[Paper Review] Adjoint quotients of reductive groups

Ting-Yu Lee|arXiv (Cornell University)|Nov 15, 2012

Advanced Algebra and Geometry9 references3 citations

TL;DR

This paper establishes that the adjoint quotient of a reductive group over any commutative ring is stable under base change and is isomorphic to the Weyl group quotient of a maximal torus. For semisimple simply connected groups of constant type, the adjoint quotient is isomorphic to the Weil restriction of affine lines over the Dynkin scheme, and Steinberg’s cross-section can be defined over arbitrary commutative rings when the group is quasi-split and has no $A_{2m}$-type components.

ABSTRACT

Let $ G$ be a reductive group over a commutative ring $k$. In this article, we prove that the adjoint quotient $\adqG$ is stable under base change. Moreover, if $ G$ has a maximal torus $ T$, then the adjoint quotient of the torus $ T$ by its Weyl group will be isomorphic to $\adqG$. Then we focus on the semisimple simply connected group $ G$ of the constant type. In this case, $\adqG$ is isomorphic to the Weil restriction $\underset{ D/\spec k}{\prod}\aff^{1}_ D$, where $ D$ is the Dynkin scheme of $ G$. Then we prove that for such $ G$, the Steinberg's cross-section can be defined over $k$ if $ G$ is quasi-split and without $ A_{2m}$-type components

Motivation & Objective

• To define and study the adjoint quotient $\mathrm{G}//\mathrm{G}$ for reductive groups over arbitrary commutative rings.
• To prove that the adjoint quotient is stable under base change and isomorphic to the Weyl group quotient of a maximal torus.
• To characterize the adjoint quotient for semisimple simply connected groups of constant type as a Weil restriction of affine lines over the Dynkin scheme.
• To establish conditions under which Steinberg’s cross-section exists over arbitrary commutative rings.
• To show the existence of semisimple regular elements and the maximality of their centralizers over semi-local rings.

Proposed method

• Define the adjoint quotient $\mathrm{G}//\mathrm{G} = \mathrm{Spec}(k[\mathrm{G}]^\mathrm{G})$ as the spectrum of $\mathrm{G}$-invariant regular functions on $\mathrm{G}$.
• Use the conjugation action of $\mathrm{G}$ on $k[\mathrm{G}]$ to induce a $\mathrm{W}$-action on $k[\mathrm{T}]$ for a maximal torus $\mathrm{T}$, leading to $\mathrm{T}//\mathrm{W} = \mathrm{Spec}(k[\mathrm{T}]^\mathrm{W})$.
• Prove that the restriction map $k[\mathrm{G}]^\mathrm{G} \to k[\mathrm{T}]^\mathrm{W}$ induces an isomorphism $\mathrm{T}//\mathrm{W} \xrightarrow{\sim} \mathrm{G}//\mathrm{G}$ over $\mathbb{Z}$, then extend to arbitrary commutative rings.
• For semisimple simply connected groups of constant type, show $\mathrm{G}//\mathrm{G} \cong \prod_{\mathrm{D}/\mathrm{Spec}\,k} \mathbb{A}^1_{\mathrm{D}}$, where $\mathrm{D}$ is the Dynkin scheme.
• Construct Steinberg’s cross-section over $k$ when $\mathrm{G}$ is quasi-split and without $A_{2m}$-type components, using the isomorphism of the adjoint quotient with a product of affine lines.
• Prove that over semi-local rings, $\mathrm{G}$ contains semisimple regular elements whose centralizers are maximal tori, using lifting from residue fields and surjectivity of unipotent group points.

Experimental results

Research questions

• RQ1 Is the adjoint quotient $\mathrm{G}//\mathrm{G}$ of a reductive group over a commutative ring stable under base change?
• RQ2 Does the isomorphism $\mathrm{T}//\mathrm{W} \xrightarrow{\sim} \mathrm{G}//\mathrm{G}$ hold over arbitrary commutative rings when $\mathrm{G}$ contains a maximal torus?
• RQ3 Can the adjoint quotient of a semisimple simply connected group of constant type be described as a Weil restriction of affine lines over the Dynkin scheme?
• RQ4 Under what conditions can Steinberg’s cross-section be defined over an arbitrary commutative ring?
• RQ5 Does a semisimple simply connected group over a semi-local ring admit a semisimple regular element whose centralizer is a maximal torus?

Key findings

• The adjoint quotient $\mathrm{G}//\mathrm{G}$ is stable under base change for any reductive group $\mathrm{G}$ over a commutative ring $k$.
• For a reductive group $\mathrm{G}$ with a maximal torus $\mathrm{T}$, the adjoint quotient $\mathrm{G}//\mathrm{G}$ is isomorphic to $\mathrm{T}//\mathrm{W}$, the quotient of $\mathrm{T}$ by its Weyl group.
• For semisimple simply connected groups of constant type over $k$, $\mathrm{G}//\mathrm{G} \cong \prod_{\mathrm{D}/\mathrm{Spec}\,k} \mathbb{A}^1_{\mathrm{D}}$, where $\mathrm{D}$ is the Dynkin scheme of $\mathrm{G}$.
• When $\mathrm{G}$ is quasi-split and has no $A_{2m}$-type components, Steinberg’s cross-section exists over $k$.
• Over a semi-local ring $k$, a semisimple simply connected group $\mathrm{G}$ of constant type contains a semisimple regular element $g$ such that $\underline{\mathrm{Centr}}_{\mathrm{G}}(g)$ is a maximal torus.
• The centralizer of a semisimple regular element in such a group is not always smooth, as demonstrated by a counterexample in $\mathrm{PGL}_{2,k}$ with $k = \mathbb{C}[[x]]$.