The development of multi-variate operator theory has been similar to

that of function theory in several complex variables except that the non-

commutating co-ordinates pose an entirely new set of challenges. The

introduction of methods of commutative algebra for studying problems

in this area by viewing a pair (H ,T ), where T := (T

_{1},...,T_{n}) is a commut-ing n-tuple of operators on the Hilbert space H , as a Hilbert module via

the multiplication (p,h) → p(T )h, p ∈ C[z

_{1},...,z_{n}], h ∈ H , was the begin-ning of a systematic new development. In this approach, the vast array

of tools from commutative algebra is available for solving problems in

multi-variate operator theory. However, these techniques don’t apply di-

rectly because of the continuity assumption of the module multiplication

either in just the ﬁrst variable, or in both the variables. A choice is made

depending on the problem at hand.

A second ingredient has been the action of a group G on the module

(H ,T ). This occurs naturally in several instances. For example, if the

joint spectrum X := σ(T ) of the d-tuple T is a G space, that is, G acts on

X by biholomorphic automorphisms, then deﬁning g · T , the classiﬁca-

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tion of imprimitivities X ,G,(H ,T ) := T :U

_{g}^{∗}p(T )U_{g }= (g · p)(T )}, U_{g }isa unitary representation of the group G and g · p = p ◦ g

^{−1}, is an impor-tant problem. One may study such imprimitivities by choosing a group G

acting on X . This choice might vary from a transitive action to an action,

where the orbits of G in X are quite big. For a typical example, consider

X = D

^{n}, the n-fold product of the unit closed disc, and the group G tobe either the bi-holomorphic automorphism group of D

^{n }or simply thepermutation group S

_{n }acting on D^{n}.The study of these problems are greatly facilitated by the analysis of

a class holomorphic curves in the Grassmannian Gr(H ,n) of rank n in

some separable complex Hilbert space H . These holomorphic curves

arise from a class of operators acting on some Hilbert space H intro-

duced in the paper of M. J. Cowen and R. G. Douglas, "Operator Theory

and Complex Geometry", Acta Math., 141 (1978), 187 - 261. The opera-

tors in this class possess an open set Ω ⊆ C of eigenvalues of (constant)

multiplicity k and characterized by the existence of a holomorphic map

γ : Ω → H such that γ(w) := (γ

_{1}(w),...,γ_{k}(w)), T γ_{i }(w) = wγ_{i }(w), 1 ≤ i ≤k, w ∈ Ω. For k = 1, one of the main features of the operator T in this class

2

¯

is that the curvature K

_{T }(w) := −∂∂logkγ_{T }(w)k of the holomorphic Her-mitian line bundle E

_{T }determined by the holomorphic map γ_{T }equippedwith the Hermitian structure kγ

_{T }(w)k is a complete unitary invariant for2

the operator T .

It is easy to see that if T is a contraction in the Cowen-Douglas class of

∗

∗

the unit disc D, then K

_{T }(w) ≤ K_{S }(w), where S is the backward unilat-2

∗

∗

eral shift acting on ` . Choosing a holomorphic frame γ

_{S }, say γ_{S }(w) =2

2

2 −1

∗

∗

(1,w,w ,...), it follows that kγ

_{S }(w)k = (1 − |w| ) and that K_{S }(w) =−(1−|w| ) , w ∈ D. Thus the the operator S

^{∗ }is an extremal operator in2 −2

the class of all contractive Cowen-Douglas operator. R. G. Douglas asked

if the curvature K

_{T }of a contraction T achieves equality in this inequalityeven at just one point, then does it follow that T must be unitarily equiv-

alent to S

^{∗}? It is easy to see that the answer is “no”, in general. However, ifT is homogeneous, namely, U

_{ϕ}^{∗}TU_{ϕ }= ϕ(T ) for each bi-holomorphic au-tomorphism ϕ of the unit disc and some unitary U

_{ϕ}, then the answer is“yes”. Of course, it is then natural to ask what are all the homogeneous

operators. These are the holomorphic imprimitivities.

The question of classifying holomorphic imprimitivities on a bounded

symmetric domain Ω amounts to classiﬁcation of commuting tuples of

homogeneous of operators in the Cowen-Douglas class of Ω. There is a

one to one correspondence between these and the holomomorphic ho-

mogeneous vector bundles on the bounded symmetric domain Ω. The

homogeneous bundles can be obtained by holomorphic induction from

representations of a certain parabolic Lie algebra on ﬁnite dimensional

inner product spaces. The representations, and the induced bundles,

have composition series with irreducible factors. In joint work with A.

Korányi [48, 61]

^{1}, our ﬁrst main result is the construction of an explicitdifferential operator intertwining the bundle with the direct sum of its

factors. Next, we study Hilbert spaces of sections of these bundles. We

use this to get, in particular, a full description and a similarity theorem

for homogeneous n-tuples of operators in the Cowen-Douglas class of

the Euclidean unit ball in C

^{m}. A different approach is in [54]. The initialstudy of these questions restricted to the case of homogeneous holomor-

phic line bundles is in [18].

