Classification of K_F-orbits of Unipotent Elements in Symmetric F-varieties of SL(n, F)

unipotent elements;Jordan decomposition;special linear group;orbit decomposition;symmetric variety;Classifications

Wang, Qiang ; Robert Bruck, Committee Member,Ernie Stitzinger, Committee Member,Kailash Misra, Committee Member,Aloysius G. Helminck, Committee Chair,Amassa Fauntleroy, Committee Co-Chair,Wang, Qiang ; Robert Bruck ; Committee Member ; Ernie Stitzinger ; Committee Member ; Kailash Misra ; Committee Member ; Aloysius G. Helminck ; Committee Chair ; Amassa Fauntleroy ; Committee Co-Chair

University:North Carolina State University

关键词: unipotent elements;    Jordan decomposition;    special linear group;    orbit decomposition;    symmetric variety;    Classifications;   

Others  :  https://repository.lib.ncsu.edu/bitstream/handle/1840.16/6172/etd.pdf?sequence=1&isAllowed=y

美国|英语

来源: null

【 摘 要 】

Richardson proved in 1982 that, given an algebraic group G and some involution, we couldhave only a finite number of K-orbits of unipotent elements in the symmetric variety P = G/K over an algebraically closed field, where K is the fixed point group of the involution.A question arises naturally: what if the field is not algebraically closed? In this thesis we tryto answer this question and go a little further by listing all K_F-orbits of unipotent elements inP explicitly. We work on the symmetric F-variety P = G_F/K_F for the special linear group overan arbitrary field F of characteristic not 2. We classify all K_F-orbits of unipotent elements in Pfor all inner involutions for the special linear group. For Cartan (outer) involution, we classifyK-orbits for small n only and illustrate how to get the canonical form for general n. Furtherproofs are still needed. We also classify G_F-orbits of unipotent elements in G_F.