, ie. : σ : involution of G over R, H = Gσ, and H(R)e ⊂ H ⊂ H(R) We propose a little extension of the setting : χ : unitary character of H, gives a line bundle Lχ over X = G/H. L2(X )χ : square integrable sections, with discrete part L2
d(X )χ
Same conjecture, with an L-group LGX ,χ which also depends on χ Theorem : In our setting, with χ trivial, the conjecture is true.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 5 / 27
The proof uses case by case considerations. In each case, we fix an H with H(R)e ⊂ H ⊂ H(R).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 6 / 27
The proof uses case by case considerations. In each case, we fix an H with H(R)e ⊂ H ⊂ H(R). If the conjecture is true for H with H(R)e ⊂ H ⊂ H(R), it is true for H1 with H ⊂ H1 ⊂ H(R).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 6 / 27
The proof uses case by case considerations. In each case, we fix an H with H(R)e ⊂ H ⊂ H(R). If the conjecture is true for H with H(R)e ⊂ H ⊂ H(R), it is true for H1 with H ⊂ H1 ⊂ H(R). the conjecture may be false for H = H(R)e and χ trivial.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 6 / 27
We recall some results of Flensted-Jensen, Oshima-Matsuki, Schlichtkrull and Vogan
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 7 / 27
We recall some results of Flensted-Jensen, Oshima-Matsuki, Schlichtkrull and Vogan θ : Cartan involution θ, σθ = θσ, K = G θ maximal compact subgroup of G g0 = k0 ⊕ p0, g0 = h0 ⊕ s0, g0 = (k0 ∩ p0) ⊕ (k0 ∩ s0) ⊕ (p0 ∩ h0) ⊕ (p0 ∩ s0).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 7 / 27
We recall some results of Flensted-Jensen, Oshima-Matsuki, Schlichtkrull and Vogan θ : Cartan involution θ, σθ = θσ, K = G θ maximal compact subgroup of G g0 = k0 ⊕ p0, g0 = h0 ⊕ s0, g0 = (k0 ∩ p0) ⊕ (k0 ∩ s0) ⊕ (p0 ∩ h0) ⊕ (p0 ∩ s0). Cartan subspace : maximal abelian subalgebra t0 of semi-simple elements in s0 rank of X = dimension of a Cartan subspace.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 7 / 27
We recall some results of Flensted-Jensen, Oshima-Matsuki, Schlichtkrull and Vogan θ : Cartan involution θ, σθ = θσ, K = G θ maximal compact subgroup of G g0 = k0 ⊕ p0, g0 = h0 ⊕ s0, g0 = (k0 ∩ p0) ⊕ (k0 ∩ s0) ⊕ (p0 ∩ h0) ⊕ (p0 ∩ s0). Cartan subspace : maximal abelian subalgebra t0 of semi-simple elements in s0 rank of X = dimension of a Cartan subspace. XK = K/K ∩ H : compact symmetric space.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 7 / 27
We recall some results of Flensted-Jensen, Oshima-Matsuki, Schlichtkrull and Vogan θ : Cartan involution θ, σθ = θσ, K = G θ maximal compact subgroup of G g0 = k0 ⊕ p0, g0 = h0 ⊕ s0, g0 = (k0 ∩ p0) ⊕ (k0 ∩ s0) ⊕ (p0 ∩ h0) ⊕ (p0 ∩ s0). Cartan subspace : maximal abelian subalgebra t0 of semi-simple elements in s0 rank of X = dimension of a Cartan subspace. XK = K/K ∩ H : compact symmetric space. Theorem : X has discrete series if and only if rank(X ) = rank(XK), ie. there exists a Cartan subspace t0 for X contained in k0 ∩ s0 (compact Cartan subspace)
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 7 / 27
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 8 / 27
X = G/H in the list. Fix t0 compact Cartan subspace of X (all choices are conjugated under K ∩ H), set L = CentG(t0), L = CentG(t0), and choose q = l ⊕ u, a θ-stable parabolic subalgebra of G.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 9 / 27
