Regular Functions on the 𝐾-nilpotent cone

Abstract

Let G G be a complex reductive algebraic group with Lie algebra g \mathfrak {g} and let G R G_{\mathbb {R}} be a real form of G G with maximal compact subgroup K R K_{\mathbb {R}} . Associated to G R G_{\mathbb {R}} is a K × C × K \times \mathbb {C}^{\times } -invariant subvariety N θ \mathcal {N}_{\theta } of the (usual) nilpotent cone N ⊂ g ∗ \mathcal {N} \subset \mathfrak {g}^* . In this article, we will derive a formula for the ring of regular functions C [ N θ ] \mathbb {C}[\mathcal {N}_{\theta }] as a representation of K × C × K \times \mathbb {C}^{\times } . Some motivation comes from Hodge theory. In [Hodge theory and unitary representations of reductive Lie groups, Frontiers of Mathematical Sciences, Int. Press, Somerville, MA, 2011, pp. 397–420], Schmid and Vilonen use ideas from Saito’s theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If G R G_{\mathbb {R}} is split, and X X is the spherical principal series representation of infinitesimal character 0 0 , then conjecturally g r ( X ) ≃ C [ N θ ] gr(X) \simeq \mathbb {C}[\mathcal {N}_{\theta }] as representations of K × C × K \times \mathbb {C}^{\times } . So a formula for C [ N θ ] \mathbb {C}[\mathcal {N}_{\theta }] is an essential ingredient for computing Hodge filtrations.