2026-09-19T19:27:52Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/415582022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
David A. Vogan, Jr.
Fang, Chuying
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2008-05-19T15:01:06Z
2008-05-19T15:01:06Z
2007
2007
http://hdl.handle.net/1721.1/41558
225066747
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 85-88).
In this thesis, we study ad-nilpotent ideals and its relations with nilpotent orbits, affine Weyl groups, sign types and hyperplane arrangements. This thesis is divided into three parts. The first and second parts deal with ad-nilpotent ideals for complex reductive Lie groups. In the first part, we study the left equivalence relation of ad-nilpotent ideals and relate it to some equivalence relation of affine Weyl groups and sign types. In the second part, we prove that for classical groups there always exist ideals of minimal dimension as conjectured by Sommers. In the third part, we define an analogous object for connected real reductive Lie groups, which is called 0-nilpotent subspaces. We relate 0-nilpotent subspaces to dominant regions of some real hyperplane arrangement and get the characteristic polynomials of the real hyperplane arrangement in the case of U(m, n) and Sp(m, n). We conjecture a general formula for other types.
by Chuying Fang.
Ph.D.
88 p.
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Ad-nilpotent ideals of complex and real reductive groups
Thesis
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<Abstract>In this thesis, we study ad-nilpotent ideals and its relations with nilpotent orbits, affine Weyl groups, sign types and hyperplane arrangements. This thesis is divided into three parts. The first and second parts deal with ad-nilpotent ideals for complex reductive Lie groups. In the first part, we study the left equivalence relation of ad-nilpotent ideals and relate it to some equivalence relation of affine Weyl groups and sign types. In the second part, we prove that for classical groups there always exist ideals of minimal dimension as conjectured by Sommers. In the third part, we define an analogous object for connected real reductive Lie groups, which is called 0-nilpotent subspaces. We relate 0-nilpotent subspaces to dominant regions of some real hyperplane arrangement and get the characteristic polynomials of the real hyperplane arrangement in the case of U(m, n) and Sp(m, n). We conjecture a general formula for other types.</Abstract>
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