X = G/H in the list. Fix t0 compact Cartan subspace of X (all choices are conjugated under K ∩ H), set L = CentG(t0), L = CentG(t0), and choose q = l ⊕ u, a θ-stable parabolic subalgebra of G. Define P(q) to be the set of unitary characters πL of L satisfying (i) and (ii) below :
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 9 / 27
X = G/H in the list. Fix t0 compact Cartan subspace of X (all choices are conjugated under K ∩ H), set L = CentG(t0), L = CentG(t0), and choose q = l ⊕ u, a θ-stable parabolic subalgebra of G. Define P(q) to be the set of unitary characters πL of L satisfying (i) and (ii) below : (i) πL is in the fair range for q : Aq(πL) = RS
q,L,G(πL) (Vogan-Zuckerman module, coho-
mological induction in degree S = dim(u ∩ k)) is unitary (possibly 0) If πL is in the good range, Aq(πL) is irreducible unitary, non-zero.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 9 / 27
X = G/H in the list. Fix t0 compact Cartan subspace of X (all choices are conjugated under K ∩ H), set L = CentG(t0), L = CentG(t0), and choose q = l ⊕ u, a θ-stable parabolic subalgebra of G. Define P(q) to be the set of unitary characters πL of L satisfying (i) and (ii) below : (i) πL is in the fair range for q : Aq(πL) = RS
q,L,G(πL) (Vogan-Zuckerman module, coho-
mological induction in degree S = dim(u ∩ k)) is unitary (possibly 0) If πL is in the good range, Aq(πL) is irreducible unitary, non-zero. (ii) : assume πL sufficiently regular (eg. good range), then µ = πL ⊗ top(u ∩ p) is the highest weight of a K-type Vµ in Aq(πL) (bottom-layer). This is a minimal K-type with multiplicity 1. The condition is then : Vµ has non-trivial K ∩ H-invariants
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 9 / 27
Reformulation of (ii) : (ii′) : µ = πL ⊗ top(u ∩ p) is trivial on K ∩ H ∩ L.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 10 / 27
Reformulation of (ii) : (ii′) : µ = πL ⊗ top(u ∩ p) is trivial on K ∩ H ∩ L. (ii) is the condition which appears in Flensted-Jensen : Aq(πL) (πL regular) occurs in L2
d(X ).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 10 / 27
Reformulation of (ii) : (ii′) : µ = πL ⊗ top(u ∩ p) is trivial on K ∩ H ∩ L. (ii) is the condition which appears in Flensted-Jensen : Aq(πL) (πL regular) occurs in L2
d(X ).
reformulation (ii′) is due to Schlichkrull, it uses Flensted-Jensen duality and Cartan-Helgason theorem
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 10 / 27
Reformulation of (ii) : (ii′) : µ = πL ⊗ top(u ∩ p) is trivial on K ∩ H ∩ L. (ii) is the condition which appears in Flensted-Jensen : Aq(πL) (πL regular) occurs in L2
d(X ).
reformulation (ii′) is due to Schlichkrull, it uses Flensted-Jensen duality and Cartan-Helgason theorem (ii′) does not assume πL regular : it gives the correct definition in the fair range.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 10 / 27
Reformulation of (ii) : (ii′) : µ = πL ⊗ top(u ∩ p) is trivial on K ∩ H ∩ L. (ii) is the condition which appears in Flensted-Jensen : Aq(πL) (πL regular) occurs in L2
d(X ).
reformulation (ii′) is due to Schlichkrull, it uses Flensted-Jensen duality and Cartan-Helgason theorem (ii′) does not assume πL regular : it gives the correct definition in the fair range. This d´ efinition is compatible with translation functors : Aq(πL) and discrete series of X come in coherent families...
