some tests and figuring out

src/groups/group_operation_action.jl

@@ -73,7 +73,7 @@ function adjoint_apply_diff_group(A::GroupOperationAction, a, X, p)
             G,
             p,
             a,
-            inv_diff(G, a, X),
+            inv_diff(G, apply(A, a, p), X),
             (direction(A), switch_side(action_side(A))),
         )
     end
@@ -90,7 +90,7 @@ function adjoint_apply_diff_group!(A::GroupOperationAction, Y, a, X, p)
             Y,
             p,
             a,
-            inv_diff(G, a, X),
+            inv_diff(G, apply(A, a, p), X),
             (direction(A), switch_side(action_side(A))),
         )
     end
@@ -118,6 +118,35 @@ function apply_diff!(A::GroupOperationAction, Y, a, p, X)
     return translate_diff!(A.group, Y, a, p, X, direction_and_side(A))
 end
 
+@doc raw"""
+    apply_diff_group(A::GroupOperationAction, a, X, p)
+
+Compute differential of [`GroupOperationAction`](@ref) `A` with respect to group element
+at tangent vector `X`:
+
+````math
+(\mathrm{d}τ^p) : T_{a} \mathcal G → T_{τ_a p} \mathcal M
+````
+
+There are four cases:
+* left action from the left side: ``L_a: p ↦ a \circ p``, where
+````math
+(\mathrm{d}L_a) : T_{a} \mathcal G → T_{a \circ p} \mathcal M.
+````
+* right action from the left side: ``L'_a: p ↦ a^{-1} \circ p``, where
+````math
+(\mathrm{d}L'_a) : T_{a} \mathcal G → T_{a^{-1} \circ p} \mathcal M.
+````
+* right action from the right side: ``R_a: p ↦ p \circ a``, where
+````math
+(\mathrm{d}R_a) : T_{a} \mathcal G → T_{p \circ a} \mathcal M.
+````
+* left action from the right side: ``R'_a: p ↦ p \circ a^{-1}``, where
+````math
+(\mathrm{d}R'_a) : T_{a} \mathcal G → T_{p \circ a^{-1}} \mathcal M.
+````
+
+"""
 function apply_diff_group(A::GroupOperationAction, a, X, p)
     G = base_group(A)
     if direction_and_side(A) === LeftForwardAction() ||

test/groups/circle_group.jl

@@ -66,6 +66,7 @@ using Manifolds:
             test_invariance=true,
             test_lie_bracket=true,
             test_adjoint_action=true,
+            test_inv_diff=true,
         )
     end
 

test/groups/group_operation_action.jl

@@ -124,5 +124,14 @@ using Manifolds:
         adjoint_apply_diff_group!(A_left_fwd, Y, a_pts[1], X_pts[1], m_pts[1])
         @test Y ≈
               inverse_translate_diff(G, a_pts[1], m_pts[1], X_pts[1], RightBackwardAction())
+
+        @test adjoint_apply_diff_group(A_right_fwd, a_pts[1], X_pts[1], m_pts[1]) ≈
+              inverse_translate_diff(
+            G,
+            m_pts[1],
+            a_pts[1],
+            inv_diff(G, a_pts[1], X_pts[1]),
+            RightBackwardAction(),
+        )
     end
 end