Conjugating a double-coset slash sum

Let g normalize Γ₁ and Γ₂, and suppose that conjugation by g carries the double coset Γ₁ δ Γ₂ into itself, i.e. g⁻¹ δ g ∈ Γ₁ δ Γ₂. Then conjugation by g permutes the right cosets Γ₁ aᵥ that Γ₁ δ Γ₂ decomposes into, Γ₁ aᵥ ↦ Γ₁ (g⁻¹ aᵥ g), and so for a Γ₁-invariant f

(∑ᵥ f ∣[k] aᵥ) ∣[k] g = ∑ᵥ (f ∣[k] g) ∣[k] (g⁻¹ aᵥ g).

The right-hand side is again the slash sum of Γ₁ δ Γ₂, now applied to f ∣[k] g, which is Γ₁-invariant because g normalizes Γ₁: the slash by g intertwines the Hecke operator [Γ₁ δ Γ₂] with itself. This is the mechanism by which an operator given by an element of the normalizer — an Atkin–Lehner or Fricke involution of Γ₀(N), say — commutes with the Hecke operators whose double cosets it fixes.

Main results

• HeckeRing.GL2.heckeSlashSum_slash_of_mem_normalizer: for g normalizing Γ₁ and Γ₂ with g⁻¹ δ g ∈ Γ₁ δ Γ₂, heckeSlashSum k D f ∣[k] g = heckeSlashSum k D (f ∣[k] g).

References

• G. Shimura, Introduction to the arithmetic theory of automorphic functions, §3.4.

theorem HeckeRing.GL2.heckeSlashSum_slash_of_mem_normalizer (k : ℤ) {Δ : Submonoid (GL (Fin 2) ℚ)} {Γ₁ Γ₂ : Subgroup (GL (Fin 2) ℚ)} (D : HeckeCoset Δ Γ₁ Γ₂) [Finite (DoubleCoset.DecompQuotient Γ₂ Γ₁ (↑(Quotient.out D))⁻¹)] {g : GL (Fin 2) ℚ} (hg₁ : g ∈ Subgroup.normalizer ↑Γ₁) (hg₂ : g ∈ Subgroup.normalizer ↑Γ₂) (hD : g⁻¹ * ↑(Quotient.out D) * g ∈ DoubleCoset.doubleCoset ↑(Quotient.out D) ↑Γ₁ ↑Γ₂) (f : UpperHalfPlane → ℂ) (hf : ∀ γ ∈ Γ₁, SlashAction.map k γ f = f) : SlashAction.map k g (heckeSlashSum k D f) = heckeSlashSum k D (SlashAction.map k g f)

The slash by an element of the normalizers commutes with a Hecke operator whose double coset it fixes. If g normalizes Γ₁ and Γ₂ and g⁻¹ δ g ∈ Γ₁ δ Γ₂, where δ represents D, then for every Γ₁-invariant f,

heckeSlashSum k D f ∣[k] g = heckeSlashSum k D (f ∣[k] g).

No positivity is asked of g: the slash action of GL(2, ℚ) is a monoid action whatever the sign of the determinant, and that is all the comparison uses.