Characteristic polynomials of reductive subgroup over C

Question

Can any one provide a hint to prove the following statement? : Let $H$ be a complex reductive subgroup (not necessarily connected) contained in $SO(n,\mathbb{C)})$. Consider the map $H \rightarrow Aff^*$, where $Aff^*$ is the affine $(n-1)$-dimensional subspace not including the hyperplane ${x_n =0 }$, defined by sending an element $h$ of $H$ to the tuple $(a_1(h), \ldots, a_{n-1}(h))$ where $X^n + a_1 X^{n-1} + \ldots + a_{n-1} X + (-1)^n $ is the characteristic polynomial of $h \in H $. Then show that the dimension of image of H is equal to rank H .

I tried to prove the result using lemma 1 and lemma 2 of Serre in his letters to K.Ribet of Collected works Volume 4 ( in the beginning of the book ), but I could not get far to have a completely satisfactory answer. Thanks for the help.

Thanks.

Answer 1

I'm not sure where your "statement" comes from, or why the specific type of embedding of $H$ is assumed here. But the closest relative of this situation I'm aware of goes back to work of Kostant (in characteristic 0) and later Steinberg (more generally), in which they consider a sort of adjoint quotient of a semisimple (or reductive) group. Only the semisimple classes of such a group are closed, so one has to settle for something less than a strict "quotient" construction.

In your concrete situation, the significance of assigning to a matrix the nontrivial coefficients of its characteristic polynomial originates with the behavior of semisimple elements under the Weyl group action: affine space of dimension $r$ provides a model of the orbit space $T/W$ when $\dim T = r$ (the rank of $H$ when $T$ is a maximal torus). It's probably useful to consult $\S6$ in Steinberg's paper on regular elements here. In my 1995 AMS monograph on conjugacy classes in semisimple algebraic groups, I wrote up some of the relevant material in Chapter 3.

For an arbitrary matrix in $H$, the only information given by its characteristic polynomial is the list of eigenvalues with multiplicity. The semisimple elements of $H$ are Zariski-dense, and their conjugacy classes are effectively parametrized in this way. By the standard structure theorems this boils down to the orbit space $T/W$. Technically it's all a bit complicated, and in an abstract formulation (say for simply connected groups) one works with the values of fundamental representations rather than the explicit characteristic polynomials here.