Lifting via the converse theorem: new results
Lifting via the converse theorem: new results Solomon Friedberg, Boston College Joint work with Yuanqing Cai, Weizmann Institute of Sciences David Ginzburg, Tel Aviv University Eyal Kaplan, Bar Ilan University
Simons Symposium, 2018
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
1 / 43
Plan for the Talk 1
Functorial Lifts
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2
Functorial Lifts The Converse Theorem
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3
Functorial Lifts The Converse Theorem Ingredients in the Integral
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4 5
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example Why This Integral?
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4 5 6
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example Why This Integral? The Integrals in More Detail
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4 5 6 7
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example Why This Integral? The Integrals in More Detail Whittaker-Speh-Shalika Representations
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4 5 6 7 8
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example Why This Integral? The Integrals in More Detail Whittaker-Speh-Shalika Representations Local Gamma Factors
References: Cai, F, Ginzburg, Kaplan: “Doubling constructions and tensor product L-functions: the linear case” (arXiv:1710.00905 ) Cai, F, Kaplan: “Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations” (arXiv:1802.02637). Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Functorial Lifts, I Groups in this talk are all split. Groups of concern: G
Gˆ
Sp2n SO2n+1 SO2n GSpin2n GSpin2n+1
SO2n+1 (C) Sp2n (C) SO2n (C) GSO2n (C) GSp2n (C)
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
3 / 43
Functorial Lifts, I Groups in this talk are all split. Groups of concern: G
Gˆ
Sp2n SO2n+1 SO2n GSpin2n GSpin2n+1
SO2n+1 (C) Sp2n (C) SO2n (C) GSO2n (C) GSp2n (C)
There is an embedding of dual groups ˆ N = GLN (C) i : Gˆ → GL
N = 2n or N = 2n + 1,
so Langlands Functoriality predicts a lifting of automorphic representations A0 (G ) → A(GLN ), the “endoscopic transfer.” Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
3 / 43
Functorial Lifts, II Goal: Explain how to obtain endoscopic lifts of automorphic representations, i.e. those corresponding to this embedding, via the converse theorem. We show: Theorem. Any irreducible unitary cuspidal automorphic representation of G (A) has a functorial lift to GLN (A).
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
4 / 43
Functorial Lifts, II Goal: Explain how to obtain endoscopic lifts of automorphic representations, i.e. those corresponding to this embedding, via the converse theorem. We show: Theorem. Any irreducible unitary cuspidal automorphic representation of G (A) has a functorial lift to GLN (A). Notes: Arthur establishes endoscopic liftings for all classical groups (not necessarily split) using the trace formula. This draws on the work of Ngˆo and Waldspurger, among others.
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
4 / 43
Functorial Lifts, II Goal: Explain how to obtain endoscopic lifts of automorphic representations, i.e. those corresponding to this embedding, via the converse theorem. We show: Theorem. Any irreducible unitary cuspidal automorphic representation of G (A) has a functorial lift to GLN (A). Notes: Arthur establishes endoscopic liftings for all classical groups (not necessarily split) using the trace formula. This draws on the work of Ngˆo and Waldspurger, among others. In fact, our work includes GSpin, not handled by Arthur. (For generic automorphic representations, this is recent independent work of Asgari, Cogdell and Shahidi.)
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
4 / 43
Functorial Lifts, II Goal: Explain how to obtain endoscopic lifts of automorphic representations, i.e. those corresponding to this embedding, via the converse theorem. We show: Theorem. Any irreducible unitary cuspidal automorphic representation of G (A) has a functorial lift to GLN (A). Notes: Arthur establishes endoscopic liftings for all classical groups (not necessarily split) using the trace formula. This draws on the work of Ngˆo and Waldspurger, among others. In fact, our work includes GSpin, not handled by Arthur. (For generic automorphic representations, this is recent independent work of Asgari, Cogdell and Shahidi.) Our results are for the split case, but that condition is almost certainly not essential for this approach. Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
4 / 43
Functorial Lifts, III More precisely, Suppose that v < ∞ and πv is an unramified irreducible representation of G (Fv ) with associated Satake parameter, a semi-simple conjugacy class, [Av ] ⊂ Gˆ. Then the representation Πv on GLN (Fv ) is a local functorial lift if it is unramified and its Satake parameter is the conjugacy class [Av ] ⊂ GLN (C).
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
5 / 43
Functorial Lifts, III More precisely, Suppose that v < ∞ and πv is an unramified irreducible representation of G (Fv ) with associated Satake parameter, a semi-simple conjugacy class, [Av ] ⊂ Gˆ. Then the representation Πv on GLN (Fv ) is a local functorial lift if it is unramified and its Satake parameter is the conjugacy class [Av ] ⊂ GLN (C). Suppose that v is archimedean and πv is an irreducible representation. Then πv is determined by an admissible homomorphism ϕv : Wv → Gˆ of the local Weil group of Fv (this is the arithmetic Langlands classification). The representation Πv is a local functorial lift if it is determined by the admissible homomorphism i ◦ ϕv .
