Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as
(1) $k[G]^G \xrightarrow{\sim} k[T]^W$
and sometimes as
(2) $k[\mathfrak{g}]^G \xrightarrow{\sim} k[\mathfrak{t}]^W$.
I think both can be proved in essentially the same way, but is there a way to directly deduce them from each other?
I think (1) implies (2) directly: equip $k[G]$ and $k[T]$ with the filtrations by the powers of the maximal ideals at the origins. Then the map in (2) is the associated graded of the map in (1). Since the map in (1) is a strict map of filtered rings, it being an isomorphism implies its associated graded is an isomorphism.
It's also clear that the injectivity of (2) implies the injectivity of (1). However I'm not quite seeing what to do about surjectivity -- is there an adjective one can put in front of "filtered" to guarantee that a map is surjective if its associated graded is? (This is the case when the filtration is finite, but that is not satisfied here...)
