Regular Functions on the $K$-nilpotent cone

Abstract:

Let $G$ be a complex reductive algebraic group with Lie algebra $\mathfrak {g}$ and let $G_{\mathbb {R}}$ be a real form of $G$ with maximal compact subgroup $K_{\mathbb {R}}$. Associated to $G_{\mathbb {R}}$ is a $K \times \mathbb {C}^{\times }$-invariant subvariety $\mathcal {N}_{\theta }$ of the (usual) nilpotent cone $\mathcal {N} \subset \mathfrak {g}^*$. In this article, we will derive a formula for the ring of regular functions $\mathbb {C}[\mathcal {N}_{\theta }]$ as a representation of $K \times \mathbb {C}^{\times }$.

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 $G_{\mathbb {R}}$ is split, and $X$ is the spherical principal series representation of infinitesimal character $0$, then conjecturally $gr(X) \simeq \mathbb {C}[\mathcal {N}_{\theta }]$ as representations of $K \times \mathbb {C}^{\times }$. So a formula for $\mathbb {C}[\mathcal {N}_{\theta }]$ is an essential ingredient for computing Hodge filtrations.