On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra

Edi Kurniadi

Abstract


In this paper, we study irreducible unitary representations of a real standard filiform Lie group with dimension equals 4 with respect to its basis. To find this representations we apply the orbit method introduced by Kirillov. The corresponding orbit of this representation is genereric orbits of dimension 2. Furthermore, we show that obtained representation of this group is square-integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar  identity operator. In our case  that scalar is equal to one.


Keywords


Duflo-Moore operator, irreducible unitary representation, square-integrable representation, standard filiform Lie algebra.

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References


A. Hadjer and A. Makhlouf, “Index of Graded Filiform and Quasi Filiform Lie Algebras,” no. May 2014, 2012.

A. A. Kirillov, “Lectures on the Orbit Method, Graduate Studies in Mathematics,” Am. Math. Soc., vol. 64, 2004.

A. . Kirillov, “Unitary representations of nilpotent Lie groups,” Uspekhi Mat.Nauk, vol. 17, pp. 57--110, 1962.

R. Berndt, Representation of linear groups. An introduction based on examples from physics and number theory. Wiesbaden: Vieweg, 2007.

J. . Lee, Introduction to smooth manifolds, Graduate Text in Mathematics,. New York: Springer-Verlag, 2003.

M. Duflo and C. C. Moore, “On the Regular Representation of a nonunimodular Locally Compact,” J. Funct. Anal., vol. 21, pp. 209–243, 1976.

A. . Carey, “Square-integrable representation of nonunimodular groups,” Bull.Austral.Math.Soc, vol. 15, pp. 1--12, 1976.

A. Grossmann, J. Morlet, and T. Paul, “Transform associated to square-integrable group of representations I,” J.Math.Phys, vol. 26, pp. 2473--2479, 1985.

A. Grossmann, J. Morlet, and T. Paul, “Trsansform associated to square-integrable group representations .II. Examples,” Ann.Inst.H.Poincare Phys.Theor, vol. 45, pp. 293--309, 1986.

L. J. Corwin and F. . Greenleaf, Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples,. Cambridge: Cambridge University Press, 1990.




DOI: https://doi.org/10.18860/ca.v6i2.9094

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