English
The IsGLPos smul construction yields a well-defined MDifferentiable map after applying SL action, preserving the cusp form structure.
Русский
Конструкция IsGLPos smul образует корректную MDifferentiable-линию после применения SL-действия, сохраняя структуру cusp form.
LaTeX
$$$\forall c,f,\; MDifferentiable( ..., (\/SL) map \cdot f) \Rightarrow \text{ cusp-form }$$$
Lean4
instance instSMul : SMul α (CuspForm Γ k) where
smul c
f :=
{ toSlashInvariantForm := c • f.1
holo' := by simpa using f.holo'.const_smul (c • (1 : ℂ))
zero_at_cusps' hc g
hg :=
by
simp_rw [IsZeroAtImInfty, Filter.ZeroAtFilter, SlashInvariantForm.toFun_eq_coe, SlashInvariantForm.coe_smul,
toSlashInvariantForm_coe, ← smul_one_smul ℂ c ⇑f, smul_slash]
exact (f.zero_at_cusps' hc g hg).smul _ }
