Let $G$ be a Lie group acting on a complex manifold $M$. Let $p$ be an isolated fixed point. Let us look at the representation of $G$ on $T\_pM$. Suppose $T\_pM = \bigoplus V\_i^{\oplus n\_i}$ where $V\_i$ is an irreducible complex representation and $V\_i\ne V\_j$. Let us denote $V\_i^{\oplus n\_i}$ by $W\_i$ and call $W\_i$ an isotypic component of the representation.


Question 1: Under what conditions on $G$ and/or on the isotypic component $W\_i$ there locally exists a $G$-invariant complex submanifold $M\_i$ such that $T\_pM\_i = W\_i$? If it exists, how one can construct it? It is clear that we should look exactly on the isotypic components and not on arbitrary subrepresentations as one can easily imagine an example when the representation is trivial but for every complex subspace of $T\_pM$ there doesn't exist an invariant submanifold tangent to it. In fact, when $W\_i$ is trivial, I suppose, many bad things can happen.


Question 2: If such a submanifold exists, is it unique?


I'd be grateful if you can answer these questions just for $\mathbb G\_m^n$ but of course it is much more interesting for arbitrary Lie groups. In the case of $\mathbb G\_m^n$ one can look on the attractor and the repellent of the fixed point but that gives as only invariant submanifolds tangent the direct sum of subrepresentations with negative and positive weights accordingly.



Invariant submanifolds tangent to isotypic subrepresentations
By Bochner's theorem, in the complex analytic setting, if $G$ is a reductive complex Lie group acting holomorphically, or is a compact Lie group, then there such submanifolds, because the action is linearisable.


