REPRESENTATIONS OF LINEAR GROUPS

By STEVEN THOMPSON

CONTENTS PROLOGUE: SOME GROUPS AND THEIR ACTIONS SEVERAL MATRIX GROUPS GROUP ACTIONS THE SYMMETRIC GROUP BASIC ALGEBRAIC CONCEPTS LINEAR REPRESENTATIONS EQUIVALENT REPRESENTATIONS FIRST EXAMPLES BASIC CONSTRUCTION PRINCIPLES SUM OF REPRESENTATIONS TENSOR PRODUCT OF REPRESENTATIONS THE CONTRAGREDIENT REPRESENTATION THE FACTOR REPRESENTATION DECOMPOSITIONS CHARACTERS REPRESENTATIONS OF FINITE GROUPS CHARACTERS AS ORTHONORMAL SYSTEMS THE REGULAR REPRESENTATION CHARACTERS AS ORTHONORMAL BASES CONTINUOUS REPRESENTATIONS TOPOLOGICAL AND LINEAR GROUPS THE CONTINUITY CONDITION INVARIANT MEASURES EXAMPLES REPRESENTATIONS OF COMPACT GROUPS BASIC FACTS THE EXAMPLE G = SU(2) THE EXAMPLE G = SO(3) REPRESENTATIONS OF ABELIAN GROUPS CHARACTERS AND THE PONTRJAGIN DUAL CONTINUOUS DECOMPOSITIONS CONTENTS THE IN?NITESIMAL METHOD LIE ALGEBRAS AND THEIR REPRESENTATIONS THE LIE ALGEBRA OF A LINEAR GROUP DERIVED REPRESENTATIONS UNITARILY INTEGRABLE REPRESENTATIONS OF SL(2, R) THE EXAMPLES SU(2) AND HEIS(R) SOME STRUCTURE THEORY SPECI?CATIONS OF GROUPS AND LIE ALGEBRAS STRUCTURE THEORY FOR COMPLEX SEMISIMPLE LIE ALGEBRAS STRUCTURE THEORY FOR COMPACT REAL LIE ALGEBRAS STRUCTURE THEORY FOR NONCOMPACT REAL LIE ALGEBRAS REPRESENTATIONS OF HIGHEST WEIGHT THE EXAMPLE SU(3) INDUCED REPRESENTATIONS THE PRINCIPLE OF INDUCTION PRELIMINARY APPROACH MACKEY’S APPROACH FINAL APPROACH SOME QUESTIONS AND TWO EASY EXAMPLES UNITARY REPRESENTATIONS OF SL(2, R) UNITARY REPRESENTATIONS OF SL(2,C) AND OF THE LORENTZ GROUP UNITARY REPRESENTATIONS OF SEMIDIRECT PRODUCTS UNITARY REPRESENTATIONS OF THE POINCAR´E GROUP INDUCED REPRESENTATIONS AND VECTOR BUNDLES GEOMETRIC QUANTIZATION AND THE ORBIT METHOD THE HAMILTONIAN FORMALISM AND ITS QUANTIZATION COADJOINT ORBITS AND REPRESENTATIONS PREQUANTIZATION EXAMPLE: CONSTRUCTION OF LINE BUNDLES OVER M = P1(C) QUANTIZATION COADJOINT ORBITS AND HAMILTONIAN G-SPACES CONSTRUCTION OF AN IRREDUCIBLE UNITARY REPRESENTATION BY AN ORBIT THE EXAMPLES SU(2) AND SL(2, R) THE EXAMPLE HEIS(R) SOME HINTS CONCERNING THE JACOBI GROUP EPILOGUE: OUTLOOK TO NUMBER THEORY THETA FUNCTIONS AND THE HEISENBERG GROUP MODULAR FORMS AND SL(2,R) THETA FUNCTIONS AND THE JACOBI GROUP HECKE’S THEORY OF L?FUNCTIONS ASSOCIATED TO MODULAR FORMS ELEMENTS OF ALGEBRAIC NUMBER THEORY AND HECKE L-FUNCTIONS ARITHMETIC L-FUNCTIONS SUMMARY AND FINAL RE?ECTIONS BIBLIOGRAPHY