On Reeder's conjecture for type B and C Lie algebras (Q2071257)

Given a complex semisimple Lie algebra \(\mathfrak{s}\), a long standing problem has been to decompose the exterior algebra \(\bigwedge \mathfrak{s}\) as a representation of the Lie algebra under the adjoint action. Important results on the action of \(\mathfrak{s}\) on its symmetric algebra were obtained by \textit{B. Kostant} [Am. J. Math. 85, 327--404 (1963; Zbl 0124.26802)], providing tools for the generalized statement of the problem. Reeder reanalyzed the case of the exterior algebra in [\textit{M. Reeder}, Can. J. Math. 49, No. 1, 133--159 (1997; Zbl 0878.20028)], although the results are not complete, and where the conjecture on the graded multiplicities of small representations in the exterior algebra was first formulated. In this paper, the author proposes a proof for the Reeder conjecture in the case of the simple Lie algebras of type B and C. The proof is based on the recursive relations introduced in [\textit{J. R. Stembridge}, IMRP, Int. Math. Res. Pap. 2005, No. 4, 183--236 (2005; Zbl 1105.22008)], as well as the closed formulae for the invariants of representations of the Weyl groups. With the results of this work, the problem is settled for the classical series A,B,C, the first case being deduced indirectly from works by Stembridge, Kirillov, Pak and Molchanov.