| Summary: | The principal motivation of this dissertation is to understand the unitary irreducible representations and characters of Dif f⁺S¹-the group of all orientation-preserving diffeomorphisms of S¹ by studying conjugacy classes of Dif f⁺S¹ and its coadjoint orbits. For this purpose, we mainly focus on the following two topics. The first is to study the relation between a real Lie group G and its associated complex semigroup S(G), which was initiated by Oshansky. We consider two particular examples: (1) PSU(1.1) and PSL(2, C)⁺ (Chapter 1); (2) D and A (Chapter 2). We have shown that (a) The equivalence classes determined by the function q on PSL(2, C)⁺ (resp. A) are the same as the conjugacy classes in PSL(2, C)⁺ (resp. A ). In fact the restriction of q to PSL(2, C)⁺ equals the square of the "smaller" of the two eigenvalues of an element in PSL(2, C)+. (b) The fact that the representation of PSL(2, C)⁺ is of trace class makes the character of PSL(2, C)⁺ well-defined. Moreover the character of PSL(2, C)⁺ has analytic continuation onto PSU(1,1) except on a set of measure zero. Surprisingly, the extended characters are exactly Harish-Chandra global characters Xᵐ(PSU)₍₁.₁₎ =Θm. Secondly, we investigate the coadjoint orbits of Virasoro group-the central extension of Dif f⁺S¹ (Chapter 3), which has been considered before by Segal, Kirillov and (later) Witten. We improved Segal's result by parameterizing coadjoint orbits precisely in terms of following conjugacy classes in P͂S͂U͂(1,1): Par⁺₀,{n,n ∈ N},{Elln, n∈N},{Par⁽⁺/⁻⁾(n),n > 0}, {Hyp(n),n ≥ }0. We also completed Kirillov's list of representatives of coadjoint orbits, and we fleshed out the connection between Segal's and Kirillov's and Witten's work by giving the correspondence between conjugacy classes in P͂S͂U͂ (1,1) and the representatives of coadjoint orbits and stabilizers.
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