## Osaka Journal of Mathematics

### The Dynkin index and conformally invariant systems associated to parabolic subalgebras of Heisenberg type

Toshihisa Kubo

#### Abstract

Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to parabolic subalgebras of Heisenberg type. When they constructed such systems of operators, two constants, which play a role for the construction, were defined as the constants of proportionality between two expressions. In this paper we give concrete and uniform expressions for these constants. To do so we introduce a new constant inspired by a formula on the Dynkin index of a finite dimensional representation of a complex simple Lie algebra.

#### Article information

Source
Osaka J. Math., Volume 51, Number 2 (2014), 359-375.

Dates
First available in Project Euclid: 8 April 2014

https://projecteuclid.org/euclid.ojm/1396966253

Mathematical Reviews number (MathSciNet)
MR3192546

Zentralblatt MATH identifier
1338.17010

Subjects
Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 22E46: Semisimple Lie groups and their representations

#### Citation

Kubo, Toshihisa. The Dynkin index and conformally invariant systems associated to parabolic subalgebras of Heisenberg type. Osaka J. Math. 51 (2014), no. 2, 359--375. https://projecteuclid.org/euclid.ojm/1396966253

#### References

• L. Barchini, A.C. Kable and R. Zierau: Conformally invariant systems of differential equations and prehomogeneous vector spaces of Heisenberg parabolic type, Publ. Res. Inst. Math. Sci. 44 (2008), 749–835.
• L. Barchini, A.C. Kable and R. Zierau: Conformally invariant systems of differential operators, Adv. Math. 221 (2009), 788–811.
• N. Bourbaki: Groupes et Algébres de Lie, Masson, Paris, 1981, Chapters 4–6.
• H.W. Braden: Integral pairings and Dynkin indices, J. London Math. Soc. (2) 43 (1991), 313–323.
• E.B. Dynkin: Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl., Ser. II, 6 (1957), 111–244.
• V.G. Kac: Infinite-Dimensional Lie Algebras, third edition, Cambridge Univ. Press, Cambridge, 1990.
• S. Kumar and M.S. Narasimhan: Picard group of the moduli spaces of $G$-bundles, Math. Ann. 308 (1997), 155–173.
• S. Kumar, M.S. Narasimhan and A. Ramanathan: Infinite Grassmannians and moduli spaces of $G$-bundles, Math. Ann. 300 (1994), 41–75.
• A.L. Onishchik: Torsion of special Lie groups, Amer. Math. Soc. Transl., Ser. II, 50 (1966), 1–4.
• D.I. Panyushev: On the Dynkin index of a principal $\mathfrak{sl}_{2}$-subalgebra, Adv. Math. 221 (2009), 1115–1121.
• W. Wang: Dimension of a minimal nilpotent orbit, Proc. Amer. Math. Soc. 127 (1999), 935–936.