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Preface 1 Introduction 1.1 Background 1.2 Main results 2 Elementary analysis of stratified Lie groups 2.1 Preliminaries on Lie groups 2.1.1 Vector fields in RN 2.1.2 Lie groups on HN 2.1.3 Homogeneous stratified Lie groups 2.2 The sub-Laplacians on stratified Lie groups 2.3 2-step stratified Lie groups 2.3.1 Characterization of 2-step stratified groups 2.3.2 Some examples 3 Harmonic analysis on 2-step stratified Lie groups 3.1 Orbit method on 2-step stratified Lie groups 3.1.1 Parametrization of coadjoint orbits 3.1.2 Polarization and unitary representation 3.2 The Fourier analysis 3.2.1 Irreducible unitary representations 3.2.2 Examples 3.2.3 The Fourier transform 3.2.4 The sub-Laplacian operator 3.3 (��,��)-Weyl transforms 3.3.1 (��,��)-Fourier-Wigner transform 3.3.2 (��,��)-Wigner transform 3.3.3 (��,��)-Weyl transform 3.3.4 The A-twisted convolution 3.4 Stone-von Neumann theorem 3.5 Hermite and special Hermite functions 3.5.1 Mehler's formula for the rescaled harmonic oscillator 3.5.2 Special Hermite functions 3.5.3 Eigenvalue problems of the A-twisted sub-Laplacian\" 3.6 Laguerre functions 3.6.1 Laguerre polynomials 3.6.2 Laguerre formulas for special Hermite functions 4 Applications 4.1 Weyl-HSrmander calculus 4.1.1 Weyl-HSrmander calculus on Rn 4.1.2 The (��,��)-Shubin classes ��(��(sh��)������ʽ) 4.1.3 (��,��)-Shubin Sobolev spaces 4.2 Heat kernels of sub-Laplacians 4.2.1 Heat kernels of H(��) 4.2.2 Heat kernels of L 5 Appendix 5.1 Abstract Lie groups 5.2 Left-invariant vector fields and the Lie algebra 5.3 Nilpotent Lie groups 5.4 Abstract and homogeneous stratified Lie groups Bibliograply Index Index of Symbols