The normalizer of a reductive subgroup - MathOverflow

The normalizer of a reductive subgroup

2010-06-07T07:02:13Z

Let $k$ be a field and $G$ a linear algebraic group over $k$. Let $H$ be a diagonalizable subgroup of $G$. Then it is a classical fact that the centralizer $C_G(H)$ of $H$ is of finite index in the normalizer $N_G(H)$ of $H$.

Now let $H$ be an arbitrary reductive subgroup of $G$. Is it true that $Z_G(H)\cdot H$ is of finite index in $N_G(H)$?

In the case of certain Lie groups, the answer is yes, see e.g. D. Poguntke, Normalizers and centralizers of reductive subgroups of almost connected Lie groups (MR1650341), which motivates my question.

Answer by Angelo for The normalizer of a reductive subgroup

2010-06-07T07:08:08Z

I think so. This is implied by the fact that the group of outer automorphisms of $H$ is finite. When $H$ is semisimple, then the group of outer automorphisms is contained in the group of automorphisms of the Dynkin diagram, so it is finite. The general case should reduce to this by considering the commutator subgroup of $H$, which is semisimple.

Answer by Jim Humphreys for The normalizer of a reductive subgroup

2010-06-07T13:28:13Z

While the comments by Angelo and Wilberd go a long way toward an answer, I'd prefer to start with a more precise formulation of the question. Up to finite index, the group $G$ and its (closed!) subgroup $H$ may be assumed to be connected with $H$ moreover reductive. The structure theory involved here is essentially independent of the characteristic of the ground field, but on the other hand a field of definition apparently plays no role and could be assumed to be algebraically closed.

Up to finite index (or isogeny), a connected reductive group is a direct product of closed connected subgroups which can be tori or products of (quasi-)simple groups of a fixed type. Due to the standard theorem on "rigidity" of tori, the centralizer of a torus in $G$ has finite index in the normalizer. On the other hand, the outer automorphism group of a simple group is finite (coming from the automorphisms of the Dynkin diagram) while other outer automorphisms of a direct product of groups of the same type come from just finitely many permutations of isomorphic simple factors.

Since an automorphism of $H$ preserves the center and the various products of simple factors of the same type (whose overall product is the derived group of $H$ ), these pieces combine to show that $N_G(H)/H \cdot C_G(H)$ is finite. (Though I haven't written it all down myself.) I don't recall a specific reference for such a general result in the literature; it may only come up in special situations.