## Involve: A Journal of Mathematics

• Involve
• Volume 12, Number 3 (2019), 451-462.

### Nilpotent orbits for Borel subgroups of $\mathrm{SO}_{5}(k)$

#### Abstract

Let $G$ be a quasisimple algebraic group defined over an algebraically closed field $k$ and $B$ a Borel subgroup of $G$ acting on the nilradical $n$ of its Lie algebra $b$ via the adjoint representation. It is known that $B$ has only finitely many orbits in only five cases: when $G$ is type $A n$ for $n ≤ 4$, and when $G$ is type $B 2$. We elaborate on this work in the case when $G = SO 5 ( k )$ (type $B 2$) by finding the defining equations of each orbit. We use these equations to determine the dimension of the orbits and the closure ordering on the set of orbits. The other four cases, when $G$ is type $A_n$, can be approached the same way and are treated in a separate paper.

#### Article information

Source
Involve, Volume 12, Number 3 (2019), 451-462.

Dates
Revised: 8 February 2018
Accepted: 10 July 2018
First available in Project Euclid: 5 February 2019

https://projecteuclid.org/euclid.involve/1549335632

Digital Object Identifier
doi:10.2140/involve.2019.12.451

Mathematical Reviews number (MathSciNet)
MR3905340

Zentralblatt MATH identifier
07033141

#### Citation

Burkhart, Madeleine; Vella, David. Nilpotent orbits for Borel subgroups of $\mathrm{SO}_{5}(k)$. Involve 12 (2019), no. 3, 451--462. doi:10.2140/involve.2019.12.451. https://projecteuclid.org/euclid.involve/1549335632

#### References

• A. Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics 126, Springer, 1991.
• N. Bourbaki, Lie groups and Lie algebras: Chapters 4–6, Springer, 2002.
• T. Brüstle, L. Hille, G. Röhrle, and G. Zwara, “The Bruhat–Chevalley order of parabolic group actions in general linear groups and degeneration for $\Delta$-filtered modules”, Adv. Math. 148:2 (1999), 203–242.
• H. Bürgstein and W. H. Hesselink, “Algorithmic orbit classification for some Borel group actions”, Compositio Math. 61:1 (1987), 3–41.
• M. Burkhart and D. Vella, “Defining equations of nilpotent orbits for Borel subgroups of modality zero in type $A_{n}$”, preprint, 2017.
• R. W. Carter, Finite groups of Lie type: conjugacy classes and complex characters, John Wiley & Sons, New York, 1985.
• D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Co., New York, 1993.
• E. M. Friedlander and B. J. Parshall, “Support varieties for restricted Lie algebras”, Invent. Math. 86:3 (1986), 553–562.
• L. Hille and G. Röhrle, “A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical”, Transform. Groups 4:1 (1999), 35–52.
• J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics 9, Springer, 1972.
• J. E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics 21, Springer, 1975.
• J. C. Jantzen, “Nilpotent orbits in representation theory”, pp. 1–211 in Lie theory, edited by J.-P. Anker and B. Orsted, Progr. Math. 228, Birkhäuser, Boston, 2004.
• V. V. Kashin, “Orbits of an adjoint and co-adjoint action of Borel subgroups of a semisimple algebraic group”, pp. 141–158 in Problems in group theory and homological algebra, edited by A. L. Onishchik, Yaroslav. Gos. Univ., Yaroslavl, 1990. In Russian.
• D. K. Nakano, B. J. Parshall, and D. C. Vella, “Support varieties for algebraic groups”, J. Reine Angew. Math. 547 (2002), 15–49.
• V. L. Popov, “A finiteness theorem for parabolic subgroups of fixed modality”, Indag. Math. $($N.S.$)$ 8:1 (1997), 125–132.
• V. Popov and G. Röhrle, “On the number of orbits of a parabolic subgroup on its unipotent radical”, pp. 297–320 in Algebraic groups and Lie groups, edited by G. Lehrer et al., Austral. Math. Soc. Lect. Ser. 9, Cambridge Univ. Press, 1997.
• G. Röhrle, “Parabolic subgroups of positive modality”, Geom. Dedicata 60:2 (1996), 163–186.
• G. Röhrle, “On the modality of parabolic subgroups of linear algebraic groups”, Manuscripta Math. 98:1 (1999), 9–20.