Basic sums of coadjoint orbits of the unitriangular group
type: Item
instance of: scholarly article
title: Basic sums of coadjoint orbits of the unitriangular group
zbMATH Open document ID: 0837.20050
author: Carlos A. M. André
published in: Journal of Algebra
publication date: 1996-04-21T00:00:00Z
review text: Let \(K = \overline {\mathbf F}_q\), \(U_n(K)\) the \(n \times n\) unitriangular group over \(K\), \(F : U_n (K) \to U_n(K)\) the standard Frobenius map and \(U = U_n(q)\) the finite group of \(F\)-stable points of \(U_n(K)\). Thus \(U\) is the group of \(n \times n\) unipotent upper triangular matrices over \({\mathbf F}_q\). The author studies the complex characters of \(U\). The classification of the irreducible characters of \(U_n(q)\) has long been known to be a difficult problem and this paper makes a nice contribution to the problem, when \(q = p^e\) and \(p \geq n\). The condition \(p \geq n\) is necessary since the author makes use of the Kirillov theory of coadjoint orbits, which was adapted to the case of \(U_n(q)\) by Kazhdan. Now \(U_n(K)\) acts on \(u_n(K)^*\), the dual of the Lie algebra \(u_n(K)\) of \(n \times n\) nilpotent matrices over \(K\), by the coadjoint representation. By the Kirillov theory the irreducible characters of \(U\) are in bijection with the \(F\)-stable \(U_n(K)\)-orbits on \(u_n(K)^*\), and the degree of an irreducible character is \(q^{{1\over 2} \dim O}\) where \(O\) is the corresponding orbit. The author defines certain ``orbit sums'' as follows. Let \(\Phi(n) = \{(i,j)\mid 1 \leq i < j \leq n\}\). A subset \(D\) of \(\Phi(n)\) is called a basic subset of \(|D \cap \{(i,j) : i < j \leq n\}|\leq 1\) for all \(i\), \(1 \leq i < n\) and \(|D \cap \{(i,j) : 1 \leq i < j\}|\leq 1\) for all \(j\), \(1 < j \leq n\). Let \(e^*_{ij} \in u_n (K)^*\) be defined by \(e^*_{ij} (a) = a_{ij}\) where \(a = (a_{ij}) \in u_n(K)\). For each \(\alpha \in K^\#\), let \(O_{ij}(\alpha)\) denote the \(U_n(K)\)-orbit of \(\alpha e^*_{ij}\). The author then defines for any non-empty basic subset \(D\) of \(\Phi(n)\) and a map \(\phi : D \to K^\#\), the ``basic sum'' \(O_D (\phi)\) to be the subset \(O_D(\phi) = \sum_{(i,j) \in D} O_{ij} (\phi(i,j))\) of \(u_n (K)^*\). Now for each \((i,j) \in \Phi(n)\) one has a subgroup \(U_{ij} (q) = \{x = (x_{ab}) …
reviewed by: Bhama Srinivasan
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 815878
zbMATH Keywords: ring of invariant polynomial functions
zbMATH Keywords: unitriangular groups
zbMATH Keywords: Frobenius map
zbMATH Keywords: finite group of \(F\)-stable points
zbMATH Keywords: unipotent upper triangular matrices
zbMATH Keywords: complex characters
zbMATH Keywords: irreducible characters
zbMATH Keywords: Kirillov theory
zbMATH Keywords: coadjoint orbits
zbMATH Keywords: degree
zbMATH Keywords: induced characters
zbMATH Keywords: basic characters
zbMATH Keywords: disjoint union of irreducible subvarieties
MaRDI profile type: Publication
full work available at URL: jabr.1995.1280
OpenAlex ID: W1986749058
DOI: 10.1006/JABR.1995.1280
description: scientific article; zbMATH DE number 815878