2019/12/01 by Dmitri I. Panyushev, Panyushev, Dmitri I. · 1 citation
Mathematics · #15A75 #17B20 #17B22 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1912.00341
openalex publication_date 2019/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let mathfrak g be a simple Lie algebra, mathfrak h a Levi subalgebra,\nand C mathfrak h\∈ U( mathfrak h) the Casimir element defined via the\nrestriction of the Killing form on mathfrak g to mathfrak h. We study\nC mathfrak h-eigenvalues in mathfrak g/ mathfrak h and related\n mathfrak h-modules. Without loss of generality, one may assume that\n mathfrak h is a maximal Levi. Then mathfrak g is equipped with the\nnatural mathbb Z-grading mathfrak g= bigoplusi\∈ mathbb Z mathfrak\ng(i) such that mathfrak g(0)= mathfrak h and mathfrak g(i) is a simple\n mathfrak h-module for i\≠ 0. We give explicit formulae for the\nC_ mathfrak h-eigenvalues in each mathfrak g(i), i\≠ 0, and relate\neigenvalues of C_ mathfrak h in bigwedge^ bullet mathfrak g(1) to the\ndimensions of abelian subspaces of mathfrak g(1). We also prove that if\n mathfrak a\⊂ mathfrak g(1) is abelian, whereas mathfrak g(1) is not,\nthen \dim mathfrak a\≤ \dim mathfrak g(1)/2. Moreover, if \dim mathfrak\na=(\dim mathfrak g(1))/2, then mathfrak a has an abelian complement. The\n mathbb Z-gradings of height \≤ 2 are closely related to involutions of\n mathfrak g, and we provide a connection of our theory to (an extension of)\nthe "strange formula" of Freudenthal-de Vries.\n