The decomposition of permutation module for infinite Chevalley groups, II

TL;DR

We address the decomposition of the permutation module $\mathbb{F}[{\bf G}/{\bf B}]$ for a connected reductive group ${\bf G}$ over an algebraically closed field, by constructing a filtration with subquotients $E_J$ indexed by $J\subset I$. We show each $E_J$ is irreducible and pairwise non-isomorphic, so there are exactly $2^{|I|}$ composition factors, each with multiplicity one. The approach hinges on a detailed analysis of the unipotent radical ${\bf U}$ via self-enclosed subgroups and a non-vanishing augmentation on $E_J$, together with a character-dependent irreducibility verification. These results generalize earlier Steinberg-irreducibility findings and provide a uniform, full decomposition of the permutation module with potential applications to algebraic geometry and number theory.

Abstract

Let $\bf G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ and ${\bf B}$ be an Borel subgroup of ${\bf G}$. In this paper we completely determine the composition factors of the permutation module $\mathbb{F}[{\bf G}/{\bf B}]$ for any field $\mathbb{F}$.

Theorems & Definitions (18)

• Theorem 1.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• proof
• Example 3.1
• proof
• Lemma 3.3
• proof