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
5 / 43
Functorial Lifts, III More precisely, Suppose that v < ∞ and πv is an unramified irreducible representation of G (Fv ) with associated Satake parameter, a semi-simple conjugacy class, [Av ] ⊂ Gˆ. Then the representation Πv on GLN (Fv ) is a local functorial lift if it is unramified and its Satake parameter is the conjugacy class [Av ] ⊂ GLN (C). Suppose that v is archimedean and πv is an irreducible representation. Then πv is determined by an admissible homomorphism ϕv : Wv → Gˆ of the local Weil group of Fv (this is the arithmetic Langlands classification). The representation Πv is a local functorial lift if it is determined by the admissible homomorphism i ◦ ϕv . Note that in each case the local standard L-functions match: L(s, Πv ) = L(s, πv ).
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
5 / 43
Functorial Lifts, IV
Definition An automorphic representation Π = ⊗0 Πv is a functorial lift of π = ⊗0 πv if Πv is the local functorial lift of πv at almost all places.
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
6 / 43
Functorial Lifts, IV
Definition An automorphic representation Π = ⊗0 Πv is a functorial lift of π = ⊗0 πv if Πv is the local functorial lift of πv at almost all places. Notes: 1
We show this is there is an automorphic representation Π of GLN (A) such that Πv is the local functorial lift of πv for all but the (finite) ramified primes (or one finite place if there are none).
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
6 / 43
Functorial Lifts, IV
Definition An automorphic representation Π = ⊗0 Πv is a functorial lift of π = ⊗0 πv if Πv is the local functorial lift of πv at almost all places. Notes: 1
We show this is there is an automorphic representation Π of GLN (A) such that Πv is the local functorial lift of πv for all but the (finite) ramified primes (or one finite place if there are none).
2
By the Strong Multiplicity One Theorem, if Π is isobaric then it is uniquely determined by this property.
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
6 / 43
The Converse Theorem Suppose that Π := ⊗Πv is an irreducible admissible representation of GLN (A) whose central Q character is automorphic, and such that the Euler product L(s, Π) = v L(s, Πv ) converges for
Simons Symposium, 2018
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
1 / 43
Plan for the Talk 1
Functorial Lifts
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2
Functorial Lifts The Converse Theorem
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3
Functorial Lifts The Converse Theorem Ingredients in the Integral
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4 5
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example Why This Integral?
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4 5 6
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example Why This Integral? The Integrals in More Detail
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4 5 6 7
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example Why This Integral? The Integrals in More Detail Whittaker-Speh-Shalika Representations
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Plan for the Talk 1 2 3 4 5 6 7 8
Functorial Lifts The Converse Theorem Ingredients in the Integral An Example Why This Integral? The Integrals in More Detail Whittaker-Speh-Shalika Representations Local Gamma Factors
References: Cai, F, Ginzburg, Kaplan: “Doubling constructions and tensor product L-functions: the linear case” (arXiv:1710.00905 ) Cai, F, Kaplan: “Doubling constructions: local and global theory, with an application to global functoriality for non-generic cuspidal representations” (arXiv:1802.02637). Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
2 / 43
Functorial Lifts, I Groups in this talk are all split. Groups of concern: G
Gˆ
Sp2n SO2n+1 SO2n GSpin2n GSpin2n+1
SO2n+1 (C) Sp2n (C) SO2n (C) GSO2n (C) GSp2n (C)
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
3 / 43
Functorial Lifts, I Groups in this talk are all split. Groups of concern: G
Gˆ
Sp2n SO2n+1 SO2n GSpin2n GSpin2n+1
SO2n+1 (C) Sp2n (C) SO2n (C) GSO2n (C) GSp2n (C)
There is an embedding of dual groups ˆ N = GLN (C) i : Gˆ → GL
N = 2n or N = 2n + 1,
so Langlands Functoriality predicts a lifting of automorphic representations A0 (G ) → A(GLN ), the “endoscopic transfer.” Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
3 / 43
Functorial Lifts, II Goal: Explain how to obtain endoscopic lifts of automorphic representations, i.e. those corresponding to this embedding, via the converse theorem. We show: Theorem. Any irreducible unitary cuspidal automorphic representation of G (A) has a functorial lift to GLN (A).
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
4 / 43
Functorial Lifts, II Goal: Explain how to obtain endoscopic lifts of automorphic representations, i.e. those corresponding to this embedding, via the converse theorem. We show: Theorem. Any irreducible unitary cuspidal automorphic representation of G (A) has a functorial lift to GLN (A). Notes: Arthur establishes endoscopic liftings for all classical groups (not necessarily split) using the trace formula. This draws on the work of Ngˆo and Waldspurger, among others.
