[Paper Review] Spherical affine cones in exceptional cases and related branching rules

Bruno Niemann|arXiv (Cornell University)|Nov 16, 2011

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper classifies all triples $(G, P, H)$ where $G$ is a complex simply connected simple algebraic group of exceptional type, $P$ is a maximal parabolic subgroup, and $H \subset G$ is a maximal reductive subgroup acting spherically on the flag variety $G/P$. It derives explicit branching rules for the restriction of dual irreducible representations $V_{k\omega_i}^*$ to $H$, showing that sphericity is equivalent to multiplicity-free branching. The key result is a complete classification of spherical actions and corresponding branching rules across all exceptional groups $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$, with $E_8$ having no spherical cases due to multiplicity in restrictions.

ABSTRACT

Given a complex simply connected simple algebraic group $G$ of exceptional type and a maximal parabolic subgroup $P \subset G$, we classify all triples $(G,P,H)$ such that $H \subset G$ is a maximal reductive subgroup acting spherically on $G/P$. In addition we derive branching rules for $ ext{res}^G_H (V^*_{kω_i})$, $k \in \N$, where $ω_i$ is the fundamental weight associated to $P$. This is the first of two parts of a project to classify all such triples and corresponding branching rules for all simply connected simple algebraic groups.

Motivation & Objective

• To classify all triples $(G, P, H)$ where $G$ is a simply connected simple algebraic group of exceptional type, $P \subset G$ is a maximal parabolic subgroup, and $H \subset G$ is a maximal reductive subgroup acting spherically on $G/P$.
• To derive explicit branching rules for the restriction of irreducible $G$-modules $V_{k\omega_i}^*$ to $H$ in the spherical cases.
• To establish a correspondence between the sphericity of the flag variety $G/P$ under $H$-action and the multiplicity-free decomposition of the coordinate ring of the affine cone $\widehat{Y}$ over $G/P$.
• To complete the first part of a two-part project to classify all such spherical triples and their branching rules across all simply connected simple algebraic groups.

Proposed method

• Use the theory of spherical varieties: a $G$-variety $G/P$ is $H$-spherical if and only if the coordinate ring $\mathbb{C}[\widehat{Y}]$ is multiplicity-free as a $G$-module.
• Work with the affine cone $\widehat{Y}$ over the flag variety $G/P$, whose homogeneous components are isomorphic to $V_{k\omega_i}^*$, and analyze their restrictions to $H$.
• Apply root system and weight space techniques to compute the structure of $\text{res}^G_H(V_{k\omega_i}^*)$ using the LiE software for character and multiplicity computations.
• Use dimension comparisons between Borel subgroups of $H$ and the flag variety $G/P$ to eliminate non-spherical cases.
• Verify sphericity via the existence of a unique open orbit under a Borel subgroup of $H$ by analyzing the orbit structure and the dimension of the $U_H$-orbit closure in the nilradical.
• For $E_7$, explicitly construct a highest weight vector $X$ in the nilradical $\mathfrak{n}$ such that $\dim[U_H, X] = 16 = \dim N$, confirming sphericity.

Experimental results

Research questions

• RQ1 Which maximal reductive subgroups $H \subset G$ act spherically on the flag variety $G/P$ for $G$ of exceptional type and $P$ a maximal parabolic subgroup?
• RQ2 For which triples $(G, P, H)$ is the restriction $\text{res}^G_H(V_{k\omega_i}^*)$ multiplicity-free, and what are the explicit branching rules?
• RQ3 How can the sphericity of $G/P$ under $H$-action be characterized in terms of the representation theory of $G$ and $H$?
• RQ4 Why does $E_8$ admit no spherical $H$-varieties $G/P$ for any maximal reductive subgroup $H$?
• RQ5 What role does the $\mathbb{C}^*$-factor play in the branching rules for $H = D_5 \times \mathbb{C}^* \subset E_6$ and $H = E_6 \times \mathbb{C}^* \subset E_7$?

Key findings

• For $G_2$, the maximal reductive subgroup $A_2 \subset G_2$ acts spherically on $G/P_1$ and $G/P_2$, with branching rules $\text{res}^{G_2}_{A_2}(V_{k\omega_1}^*) = \bigoplus_{a_1+a_2 \leq k} V_{a_1\lambda_1 + a_2\lambda_2}$ and $\text{res}^{G_2}_{A_2}(V_{k\omega_2}^*) = \bigoplus_{a_1+a_2+a_3=k} V_{(a_1+a_3)\lambda_1 + (a_2+a_3)\lambda_2}$.
• For $F_4$, the subgroups $B_4 \subset F_4$ yield spherical actions on $G/P_1$, $G/P_2$, $G/P_3$, and $G/P_4$, with explicit branching rules involving sums over partitions of $k$ with coefficients in the fundamental weights of $B_4$.
• For $E_6$, the subgroup $A_5 \times A_1 \subset E_6$ acts spherically on $G/P_1$, with $\text{res}^{E_6}_{A_5 \times A_1}(V_{k\omega_1}^*) = \bigoplus_{a_1 + 2a_2 + a_3 = k} V_{a_1\lambda_2 + a_2\lambda_4 + a_3\lambda_5} \otimes V_{a_1\lambda_6}$.
• For $E_7$, the subgroup $D_6 \times A_1 \subset E_7$ acts spherically on $G/P_7$, with $\text{res}^{E_7}_{D_6 \times A_1}(V_{k\omega_7}^*) = \bigoplus_{a_1 + 2a_2 + a_3 = k} V_{a_1\lambda_1 + a_2\lambda_2 + a_3\lambda_6} \otimes V_{a_1\lambda_7}$.
• For $E_8$, no maximal reductive subgroup $H$ acts spherically on any $G/P_i$, as shown by multiplicity in $\text{res}^{E_8}_{H}(V_{k\omega_8}^*)$ for $H = E_7 \times A_1$ and $H = D_8$, with $2(V_{1\lambda_1 + 2\lambda_7} \otimes V_{2\lambda_8})$ and $2V_{\lambda_8}$ appearing in the decomposition.
• The sphericity of $G/P$ is equivalent to the multiplicity-free restriction of $\mathbb{C}[\widehat{Y}]$ to $H$, and this condition is both necessary and sufficient for the branching rules to be multiplicity-free.