A little more can be said beyond holomorphic imprimitivities in the

case of a single operator T acting on a Hilbert space H . Assume that

there is a projective unitary representation σ of Möb such that ϕ(T ) =

σ(ϕ)

^{?}T σ(ϕ) for all ϕ in Möb. If this is the case, the operator T is ho-mogeneous and we say that σ(ϕ) is associated with the operator T . A

Möbius equivariant version of the Sz.-Nagy–Foias model theory for com-

pletely non-unitary (cnu) contractions is developed in [67]. As an appli-

cation, we prove that if T is a cnu contraction with associated (projective

1

The numbers in square brackets refer to the list on the Publications page.

unitary) representation σ, then there is a unique projective unitary rep-

ˆ

resentation σ, extending σ, associated with the minimal unitary dilation

ˆ

of T . The representation σ is given in terms of σ by the formula

+

−

ˆ

σ = (π⊗D

_{1 })⊕σ⊕(π_{? }⊗D_{1 }),where D

_{1}^{± }are the two Discrete series representations (one holomorphicand the other anti-holomorphic) living on the Hardy space H

^{2}(D), andπ,π

_{? }are representations of Möb living on the two defect spaces of T anddeﬁned explicitly in terms of σ and T . Moreover, a cnu contraction T has

an associated representation if and only if its Sz.-Nagy–Foias character-

istic function θ

_{T }has the product form θ_{T }(z) = π_{?}(ϕ_{z})^{∗}θ_{T }(0)π(ϕ_{z}), z ∈ D,where ϕ

_{z }is the involution in Möb mapping z to 0. We obtain a concreterealization of this product formula for a large subclass of homogeneous

cnu contractions from the Cowen-Douglas class. These are the holomor-

phic imprimitivities among the homogeneous contractions.

Typically, contractive homomorphisms induced by an operator (or, even

a commuting tuple of operators) in the Cowen-Douglas class give rise to

curvature inequalities. However, given the curvature inequality, it is clear

that it can only provide information about the second order jets of the

holomorphic curve. Strengthening the curvature inequality to obtain ad-

ditional information about the holomorphic curve remains an intriguing

problem. Some partial answers to this question are in [51].

A complete set of invariants, originally obtained by Cowen and Dou-

glas, have been reﬁned to provide a tractable set of invariants for large

class of Cowen-Douglas operators [55,56]. In these papers, after impos-

ing a mild condition on the Cowen-Douglas bundles, it is shown that the

curvature together with the second fundamental form serves as a com-

plete set of invariants.

In general, determining the moduli space for the isomorphism classes

of sub-modules of a Hilbert module is a difﬁcult problem. Thanks to

Beurling’s theorem, the moduli space is a singleton for the Hardy module

H

^{2}(D) of the unit disc, that is all sub-modules of H^{2}(D) are isomorphic.However, a rigidity phenomenon occurs in H

^{2}(D^{n}), namely, no two sub-modules of H

^{2}(D^{n}) are isomorphic barring a very few exceptions. Thisis typical of the multi-variable situation. The determination of the iso-

morphism classes of sub-modules of a Hilbert module for anlytic Hilbert

modules seems intractable at the moment. Some partial results have

been obtained by using the monoidal transform to resolve the singulari-

ties of the holomorphic curves corresponding to sub-modules of an an-

alytic Hilbert module. More recently, the submodules of analytic Hilbert

modules deﬁned over certain algebraic varieties in bounded symmetric

domains, the so-called Jordan-Kepler varieties V

_{` }of arbitrary rank ` havebeen studied. For ` > 1, the singular set of V

_{` }is not a complete intersec-tion. Hence the usual monoidal transformations do not sufﬁce for the

resolution of the singularities. Instead, we describe a new higher rank

version of the blow-up process, deﬁned in terms of Jordan algebraic de-

terminants, and apply this resolution to obtain the rigidity of the sub-

modules vanishing on the singular set [44, 47, 64]. The accompanying

question of ﬁnding a model for the quotient modules might be thought

of the model theory of Sz. - Nagy and Foias to the more general con-

text of Hilbert modules over function algebras. Finding a complete set of

invariants for the unitary equivalence classes of such canonical models

is an important problem. Both of these questions have been addressed

partially in [22, 37].

Given a pair of positive real numbers α,β and a sesqui-analytic func-

tion K on a bounded domain Ω ⊆ C

^{m}, we investigate the properties of thereal-analytic function

¡¡

¢¢

2

K

^{(α,β)}(z,z) :=K

^{(α+β) }logK (z,z)_{1≤i,j≤m}, z ∈ Ω,∂

∂z ∂z

j

i

taking values in m × m matrices. The kernel K

^{(α,β) }is non-negative def-inite whenever K

^{α }and K^{β }are non-negative deﬁnite. In this case, a re-alization of the Hilbert module determined by the kernel K

^{(α,β) }is ob-tained. Let M

_{i }, i = 1,2, be two Hilbert modules over the polynomial ringC[z

_{1},...,z_{m}]. The tensor product M_{1 }⊗ M_{2 }is clearly a module over thering C[z

_{1},...,z_{2m}]. This module multiplication restricts to C[z_{1},...,z_{m}]via the diagonal map z → (z,z), z ∈ Ω. Now, a natural decomposition

of the tensor product M

_{1 }⊗ M_{2 }very similar to the Clebsch-Gordon de-composition of the tensor product of two irreducible unitary representa-

tions occurs relative to the multiplication restricted to C[z

_{1},...,z_{m}]. Twoof the initial pieces in this decomposition have been identiﬁed in [65].

The ﬁrst of these is simply the restriction of M

_{1 }⊗ M_{2 }to the diagonalset 4 := {(z,z) : z ∈ Ω}, while the second piece in the decomposition is

the module determined by the kernel K

^{(α,β)}. Moreover, if Ω is a boundedsymmetric domain, then K

^{(α,β) }is covariant whenever K is covariant un-der the action of the bi-holomorphic automorphism group of Ω.