# N-strongly quasi-invariant measure on double coset spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, Center of Excellecy in Analysis on Algebric Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran

2 Department of Mathematics, Bojnourd Branch, Islamic Azad university, Bojnourd, Iran

10.22072/wala.2022.550186.1370

Abstract

Let $G$ be a locally compact group, $H$ and $K$ be two closed subgroups of $G$, and $N$ be the normalizer group of $K$ in $G$. In this paper, the existence and properties of a rho-function for the triple $(K, G, H)$ and an $N$-strongly quasi-invariant measure of double coset space $K \backslash G /H$ is investigated. In particular, it is shown that any such measure arises from a rho-function. Furthermore, the conditions under which an $N$-strongly quasi-invariant measure arises from a rho-function are studied.

Keywords

#### References

[1] M. Amini and A. Medghalchi, Harmonic analysis on double coset spaces, Functional Analysis, (math.FA), 2002.
[2] N. Bourbaki, {\em Elements of Mathematics Integration}, Mathematics, 2004.
[3] F. Bruhat, Sur les representations induites des groups de Lie, Bull. Soc. Math. Fr., 81 (1956), 97-205.
[4] Ch.-H. Chu and A.T.-M. Lau, Jordan structures in Harmonic functions and Fourier algebras on homogeneous spaces,
Math. Ann., 336(4) (2006), 803-840.
[5] A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, Springer Science Business, LLC, 2009.
[6] F. Esmaeelzadeh and R.A. Kamyabi-Gol, Homogeneous spaces and square-integrable representations, Ann. Funct.
Anal., 7(1) (2016), 9-16.
[7] F. Fahimian and R.A. Kamyabi-Gol and F. Esmaeelzadeh, N- Relatively invariant and N-invariant measure on double
coset spaces, Bull. Iran. Math. Soc., 45 (2019), 515-525.
[8] G. Folland, A Course In Abstract Harmonic Analysis, CRC press, Inc, 1995.
[9] A.G. Farashahi, Absolutely convergent Fourier series of functions over homogeneous spaces of compact groups, Mich.
Math. J., 69(1) (2020), 179-200.
[10] A.G. Farashahi, A class of abstract linear representations for convolution function algebras over homogeneous spaces
of compact groups, Can. J. Math., 70(1) (2018), 97-116.
[11] A.G. Farashahi, Fourier-Stieltjes transforms over homogeneous spaces of compact groups, Groups Geom. Dyn., 13(2)
(2019), 511-547.
[12] R. Jewett, Spaces With an Abstract Convolution of Measures, Advances in Math, 1975.
[13] T.-s. Liu, Invariant Measure on Double Coset Spaces, University of pennsylvania and university of Massachusetts,
1965.
[14] L.H. Loomis, Positive definite functions and induced representations, Duke Math, J., 27 (1960), 569-579.
[15] G. Mackey, Induced representations of locally compact groups I, Ann. of Math., 55 (1959), 101-139.
[16] H. Reiter and J.D. Stegeman, Classical Harmonic Analysis on Locally Compact Groups, Oxford University Press, 2000.