By his own account, in his Notice sur les travaux scientifiques, the main theme of his works (numbering 186 and published throughout the period 1893–1947) was the theory of Lie groups.
The transformation group of such transformations has 15 parameters and extends both the Poincaré group and the Lorentz group.
Lie groups are used to study space, structure, and change.
Michel Duflo (born 1943) is a French mathematician working in the representation theory of Lie groups.
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In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO(3).
Stasheff's research contributions include the study of associativity in loop spaces and the construction of the associahedron (also called the Stasheff polytope), ideas leading to the theory of operads; homotopy theoretic approaches to Hilbert's fifth problem on the characterization of Lie groups; and the study of Poisson algebras in mathematical physics.
Analogous statements hold for simply connected Lie groups, and, as shown by George Mostow, for algebraic Lie algebras and simply connected algebraic groups over a field of characteristic zero.
Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to Hurwitz, and in the Lie group cases can always be given by an invariant differential form.
Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 (July 1963)
Harish-Chandra's Ξ function, a special spherical function on a semisimple Lie group