A bounded operator T on a complex separable Hilbert space is said to be homogeneous if '(T ) is unitarily equivalent to T for all ' in M•ob, where M•ob is the M•obius group. A complete description of all homogeneous weighted shifts was obtained by Bagchi and Misra. The first examples of irreducible bi-lateral homogeneous 2-shifts were given by Koranyi. We describe all irreducible homogeneous 2-shifts up to unitary equivalence completing the list of homogeneous 2-shifts of Koranyi. After completing the list of all irreducible homogeneous 2-shifts, we show that every homogeneous operator whose associated representation is a direct sum of three copies of a Complementary series representation, is reducible. Moreover, we show that such an operator is either a direct sum of three bi-lateral weighted shifts, each of which is a homogeneous operator or a direct sum of a homogeneous bi-lateral weighted shift and an irreducible bi-lateral 2-shift. It is known that the characteristic function T of a homogeneous contraction T with an associated representation is of the form T (a) = L( a) T (0) R( a); where L and R are projective representations of the M•obius group M•ob with a common multiplier. We give another proof of the \product formula". We point out that the defect operators of a homogeneous contraction in B2(D) are not always quasi-invertible (recall that an operator T is said to be quasi-invertible if T is injective and ran(T ) is dense). We prove that when the defect operators of a homogeneous contraction in B2(D) are not quasi-invertible, the projective representations L and R are unitarily equivalent to the holomorphic Discrete series representations D+ 1 and D++3, respectively. Also, we prove that, when the defect operators of a homogeneous contraction in B2(D) are quasi-invertible, the two representations L and R are unitarily equivalent to certain known pairs of representations D 1; 2 and D +1; 1 ; respectively. These are described explicitly. Let G be either (i) the direct product of n-copies of the bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic automorphism group of the polydisc Dn: A commuting tuple of bounded operators T = (T1; T2; : : : ; Tn) is said to be homogeneous with respect to G if the joint spectrum of T lies in Dn and '(T); defined using the usual functional calculus, is unitarily equivalent to T for all ' 2 G: We show that a commuting tuple T in the Cowen-Douglas class of rank 1 is homogeneous with respect to G if and only if it is unitarily equivalent to the tuple of the multiplication operators on either the reproducing kernel Hilbert space with reproducing kernel n 1 i=1 (1 ziwi) i or Q n i i n; are positive real numbers, according asQG is as in (i) or 1 ; where ; i, 1 i i=1 (1 z w ) (ii). Finally, we show that a commuting tuple (T1; T2; : : : ; Tn) in the Cowen-Douglas class of rank 2 is homogeneous with respect to M•obn if and only if it is unitarily equivalent to the tuple of the multiplication operators on the reproducing kernel Hilbert space whose reproducing kernel is a product of n 1 rank one kernels and a rank two kernel. We also show that there is no irreducible tuple of operators in B2(Dn), which is homogeneous with respect to the group Aut(Dn):

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The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class. Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ). Clearly, finding relationships amongs the complex geometric invariants inherent in the short exact sequence 0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0 is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )). We obtain a refinement of this formula: trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).