A characterization of Dynkin elements
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We give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element is the unique point of minimal length in its $N$-region. In type $A_n$ this translates into a statement about the regions determined by the canonical left Kazhdan-Lusztig cells, which leads to some conjectures in representation theory.
PE Gunnells and E Sommers. "A characterization of Dynkin elements" Mathematical Research Letters 10.2-3 (2003): 363-373.
Available at: http://works.bepress.com/eric_sommers/9