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Abstract
This thesis is concerned with the intrinsic structure of (split) real flag varieties and relationships with total positivity for reductive algebraic groups. In the first two chapters we study intersections of real Bruhat cells. Our main result is a combinatorial parameterization of the connected components of the intersection of two opposed big cells. We also give a formula for the Euler characteristic of the intersection of two general Bruhat cells.
The remaining chapters are related to Lusztig's theory of total positivity for reductive algebraic groups. In Chapter 3 we study the infinitesimal cone of the totally nonnegative semigroup in a reductive group using the canonical basis of the adjoint representation. Then the intersections of Bruhat cells from the earlier chapters are shown to induce an algebraic cell decomposition on the totally nonnegative part of the flag variety. We conjecture a description of these cells in terms of canonical bases, generalizing Lusztig's characterization of the totally positive part of the flag variety. Finally we prove a characterization of the totally positive part of a partial flag variety in terms of canonical bases. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)