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 10 / 27
{qj}j : system of representatives of θ-stable parabolic subalgebra qj = l ⊕ uj under NormK(t0).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 11 / 27
{qj}j : system of representatives of θ-stable parabolic subalgebra qj = l ⊕ uj under NormK(t0). THM (Flensted-Jensen, Oshima-Matsuki -Schlichtkrull-Vogan) : L2
d(X ) = j
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 11 / 27
{qj}j : system of representatives of θ-stable parabolic subalgebra qj = l ⊕ uj under NormK(t0). THM (Flensted-Jensen, Oshima-Matsuki -Schlichtkrull-Vogan) : L2
d(X ) = j
Aqj(πL) are unitary irreducible (Vogan), or 0
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 11 / 27
From our previous work on Arthur packets for classical groups, we are able to give, for each π = Aqj(πL) in the theorem, an Arthur parameter ψπ = ψ : WR × SL(2, C) → LG with π ∈ Π(G, ψ).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 12 / 27
From our previous work on Arthur packets for classical groups, we are able to give, for each π = Aqj(πL) in the theorem, an Arthur parameter ψπ = ψ : WR × SL(2, C) → LG with π ∈ Π(G, ψ). All we need to give the form of ψπ is L = CentrG(t0). This group is conjugated to a group denoted by L(X ) in Sakellaridis-Venkatesh, Levi factor of a minimal σ-split parabolic P(X ) of G, and computed in all 13 cases by Knop-Schalke.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 12 / 27
Recall that we want to factorize Arthur’s parameter ψπ = ψ : WR × SL(2, C) − → LG associated to discrete series π of X , and write ψ = ϕ ◦ φd with ϕ : LGX × SL(2, C) − → LG : L-morphism and φd : WR → LGX : discrete Langlands parameter
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 13 / 27
Recall that we want to factorize Arthur’s parameter ψπ = ψ : WR × SL(2, C) − → LG associated to discrete series π of X , and write ψ = ϕ ◦ φd with ϕ : LGX × SL(2, C) − → LG : L-morphism and φd : WR → LGX : discrete Langlands parameter Notice that ψ|SL(2) = ϕ|SL(2) [SV] are specific about this restriction : ϕ|SL(2) : SL(2, C) → LG is a Jacobson-Morozov morphism associated to the principal unipotent orbit of L(X )∨ ⊂ G ∨, where L(X )∨ is dual to the Levi L(X ) introduced above.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 13 / 27
Prop : The Arthur parameter ψπ we associate to π = Aq(πL) discrete series of X by our results on Arthur parameters for real classical groups has restriction to SL(2) the Jacobson-Morozov morphism associated to the principal unipotent orbit of L(X )∨ ⊂ G ∨, as prescribed by [SV].
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 14 / 27
Prop : The Arthur parameter ψπ we associate to π = Aq(πL) discrete series of X by our results on Arthur parameters for real classical groups has restriction to SL(2) the Jacobson-Morozov morphism associated to the principal unipotent orbit of L(X )∨ ⊂ G ∨, as prescribed by [SV]. Proof : the cohomological induction for π = Aq(πL) is from L = L(R), with L conjugated in G with L(X ), so the dual of L is also L(X )∨. Parameters ψπ have restriction to SL(2) equal to the Jacobson-Morozov morphism associated to the principal unipotent orbit of L(X )∨ ⊂ G ∨.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 14 / 27
Prop : The Arthur parameter ψπ we associate to π = Aq(πL) discrete series of X by our results on Arthur parameters for real classical groups has restriction to SL(2) the Jacobson-Morozov morphism associated to the principal unipotent orbit of L(X )∨ ⊂ G ∨, as prescribed by [SV]. Proof : the cohomological induction for π = Aq(πL) is from L = L(R), with L conjugated in G with L(X ), so the dual of L is also L(X )∨. Parameters ψπ have restriction to SL(2) equal to the Jacobson-Morozov morphism associated to the principal unipotent orbit of L(X )∨ ⊂ G ∨. In all cases of the list, L(X ) and thus also L(X )∨ are explicitely computed by Knopp- Schalke
