Kostant's problem and parabolic subgroups

2008/06/18 by Johan Kåhrström, Kåhrström, Johan · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.0806.2917

arxiv created 2008/06/18 · openalex publication_date 2008/06/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let \frak g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I⊆ S let \frak g_I be the corresponding semi-simple subalgebra of \frak g. Denote by W_I the Weyl group of \frak g_I and let w_o and w^I_o be the longest elements of W and W_I, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight \frak g_I-module L_I(x) of highest weight x⋅ 0, x∈ W_I, as the answer for the simple highest weight \frak g-module L(x w^I_o w_o) of highest weight (x w^I_o w_o)⋅ 0. We also give a new description of the unique quasi-simple quotient of the Verma module Δ(e) with the same annihilator as L(y), y∈ W.

Cited by

• Hecke combinatorics, Kåhrström's conditions and Kostant's problem

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