Double cosets in algebraic groups
File(s)
Author(s)
Brundan, Jonathan
Type
Thesis
Abstract
In this thesis, I consider the problem of classifying certain orbits of algebraic groups - double cosets. Let H and K be closed subgroups of a simple algebraic group G, defined over an algebraically closed field of arbitrary characteristic p. Then, H K acts on G by (h, k).g = hgk-1, for (h,k) H K,g G, and the orbits are the (H,K)-double cosets in G. I consider the following properties:
(D1) G = HK is a factorisation of G..
(D2) There are finitely many (H,K)-double cosets in G..
(D3) There is a dense (H,K)-double coset in G..
(D4) There are finitely many closed (H,K)-double cosets in G..
Note that (D1)(D2)(D3). Factorisations - property (D1) - with H, K either reductive or parabolic, have recently been classified by Liebeck, Saxi and Seitz. In the first four chapters of the thesis, I consider the case when both H, K are reductive subgroups of G. In characteristic 0, this implies that all four of these properties are equivalent. I prove that in arbitrary characteristic, (D1)-(D3) are equivalent, modulo a certain week technical condition.
In Chapter 5 and Chapter 6, I consider the case when K = P is parabolic. Here, there are examples in which each of the implications (D4)(D3), (D3)(D2), and (D2)(D1) fail. The study of such double cosets reduces the certain 'branching rules' which describe the restriction of coWeyl modules for G to the subgroup H. Using this, I classify a special case of spherical double cosets, leading to a definition of multiplicity-free subgroups in arbitrary characteristic.
In the final two chapters, I consider a very special branching rule describing the restriction of modules from GLn(k) to GLn-1(k), closely connected with the work of Kleshchev.
(D1) G = HK is a factorisation of G..
(D2) There are finitely many (H,K)-double cosets in G..
(D3) There is a dense (H,K)-double coset in G..
(D4) There are finitely many closed (H,K)-double cosets in G..
Note that (D1)(D2)(D3). Factorisations - property (D1) - with H, K either reductive or parabolic, have recently been classified by Liebeck, Saxi and Seitz. In the first four chapters of the thesis, I consider the case when both H, K are reductive subgroups of G. In characteristic 0, this implies that all four of these properties are equivalent. I prove that in arbitrary characteristic, (D1)-(D3) are equivalent, modulo a certain week technical condition.
In Chapter 5 and Chapter 6, I consider the case when K = P is parabolic. Here, there are examples in which each of the implications (D4)(D3), (D3)(D2), and (D2)(D1) fail. The study of such double cosets reduces the certain 'branching rules' which describe the restriction of coWeyl modules for G to the subgroup H. Using this, I classify a special case of spherical double cosets, leading to a definition of multiplicity-free subgroups in arbitrary characteristic.
In the final two chapters, I consider a very special branching rule describing the restriction of modules from GLn(k) to GLn-1(k), closely connected with the work of Kleshchev.
Version
Open Access
Date Issued
1996
Date Acceptance
1996
Copyright Statement
Attribution-Non Commercial-No Derivatives 4.0 International Licence (CC BY-NC-ND)