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 14 / 27
Prop : The Arthur parameter ψπ we associate to π = Aq(πL) discrete series of X by our results on Arthur parameters for real classical groups has restriction to SL(2) the Jacobson-Morozov morphism associated to the principal unipotent orbit of L(X )∨ ⊂ G ∨, as prescribed by [SV]. Proof : the cohomological induction for π = Aq(πL) is from L = L(R), with L conjugated in G with L(X ), so the dual of L is also L(X )∨. Parameters ψπ have restriction to SL(2) equal to the Jacobson-Morozov morphism associated to the principal unipotent orbit of L(X )∨ ⊂ G ∨. In all cases of the list, L(X ) and thus also L(X )∨ are explicitely computed by Knopp- Schalke So far, we need only L(X ) ≃ L as a complex group. For more precise information, we will need later the real form L of L used for cohomological induction.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 14 / 27
L(X )∨ = GL(2)(p+q)/2 × SO(2), p + q odd
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 15 / 27
What we want now is ψ|WR : WR
φd
− → LGX
ϕ
− → LG
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 16 / 27
What we want now is ψ|WR : WR
φd
− → LGX
ϕ
− → LG The image is in the commutant G of ϕ|SL(2) in LG
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 16 / 27
What we want now is ψ|WR : WR
φd
− → LGX
ϕ
− → LG The image is in the commutant G of ϕ|SL(2) in LG The L-group LGX is in fact already known : it is a semi-direct product LGX = G ∨
X ⋊ WR.
Knopp and Schalke have computed G ∨
X .
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 16 / 27
What we want now is ψ|WR : WR
φd
− → LGX
ϕ
− → LG The image is in the commutant G of ϕ|SL(2) in LG The L-group LGX is in fact already known : it is a semi-direct product LGX = G ∨
X ⋊ WR.
Knopp and Schalke have computed G ∨
X .
If we want an L-group which admits discrete Langlands parameters φd, the semi-direct product structure is fixed.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 16 / 27
What we want now is ψ|WR : WR
φd
− → LGX
ϕ
− → LG The image is in the commutant G of ϕ|SL(2) in LG The L-group LGX is in fact already known : it is a semi-direct product LGX = G ∨
X ⋊ WR.
Knopp and Schalke have computed G ∨
X .
If we want an L-group which admits discrete Langlands parameters φd, the semi-direct product structure is fixed. In some cases, we find that G ≃ LGX . This gives ϕ in those cases and proves the conjecture.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 16 / 27
What we want now is ψ|WR : WR
φd
− → LGX
ϕ
− → LG The image is in the commutant G of ϕ|SL(2) in LG The L-group LGX is in fact already known : it is a semi-direct product LGX = G ∨
X ⋊ WR.
Knopp and Schalke have computed G ∨
X .
If we want an L-group which admits discrete Langlands parameters φd, the semi-direct product structure is fixed. In some cases, we find that G ≃ LGX . This gives ϕ in those cases and proves the conjecture. these are cases 2, 5, 6, 9 (n even), 10 (n even), 11, 12, 13
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 16 / 27
G ≃ LGX = Sp(2p) × WR
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 17 / 27
In those cases, we have G ≃ LGX × {±1}.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 18 / 27
In those cases, we have G ≃ LGX × {±1}.
L(X )∨ = GL(1)r+s × Sp(2(r′ + s′ − r − s)),
LGX = Sp(2(r + s)) × WR, LG = Sp(2n) × WR
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 18 / 27
In those cases, we have G ≃ LGX × {±1}.
L(X )∨ = GL(1)r+s × Sp(2(r′ + s′ − r − s)),
LGX = Sp(2(r + s)) × WR, LG = Sp(2n) × WR
ϕ : natural inclusion Sp(2(r + s)) ֒ → Sp(2n), extended in the obvious way to : ϕ : LGX = Sp(2(r + s)) × WR ֒ → G ⊂ Sp(2n) × WR = LG,
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 18 / 27
In those cases, we have G ≃ LGX × {±1}.
