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1. The Subject Matter. Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W. In this book, we present a geometric perspective on the following circle of ideas: polynomials The "vertices" of this graph are some of the most important objects in representation theory. Each has a theory in its own right, and each has had its own independent historical development. - A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. (For the special linear group 2 G = SL(n,C), whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; such an orbit is nilpotent if the Jordan form has only zeros on the diagonal. In this case, the nilpotent orbits are classified by partitions of n, given by the sizes of the Jordan blocks.) The closures of the nilpotent orbits are singular in general, and understanding their singularities is an important problem. - The classification of irreducible Weyl group representations is quite old.
To talk about values and ideals is easy. To live them is much more difficult, because no one is perfect. Like all good things, it requires effort. At times we all fall short of our ideals and values. The question is: Do we have ideals and values? I hope this book will be used by individuals, families and schools as a starting point for discussing character ideals in personal development. Values and ideals are as important as any other subject taught in school because without them your other skills may bring little personal satisfaction. Although I've called this a book about values, it is really about personal happiness. Your happiness will come from the values and ideals you choose for yourself. If you choose wisely, your values will bring you strength and a foundation to build a satisfying life. Your values will shape your life. This book is not intended to "teach" you values and ideals. Family, culture and faith traditions may be the best teachers. Rather, it is intended to share with you values and ideals that men and women have respected as long as history has been recorded, and to encourage discussion about them.
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