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.
References
- Jeffrey D. Adams, Marc A. A. van Leeuwen, Peter E. Trapa, and David A. Vogan Jr., Unitary representations of real reductive groups, Astérisque 417 (2020), x + 174 (English, with English and French summaries). MR 4146144, DOI 10.24033/ast
- Jeffrey Adams and David A. Vogan Jr., Associated varieties for real reductive groups, Pure Appl. Math. Q. 17 (2021), no. 4, 1191–1267. MR 4359259, DOI 10.4310/PAMQ.2021.v17.n4.a2
- Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Modern Birkhäuser Classics, Birkhäuser Boston, Ltd., Boston, MA, 2010. Reprint of the 1997 edition. MR 2838836, DOI 10.1007/978-0-8176-4938-8
- Hans Grauert and Oswald Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292 (German). MR 302938, DOI 10.1007/BF01403182
- Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
- Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4
- B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 311837, DOI 10.2307/2373470
- George Lusztig, Singularities, character formulas, and a $q$-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR 737932
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- William M. McGovern, Rings of regular functions on nilpotent orbits and their covers, Invent. Math. 97 (1989), no. 1, 209–217. MR 999319, DOI 10.1007/BF01850661
- Wilfried Schmid and Kari Vilonen, Hodge theory and unitary representations of reductive Lie groups, Frontiers of mathematical sciences, Int. Press, Somerville, MA, 2011, pp. 397–420. MR 3050836
- Birgit Speh and David A. Vogan Jr., Reducibility of generalized principal series representations, Acta Math. 145 (1980), no. 3-4, 227–299. MR 590291, DOI 10.1007/BF02414191
- R. W. Thomason, Algebraic $K$-theory of group scheme actions, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563. MR 921490
- David A. Vogan Jr., Branching to a maximal compact subgroup, Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 321–401. MR 2401817, DOI 10.1142/9789812770790_{0}010
- David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
- David A. Vogan Jr., Associated varieties and unipotent representations, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR 1168491
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