On the Left and Right Brylinski-Kostant Filtrations

Abstract

Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, \(\mathfrak{b}\) a Borel subalgebra, and \(\mathfrak{h}\subset\mathfrak{b}\) a Cartan subalgebra. Let V be a finite dimensional simple \(U(\mathfrak{g})\) module. Based on a principal s-triple (e,h,f) and following work of Kostant, Brylinski (J Amer Math Soc 2(3):517–533, 1989) defined a filtration \(\mathcal{F}_e\) for all weight subspaces Vμ of V and calculated the dimensions of the graded subspaces for μ dominant. In Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) these dimensions were calculated for all μ. Let δM(0) be the finite dual of the Verma module of highest weight 0. It identifies with the global functions on a Weyl group translate of the open Bruhat cell and so inherits a natural degree filtration. On the other hand, up to an appropriate shift of weights, there is a unique \(U(\mathfrak{b})\) module embedding of V into δM(0) and so the degree filtration induces a further filtration \(\mathcal{F}\) on each weight subspace Vμ. A casual reading of Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) might lead one to believe that \(\mathcal{F}\) and \(\mathcal{F}_e\) coincide. However this is quite false. Rather one should view \(\mathcal{F}_e\) as coming from a left action of \(U(\mathfrak{n})\) and then there is a second (Brylinski-Kostant) filtration \(\mathcal{F}'_e\) coming from a right action. It is \(\mathcal{F}'_e\) which may coincide with \(\mathcal{F}\). In this paper the above claim is made precise. First it is noted that \(\mathcal{F}\) is itself not canonical, but depends on a choice of variables. Then it is shown that a particular choice can be made to ensure that \(\mathcal{F}=\mathcal{F}'_e\). An explicit form for the unique left \(U(\mathfrak{b})\) module embedding \(V\hookrightarrow\delta M(0)\) is given using the right action of \(U(\mathfrak{n})\). This is used to give a purely algebraic proof of Brylinski’s main result in Brylinski (J Amer Math Soc 2(3):517–533, 1989) which is much simpler than Joseph et al. (J Amer Math Soc 13(4):945–970, 2000). It is noted that the dimensions of the graded subspaces of \(\rm{gr}_{\mathcal{F}_e} V_{\!\mu}\) and \(\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}\) can be very different. Their interrelation may involve the Kashiwara involution. Indeed a combinatorial formula for multiplicities in tensor products involving crystal bases and the Kashiwara involution is recovered. Though the dimensions for the graded subspaces of \(\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}\) are determined by polynomial degree, their values remain unknown.