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
4 / 43
Functorial Lifts, II Goal: Explain how to obtain endoscopic lifts of automorphic representations, i.e. those corresponding to this embedding, via the converse theorem. We show: Theorem. Any irreducible unitary cuspidal automorphic representation of G (A) has a functorial lift to GLN (A). Notes: Arthur establishes endoscopic liftings for all classical groups (not necessarily split) using the trace formula. This draws on the work of Ngˆo and Waldspurger, among others. In fact, our work includes GSpin, not handled by Arthur. (For generic automorphic representations, this is recent independent work of Asgari, Cogdell and Shahidi.)
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
4 / 43
Functorial Lifts, II Goal: Explain how to obtain endoscopic lifts of automorphic representations, i.e. those corresponding to this embedding, via the converse theorem. We show: Theorem. Any irreducible unitary cuspidal automorphic representation of G (A) has a functorial lift to GLN (A). Notes: Arthur establishes endoscopic liftings for all classical groups (not necessarily split) using the trace formula. This draws on the work of Ngˆo and Waldspurger, among others. In fact, our work includes GSpin, not handled by Arthur. (For generic automorphic representations, this is recent independent work of Asgari, Cogdell and Shahidi.) Our results are for the split case, but that condition is almost certainly not essential for this approach. Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
4 / 43
Functorial Lifts, III More precisely, Suppose that v < ∞ and πv is an unramified irreducible representation of G (Fv ) with associated Satake parameter, a semi-simple conjugacy class, [Av ] ⊂ Gˆ. Then the representation Πv on GLN (Fv ) is a local functorial lift if it is unramified and its Satake parameter is the conjugacy class [Av ] ⊂ GLN (C).
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
5 / 43
Functorial Lifts, III More precisely, Suppose that v < ∞ and πv is an unramified irreducible representation of G (Fv ) with associated Satake parameter, a semi-simple conjugacy class, [Av ] ⊂ Gˆ. Then the representation Πv on GLN (Fv ) is a local functorial lift if it is unramified and its Satake parameter is the conjugacy class [Av ] ⊂ GLN (C). Suppose that v is archimedean and πv is an irreducible representation. Then πv is determined by an admissible homomorphism ϕv : Wv → Gˆ of the local Weil group of Fv (this is the arithmetic Langlands classification). The representation Πv is a local functorial lift if it is determined by the admissible homomorphism i ◦ ϕv .
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
5 / 43
Functorial Lifts, III More precisely, Suppose that v < ∞ and πv is an unramified irreducible representation of G (Fv ) with associated Satake parameter, a semi-simple conjugacy class, [Av ] ⊂ Gˆ. Then the representation Πv on GLN (Fv ) is a local functorial lift if it is unramified and its Satake parameter is the conjugacy class [Av ] ⊂ GLN (C). Suppose that v is archimedean and πv is an irreducible representation. Then πv is determined by an admissible homomorphism ϕv : Wv → Gˆ of the local Weil group of Fv (this is the arithmetic Langlands classification). The representation Πv is a local functorial lift if it is determined by the admissible homomorphism i ◦ ϕv . Note that in each case the local standard L-functions match: L(s, Πv ) = L(s, πv ).
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
5 / 43
Functorial Lifts, IV
Definition An automorphic representation Π = ⊗0 Πv is a functorial lift of π = ⊗0 πv if Πv is the local functorial lift of πv at almost all places.
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
6 / 43
Functorial Lifts, IV
Definition An automorphic representation Π = ⊗0 Πv is a functorial lift of π = ⊗0 πv if Πv is the local functorial lift of πv at almost all places. Notes: 1
We show this is there is an automorphic representation Π of GLN (A) such that Πv is the local functorial lift of πv for all but the (finite) ramified primes (or one finite place if there are none).
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
6 / 43
Functorial Lifts, IV
Definition An automorphic representation Π = ⊗0 Πv is a functorial lift of π = ⊗0 πv if Πv is the local functorial lift of πv at almost all places. Notes: 1
We show this is there is an automorphic representation Π of GLN (A) such that Πv is the local functorial lift of πv for all but the (finite) ramified primes (or one finite place if there are none).
2
By the Strong Multiplicity One Theorem, if Π is isobaric then it is uniquely determined by this property.
Solomon Friedberg (Boston College)
Lifting via the Converse Theorem
Simons Symposium, 2018
6 / 43
The Converse Theorem Suppose that Π := ⊗Πv is an irreducible admissible representation of GLN (A) whose central Q character is automorphic, and such that the Euler product L(s, Π) = v L(s, Πv ) converges for