L(X )∨ = GL(1)r+s × Sp(2(r′ + s′ − r − s)),
LGX = Sp(2(r + s)) × WR, LG = Sp(2n) × WR
ϕ : natural inclusion Sp(2(r + s)) ֒ → Sp(2n), extended in the obvious way to : ϕ : LGX = Sp(2(r + s)) × WR ֒ → G ⊂ Sp(2n) × WR = LG, Also, we can twist ϕ by the sign character of WR, with value in the {±1} factor of G and get : ϕ′ : LGX = Sp(2(r + s)) × WR ֒ → G ⊂ Sp(2n) × WR = LG,
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 18 / 27
In those cases, we have G ≃ LGX × {±1}.
L(X )∨ = GL(1)r+s × Sp(2(r′ + s′ − r − s)),
LGX = Sp(2(r + s)) × WR, LG = Sp(2n) × WR
ϕ : natural inclusion Sp(2(r + s)) ֒ → Sp(2n), extended in the obvious way to : ϕ : LGX = Sp(2(r + s)) × WR ֒ → G ⊂ Sp(2n) × WR = LG, Also, we can twist ϕ by the sign character of WR, with value in the {±1} factor of G and get : ϕ′ : LGX = Sp(2(r + s)) × WR ֒ → G ⊂ Sp(2n) × WR = LG,
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 18 / 27
L(X )∨ = GL(1)r+s × SO(2(r′ + s′ − r − s)) this case is similar to the previous one with
LGX = SO(2(r + s) + 1) × WR
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 19 / 27
L(X )∨ = GL(1)r+s × SO(2(r′ + s′ − r − s)) this case is similar to the previous one with
LGX = SO(2(r + s) + 1) × WR
we also get two possible morphisms ϕ, ϕ′ : LGX = SO(2(r + s) + 1) × WR ֒ → G ⊂ SO(2n) ⋊ WR = LG, and the one we want is the first one.
in the description of discrete series π of X as Aq(πL).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 19 / 27
L(X )∨ = GL(1)r+s × SO(2(r′ + s′ − r − s)) this case is similar to the previous one with
LGX = SO(2(r + s) + 1) × WR
we also get two possible morphisms ϕ, ϕ′ : LGX = SO(2(r + s) + 1) × WR ֒ → G ⊂ SO(2n) ⋊ WR = LG, and the one we want is the first one.
in the description of discrete series π of X as Aq(πL). ϕ′ could be used for the generalized conjecture with L2(X )χ.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 19 / 27
In those cases, we have G ≃ LGX × SO(2).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 20 / 27
In those cases, we have G ≃ LGX × SO(2).
LGX = Sp(p + q − 1) × WR, LG = SO(2p + 2q) × WR
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 20 / 27
In those cases, we have G ≃ LGX × SO(2).
LGX = Sp(p + q − 1) × WR, LG = SO(2p + 2q) × WR
in fact, in that case, ϕ is fixed G ≃ (Sp(p + q − 1) × SO(2)) ⋊ WR
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 20 / 27
In those cases, we have G ≃ LGX × SO(2).
LGX = Sp(p + q − 1) × WR, LG = SO(2p + 2q) × WR
in fact, in that case, ϕ is fixed G ≃ (Sp(p + q − 1) × SO(2)) ⋊ WR
G ≃ LGX = Sp(n − 1) × WR, LG = SO(2n) × WR ϕ, ϕ′ : LGX = Sp(n − 1) × WR ֒ → G ⊂ SO(2n) × WR = LG,
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 20 / 27
In those cases, we have G ≃ LGX × SO(2).
