Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types

2025/07/27 by Ignatev, Mikhail, Petukhov, Alexey
#16D60 #22E25 #22E30 #22E65 #53D17 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2507.20332

Abstract

Let \mathfrak g be a classical simple Lie algebra over an algebraically closed field \mathbb F of characteristic zero or large enough, and let \mathfrak n be a maximal nilpotent subalgebra of \mathfrak g. The main tool in representation theory of \mathfrak n is the orbit method, which classifies primitive ideals in the universal enveloping algebra \rm U(\mathfrak n) and unitary representations of the unipotent group N=exp(\mathfrak n) in terms of coadjoint orbits on the dual space \mathfrak n^*. In the paper, we describe explicitly coadjoint orbits of low dimension for \mathfrak n as above. The answer is given in terms of subsets of positive roots. As a corollary, we provide a way to calculate the number of irreducible complex representations of dimensions q, q² and q³ for a maximal unipotent subgroup N(q) in a classical Chevalley group G(q) over a finite field \mathbb F_q with q elements. It turned out that this number is a polynomial in q-1 with nonnegative integer coefficients, which agrees with Isaac's conjecture.

Citations

• The orbit method for locally nilpotent infinite-dimensional Lie algebras
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• On the coadjoint orbits of maximal unipotent subgroups of reductive\n groups
• ON THE COADJOINT ORBITS OF MAXIMAL UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS
• Center of U(n), Cascade of Orthogonal Roots, and a Construction of Lipsman-Wolf
• The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group
• Reduction for characters of finite algebra groups
• Counting irreducible representations of large degree of the upper triangular groups
• Coadjoint orbits of the group UT(7,K)

Related

• On some representations of nilpotent Lie algebras and superalgebras
• Computing with nilpotent orbits in simple Lie algebras of exceptional type
• Defining Equations of Nilpotent Orbits for Borel Subgroups of Modality Zero in Type A_n
• On the coadjoint orbits of maximal unipotent subgroups of reductive\n groups
• On adjoint orbits in nilpotent ideals of a Borel subalgebra