IBITOWA AJEWOLE RASHEED

Mathematics

At the Mathematics department office

Appointment on Visitation important

Topic: Harmonic Analysis Of Orbital Integrals

Description:

Harmonic analysis and representation theory are two broad areas of Functional analysis. Orbital integrals are integrals of class functions over the conjugacy classes of a given topological group. 

This study aims to:

1. find the transformation of orbital integrals through various integral transformations such as Fourier transform, Radon transform, etc

2. restrict orbital integral from a given locally compact group to its subgroup

3. transfer orbital integral from a reductive group G to a reductive group G'.

To achieve these, we employ the following:

1. the application of the Fourier transform 

2. make use of a reductive Lie group and its closed subgroup

3. make use of two distinct reductive groups G and G'

At the end of the study, 

1. we obtain that  the application of The Fourier transform on orbital integral is an entire function in the complex domain

2. we obtain that the restriction of orbital integral from a locally compact group G to its subgroup is possible  if the subgroup is endoscopic.

3. we obtain that the transfer of orbital integrals from G to G' is possible if there is an isomorphism between G and G'.

we have shown that the analysis of orbital integrals on real reductive groups is a topic of great research and yet to be completed. 

In future, the following can also be examined on orbital integrals on real reductive groups:

1.the Radon transform or Helgason transform of orbital integrals

2. the transfer of weighted orbital integrals

 

#	Certificate	School	Year
1.	M.Sc (Mathematics/Functional Analysis)	University of Ibadan, Ibadan	2013

Annihilation of orbital integrals

The classical Paley-Wiener theorem has been stated by late Raymond E.A.C. Paley and Norbert Wiener in 1934. It is an essential theorem in Harmonic Analysis. The distributional version of the Paley-Wiener theorem was stated by Laurent Schwartz in 1954. The Paley-Wiener theorem constitutes of taking the Fourier transformation in the complex domain. Sequel to the research on the Paley-Wiener theorem of orbital integrals, we are intending to restrict orbital integrals from a reductive Lie group G to another reductive Lie group G'. To this In all recent researches, it came to a surprise that the annihilation of orbital integrals is sstill yet to be performed. This research is to annihilate the orbital integrals on the real reductive Lie algebra of the real reductive Lie group G. To this end, an open invariant subset of a real reductive Lie algebra g  is constructed and its properties are fully demonstrated. A new tempered invariant distribution gT is formed on the invariant open set , given a tempered invariant distribution T such that gT= T on the invariant set. The analytic invariant differential operators are introduced on the invariant set. The analytic invariant differential operators are proven to annihilate orbital integrals.

IBITOWA RASHEED is a Assistant Lecturer at the Department of Mathematics

IBITOWA has a M.Sc in Mathematics/Functional Analysis from University of Ibadan, Ibadan