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- W4302014204 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. 397â420], Schmid and Vilonen use ideas from Saitoâs 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." @default.
- W4302014204 created "2022-10-06" @default.
- W4302014204 date "2022-10-05" @default.
- W4302014204 modified "2023-10-15" @default.
- W4302014204 doi "https://doi.org/10.1090/ert/2022-26-33" @default.
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