#### Title

A Canonical Description of the Plancherel Measure for a General Two-Step Free Nilpotent Lie Group.

#### Date of Award

2000

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Leonard F. Richardson

#### Abstract

In this work on g=Fn,2 , free 2-step nilpotent Lie algebra on n generators, we use the group of automorphisms to give a basis-free description of the Fourier Inversion Formula, thereby generalizing and strengthening an example discussed by Corwin & Greenleaf. In the Introduction we discuss Example 4.3.14 in Corwin & Greenleaf's book. It demonstrates how different bases of F3,2 lead to different inversion formulas. But the third "more" invariant formula describes Plancherel measure on a support expressed in terms of rotations, dilations, and translations. Actually it is not canonical since it still depends on choices of bases for F3,2 . Our goal is to redescribe Plancherel measure on a support expressed in terms of Aut*( g ). We accomplish this in the following two chapters. Chapter 2 provides a procedure for reparametrizing the family of generic orbits by establishing a 1-1 correspondence between the maximum-dimensional orbits and Ad*G \Aut*( g )/Stab(l0). Chapter 3 provides background material about relatively invariant measures. Then we prove that Plancherel measure, modeled on Ad*G \Aut*( g )/Stab(l0), is the essentially unique relatively invariant measure corresponding to a specific homomorphism. Chapter 4 demonstrates that there does not exist an Aut*g -invariant measure on Ad*G \Aut*( g )/Stab(l0) for the example g=F3,2 . Our explicit calculations for F3,2 are done in Chapter 5 and are in agreement with the results of the first four chapters. We start by finding an almost global coordinate patch for AdG\AdG · Stab*( Z*3 ); use this patch to construct a right and left Haar measure on this quotient space. Thus we get its modular function delta. A similar process applies to AdG\Aut( g ), thus obtaining its modular function Delta. Hence the ratio of Delta to delta is the restriction of any modular function for any relatively invariant measure on AdG\Aut( g )/Stab*( Z*3 ). Furthermore, we find a cross-section X for Ad G\Aut( g )/Stab*(l0) with general l0∈g*Max . Then we use X to verify the relative invariance for the measure corresponding to the example of Corwin and Greenleaf for g=F3,2 . Chapter 6 illustrates three general properties for Fn,2 and one additional result for linear algebra, which are used in Chapter 5.

#### Recommended Citation

Chu, Chin-te, "A Canonical Description of the Plancherel Measure for a General Two-Step Free Nilpotent Lie Group." (2000). *LSU Historical Dissertations and Theses*. 7347.

https://digitalcommons.lsu.edu/gradschool_disstheses/7347

#### ISBN

9780493070018

#### Pages

104