This ebook is the 1st person who brings jointly fresh effects at the harmonic research of exponential solvable Lie teams. There nonetheless are many fascinating open difficulties, and the ebook contributes to the longer term growth of this study box. to boot, numerous comparable themes are offered to inspire younger researchers.
The orbit strategy invented through Kirillov is utilized to check uncomplicated difficulties within the research on exponential solvable Lie teams. this system tells us that the unitary twin of those teams is learned because the house in their coadjoint orbits. This truth is proven utilizing the Mackey concept for caused representations, and that mechanism is defined first. one of many primary difficulties within the illustration conception is the irreducible decomposition of caused or limited representations. hence, those decompositions are studied intimately ahead of continuing to numerous similar difficulties: the multiplicity formulation, Plancherel formulation, intertwining operators, Frobenius reciprocity, and linked algebras of invariant differential operators.
The major reasoning within the evidence of the assertions made this is induction, and for this there usually are not many instruments on hand. hence an in depth research of the gadgets indexed above is hard even for exponential solvable Lie teams, and it is usually assumed that G is nilpotent. To make the placement clearer and destiny improvement attainable, many concrete examples are supplied. quite a few issues awarded within the nilpotent case nonetheless need to be studied for solvable Lie teams that aren't nilpotent. all of them current fascinating and critical yet tricky difficulties, despite the fact that, which may be addressed within the close to destiny. past the exponential case, holomorphically brought on representations brought through Auslander and Kostant are wanted, and for this reason they're integrated during this book.