The Petersson adjoint of the Hecke operators Tₙ at indices prime to the level

For n coprime to N, the Hecke operator Tₙ on S_k(Γ₁(N)) has Petersson adjoint ⟨n⟩⁻¹ Tₙ:

⟪Tₙ f, g⟫ = ⟪f, ⟨n⟩⁻¹ (Tₙ g)⟫.

When the second argument has nebentypus χ, the inverse diamond acts by the scalar χ(n)⁻¹, so the formula simplifies to ⟪Tₙ f, g⟫ = ⟪f, χ(n)⁻¹ Tₙ g⟫. This is the adjoint input for the subsequent character-space stability, normality, and simultaneous-diagonalization results.

The purely Hecke-theoretic identification of the adjugate double coset, including the commutation of Tₙ with ⟨n⟩⁻¹, is in HeckeSlash/Diamond.lean. This module only applies the Petersson trace adjunction and specializes the result to a character space.

Main results

• HeckeRing.GL2.peterssonInnerCosets_heckeTCuspNat_left: the adjoint formula on S_k(Γ₁(N)).
• HeckeRing.GL2.peterssonInnerCosets_heckeTCuspNat_left_of_mem_cuspFormCharSpace: the formula when the second argument has nebentypus χ.

References

• F. Diamond and J. Shurman, A first course in modular forms, Theorem 5.5.3.
• T. Miyake, Modular forms, Theorem 4.5.4.

theorem HeckeRing.GL2.peterssonInnerCosets_heckeTCuspNat_left {N n : ℕ} [NeZero N] [NeZero n] (k : ℤ) (hn : n.Coprime N) (f g : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) : ((heckeTCuspNat k n) f).peterssonInnerCosets g = f.peterssonInnerCosets ((diamondOpCusp k (ZMod.unitOfCoprime n hn)⁻¹) ((heckeTCuspNat k n) g))

The Petersson adjoint of Tₙ (Diamond–Shurman, Theorem 5.5.3). For n coprime to the level N and cusp forms f, g on Γ₁(N),

⟪Tₙ f, g⟫ = ⟪f, ⟨n⟩⁻¹ (Tₙ g)⟫.

theorem HeckeRing.GL2.peterssonInnerCosets_heckeTCuspNat_left_of_mem_cuspFormCharSpace {N n : ℕ} [NeZero N] [NeZero n] (k : ℤ) {χ : (ZMod N)ˣ →* ℂˣ} (hn : n.Coprime N) (f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k) {g : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hg : g ∈ cuspFormCharSpace k χ) : ((heckeTCuspNat k n) f).peterssonInnerCosets g = f.peterssonInnerCosets ((↑(χ (ZMod.unitOfCoprime n hn)))⁻¹ • (heckeTCuspNat k n) g)

For g of nebentypus χ, the adjoint formula simplifies to ⟪Tₙ f, g⟫ = ⟪f, χ(n)⁻¹ Tₙ g⟫.