## Regular Functions on the $K$-nilpotent cone

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- by Lucas Mason-Brown
- Represent. Theory
**26**(2022), 1047-1062 - DOI: https://doi.org/10.1090/ert/629
- Published electronically: October 5, 2022

## 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.

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## Bibliographic Information

**Lucas Mason-Brown**- Affiliation: The Mathematical Institute, University of Oxford, Oxford, United Kingdom
- MR Author ID: 1467580
- Received by editor(s): April 22, 2022
- Received by editor(s) in revised form: July 27, 2022
- Published electronically: October 5, 2022
- © Copyright 2022 Copyright by Lucas Mason-Brown
- Journal: Represent. Theory
**26**(2022), 1047-1062 - MSC (2020): Primary 22E46
- DOI: https://doi.org/10.1090/ert/629
- MathSciNet review: 4492621