LGX = Sp(p + q − 1) × WR, LG = SO(2p + 2q) × WR
in fact, in that case, ϕ is fixed G ≃ (Sp(p + q − 1) × SO(2)) ⋊ WR
G ≃ LGX = Sp(n − 1) × WR, LG = SO(2n) × WR ϕ, ϕ′ : LGX = Sp(n − 1) × WR ֒ → G ⊂ SO(2n) × WR = LG,
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 20 / 27
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 21 / 27
G ≃ (GL(2p) × Z(GL(n − 2p)) × WR is much bigger than LGX = Sp(2p) × R. This is due to the fact that some spherical roots (Knop) are not roots
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 21 / 27
G ≃ (GL(2p) × Z(GL(n − 2p)) × WR is much bigger than LGX = Sp(2p) × R. This is due to the fact that some spherical roots (Knop) are not roots In this case, we check directly that Arthur parameters ψ = ψπ factorise through the natural inclusion ϕ : LGX = Sp(2p) × WR ֒ → G ≃ (GL(2p) × Z(GL(n − 2p)) × WR ⊂ LG and discrete Langlands parameters φd : WR → LGX = Sp(2p) × WR
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 21 / 27
G ≃ (GL(2p) × Z(GL(n − 2p)) × WR is much bigger than LGX = Sp(2p) × R. This is due to the fact that some spherical roots (Knop) are not roots In this case, we check directly that Arthur parameters ψ = ψπ factorise through the natural inclusion ϕ : LGX = Sp(2p) × WR ֒ → G ≃ (GL(2p) × Z(GL(n − 2p)) × WR ⊂ LG and discrete Langlands parameters φd : WR → LGX = Sp(2p) × WR The proof uses condition (ii) (or (ii′)) in the description of discrete series π of X as Aq(πL).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 21 / 27
Then π = IndG
P=MN((⊗iδi) ⊗ TrivGL(n−2p,R)) where P = MN is the standard parabolic
subgroup with Levi M ≃ GL(2, R)p × GL(n − 2p, R), and δi is a discrete series of the i-th GL(2, R) with trivial central character.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 22 / 27
Then π = IndG
P=MN((⊗iδi) ⊗ TrivGL(n−2p,R)) where P = MN is the standard parabolic
subgroup with Levi M ≃ GL(2, R)p × GL(n − 2p, R), and δi is a discrete series of the i-th GL(2, R) with trivial central character. δi : WR → GL(2, C) (Langlands correspondance, same notation for the representation and its Langlands parameter)
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 22 / 27
Then π = IndG
P=MN((⊗iδi) ⊗ TrivGL(n−2p,R)) where P = MN is the standard parabolic
subgroup with Levi M ≃ GL(2, R)p × GL(n − 2p, R), and δi is a discrete series of the i-th GL(2, R) with trivial central character. δi : WR → GL(2, C) (Langlands correspondance, same notation for the representation and its Langlands parameter) φd = ⊕iδi → GL(2p, C) is symplectic, factorizes through Sp(2p, C).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 22 / 27
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 23 / 27
Here LGX = Sp(2(r + s)) × WR (or Sp(2n) × WR)
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 23 / 27
Here LGX = Sp(2(r + s)) × WR (or Sp(2n) × WR) L ≃ U(1)2(r+s) × U(r′ − r, s′ − s), L ∩ H ≃ U(1)(r+s) × U(r′ − r, s′ − s) with factors U(1) diagonally embedded in U(1) × U(1).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 23 / 27
Here LGX = Sp(2(r + s)) × WR (or Sp(2n) × WR) L ≃ U(1)2(r+s) × U(r′ − r, s′ − s), L ∩ H ≃ U(1)(r+s) × U(r′ − r, s′ − s) with factors U(1) diagonally embedded in U(1) × U(1). Condition (ii′) translate here to πL trivial on L ∩ H, this is what we need for the proof.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 23 / 27
This case is problematic, [KS] predicts LGX = Sp(2(r + s)) × WR but it doesn’t work, wrong parity for infinitesimal character of π, ψπ, φd...
