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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Regular Functions on the $K$-nilpotent cone
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by Lucas Mason-Brown
Represent. Theory 26 (2022), 1047-1062
Published electronically: October 5, 2022


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:
  • MathSciNet review: 4492621