## Tohoku Mathematical Journal

### $K$-finite solutions to conformally invariant systems of differential equations

Anthony C. Kable

#### Abstract

Let $G$ be a connected semisimple linear real Lie group, and $Q$ (resp. $K$) a real parabolic subgroup (resp. maximal compact subgroup) of $G$. The space of $K$-finite solutions to a conformally invariant system of differential equations on a line bundle over the real flag manifold $G/Q$ is studied. The general theory is then applied to certain second order systems on the flag manifold that corresponds to the Heisenberg parabolic subgroup in a split simple Lie group.

#### Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 539-559.

Dates
First available in Project Euclid: 6 January 2012

https://projecteuclid.org/euclid.tmj/1325886280

Digital Object Identifier
doi:10.2748/tmj/1325886280

Mathematical Reviews number (MathSciNet)
MR2872955

Zentralblatt MATH identifier
1236.22011

#### Citation

Kable, Anthony C. $K$-finite solutions to conformally invariant systems of differential equations. Tohoku Math. J. (2) 63 (2011), no. 4, 539--559. doi:10.2748/tmj/1325886280. https://projecteuclid.org/euclid.tmj/1325886280

#### 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.
• D. H. Collingwood and B. Shelton, A duality theorem for extensions of induced highest weight modules, Pacific J. Math. 146 (1990), 227–237.
• A. C. Kable, Conformally invariant systems of differential equations on flag manifolds for $\mathrm{G}_2$ and their $K$-finite solutions, preprint (2010).
• A. C. Kable, The Heisenberg ultrahyperbolic equation, preprint (2010).