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 24 / 27
This case is problematic, [KS] predicts LGX = Sp(2(r + s)) × WR but it doesn’t work, wrong parity for infinitesimal character of π, ψπ, φd... Solution suggested by Sakellaridis : take LGX = Sp(2(r + s)) ⋊ WR
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 24 / 27
This case is problematic, [KS] predicts LGX = Sp(2(r + s)) × WR but it doesn’t work, wrong parity for infinitesimal character of π, ψπ, φd... Solution suggested by Sakellaridis : take LGX = Sp(2(r + s)) ⋊ WR But this is not an L-group, in the sense of Langlands... This is an E-group in the sense of Adams-Barbasch-Vogan... Sakellaridis is ok with this, so are we...
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 24 / 27
So, for each π discrete series of one of our spaces X , we found an Arthur parameter ψ = ψπ satisfying the SV-conjecture.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 25 / 27
So, for each π discrete series of one of our spaces X , we found an Arthur parameter ψ = ψπ satisfying the SV-conjecture. Let us change the point of view : suppose ψ is given, what can we say about the members
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 25 / 27
So, for each π discrete series of one of our spaces X , we found an Arthur parameter ψ = ψπ satisfying the SV-conjecture. Let us change the point of view : suppose ψ is given, what can we say about the members
We study this question through the invariant ǫ(π) attached by Arthur to any π ∈ Π(ψ) :
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 25 / 27
So, for each π discrete series of one of our spaces X , we found an Arthur parameter ψ = ψπ satisfying the SV-conjecture. Let us change the point of view : suppose ψ is given, what can we say about the members
We study this question through the invariant ǫ(π) attached by Arthur to any π ∈ Π(ψ) : ψ : WR × SL(2) → LG Arthur parameter A(ψ) = CentG ∨(ψ)/(CentG ∨(ψ))0 : for G classical, this is a finite 2-group Arthur’s ǫ(π) is an invariant attached to π ∈ Π(ψ), a finite dimensional representation of A(ψ).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 25 / 27
So, for each π discrete series of one of our spaces X , we found an Arthur parameter ψ = ψπ satisfying the SV-conjecture. Let us change the point of view : suppose ψ is given, what can we say about the members
We study this question through the invariant ǫ(π) attached by Arthur to any π ∈ Π(ψ) : ψ : WR × SL(2) → LG Arthur parameter A(ψ) = CentG ∨(ψ)/(CentG ∨(ψ))0 : for G classical, this is a finite 2-group Arthur’s ǫ(π) is an invariant attached to π ∈ Π(ψ), a finite dimensional representation of A(ψ). We have shown in our previous work that for classical group, ǫ(π) is one-dimensional
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 25 / 27
So here ǫ(π) ∈ A(ψ). From a computation of the precise form of the group L in π = Aq(πL), we get ǫ(π).
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 26 / 27
So here ǫ(π) ∈ A(ψ). From a computation of the precise form of the group L in π = Aq(πL), we get ǫ(π). I won’t give here the case by case answer... but
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 26 / 27
So here ǫ(π) ∈ A(ψ). From a computation of the precise form of the group L in π = Aq(πL), we get ǫ(π). I won’t give here the case by case answer... but
series of X if and only if π′ is.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 26 / 27
So here ǫ(π) ∈ A(ψ). From a computation of the precise form of the group L in π = Aq(πL), we get ǫ(π). I won’t give here the case by case answer... but
series of X if and only if π′ is.
and a parameter ψ for G. Then, π ∈ Π(G, ψ) is a discrete series for at most one of the symmetric spaces X1, ... , Xt.
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 26 / 27
Examples : G = U(p, q), Hr,s = U(r, s) × U(r′, s′)) G = SO(p, q), Hr,s = SO(r, s) × (SO(r′, s′)) with various r, s with r + s fixed U(2r, 2s)/U(r, s) × U(r, s) and U(2n, 2n)/GL(n, C), n = r + s U(2n, 2n)/Sp(n, n) and U(2n, 2n)/Sp(2n, R) SO(2n, 2n)/U(n, n) and SO(2n, 2n)/GL(2n, R) Sp(4n, R)/Sp(2n, C) and Sp(4n, R)/Sp(2n, R) × Sp(2n, R)
David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 27 / 27