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 ÉCOLE POLYTECHNIQUE – David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 1 / 27

The conjectures (general setting, vague statement) 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

The conjectures (general setting, vague statement) 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 ). L 2 ( X ) : unitary rep. of G = G ( F ) L 2 d ( X ) : discrete spectrum, sum of irreducible unitary representations of G in L 2 ( X ) David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 2 / 27

The conjectures (general setting, vague statement) 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 ). L 2 ( X ) : unitary rep. of G = G ( F ) L 2 d ( X ) : discrete spectrum, sum of irreducible unitary representations of G in L 2 ( X ) Goal : describe L 2 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

The conjectures (general setting, vague statement) 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 ). L 2 ( X ) : unitary rep. of G = G ( F ) L 2 d ( X ) : discrete spectrum, sum of irreducible unitary representations of G in L 2 ( X ) Goal : describe L 2 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

The conjectures (general setting, vague statement) More precisely : define an L -group L G X , together with an L -morphism ϕ : L G X × SL (2 , C ) − → L G David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 3 / 27

The conjectures (general setting, vague statement) More precisely : define an L -group L G X , together with an L -morphism ϕ : L G X × SL (2 , C ) − → L G Any discrete Langlands parameter φ d : W F → L G X extends to → L G X × SL (2 , C ) φ d : W F × 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

The conjectures (general setting, vague statement) More precisely : define an L -group L G X , together with an L -morphism ϕ : L G X × SL (2 , C ) − → L G Any discrete Langlands parameter φ d : W F → L G X extends to → L G X × SL (2 , C ) φ d : W F × SL (2 , C ) − (identically on SL (2 , C )), and composing with ϕ gives → L G , ψ = ϕ ◦ φ d : W F × SL (2 , C ) − an Arthur parameter. David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 3 / 27

The conjectures (general setting, vague statement) 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

The conjectures (general setting, vague statement) Π( G , ψ ) : Arthur packet attached to ψ Conjecture : Any irreducible unitary rep. π of G occuring in L 2 ( X ) should be in an Arthur packet Π( G , ψ ) with ψ = ϕ ◦ φ d as above. David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 4 / 27

The conjectures (general setting, vague statement) Π( G , ψ ) : Arthur packet attached to ψ Conjecture : Any irreducible unitary rep. π of G occuring in L 2 ( X ) should be in an Arthur packet Π( G , ψ ) with ψ = ϕ ◦ φ d as above. L G X × SL (2 , C ) − → L G is fixed, and for each π ⊂ L 2 ( X ), there exists Given X , ϕ : φ d = φ d ( 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

Our setting, with a twist F = R , G classical (meaning GL ( . ), SO ( . ), Sp ( . ), or U ( . )) X = G / H symmetric space, ie. : σ : involution of G over R , H = G σ , 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

Our setting, with a twist F = R , G classical (meaning GL ( . ), SO ( . ), Sp ( . ), or U ( . )) X = G / H symmetric space, ie. : σ : involution of G over R , H = G σ , 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

Our setting, with a twist F = R , G classical (meaning GL ( . ), SO ( . ), Sp ( . ), or U ( . )) X = G / H symmetric space, ie. : σ : involution of G over R , H = G σ , and H ( R ) e ⊂ H ⊂ H ( R ) We propose a little extension of the setting : χ : unitary character of H , gives a line bundle L χ over X = G / H . L 2 ( X ) χ : square integrable sections, with discrete part L 2 d ( X ) χ David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 5 / 27

Our setting, with a twist F = R , G classical (meaning GL ( . ), SO ( . ), Sp ( . ), or U ( . )) X = G / H symmetric space, ie. : σ : involution of G over R , H = G σ , and H ( R ) e ⊂ H ⊂ H ( R ) We propose a little extension of the setting : χ : unitary character of H , gives a line bundle L χ over X = G / H . L 2 ( X ) χ : square integrable sections, with discrete part L 2 d ( X ) χ Same conjecture, with an L -group L G X ,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

Our setting, with a twist F = R , G classical (meaning GL ( . ), SO ( . ), Sp ( . ), or U ( . )) X = G / H symmetric space, ie. : σ : involution of G over R , H = G σ , and H ( R ) e ⊂ H ⊂ H ( R ) We propose a little extension of the setting : χ : unitary character of H , gives a line bundle L χ over X = G / H . L 2 ( X ) χ : square integrable sections, with discrete part L 2 d ( X ) χ Same conjecture, with an L -group L G X ,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

Choice of H 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

Choice of H 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 H 1 with H ⊂ H 1 ⊂ H ( R ). David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 6 / 27

Choice of H 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 H 1 with H ⊂ H 1 ⊂ 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

Discrete series of real symmetric spaces 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

Discrete series of real symmetric spaces 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 g 0 = k 0 ⊕ p 0 , g 0 = h 0 ⊕ s 0 , g 0 = ( k 0 ∩ p 0 ) ⊕ ( k 0 ∩ s 0 ) ⊕ ( p 0 ∩ h 0 ) ⊕ ( p 0 ∩ s 0 ) . David Renard, joint work with C. Moeglin Sakellaridis-Venkatesh conjectures for real classical symmetric spaces 27 juin 2019 7 / 27