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We introduce a new form of the Segal–Bargmann transform for a connected Lie group K of compact type. We show that the heat kernel ( ρ t ( x ) ) t > 0 , x ∈ K has a space-time analytic continuation to a holomorphic function ( ρ C ( τ , z ) ) Re τ > 0 , z ∈ K C , where K C is the complexification of K. The new transform is defined by the integral ( B τ f ) ( z ) = ∫ K ρ C ( τ , z k − 1 ) f ( k ) d k , z ∈ K C . If s > 0 and τ ∈ D ( s , s ) (the disk of radius s centered at s), this integral defines a holomorphic function on K C for each f ∈ L 2 ( K , ρ s ) . We construct a heat kernel density μ s , τ on K C such that, for all s , τ as above, B s , τ : = B τ | L 2 ( K , ρ s ) is an isometric isomorphism from L 2 ( K , ρ s ) onto the space of holomorphic functions in L 2 ( K C , μ s , τ ) . When τ = t = s , the transform B t , t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When τ = t ∈ ( 0 , 2 s ) , the transform B s , t coincides with the one introduced by the first two authors.

We consider the (projective) representations of the group of holomorphic automorphisms of a symmetric tube domain $V\oplus i\Omega$ that are obtained by analytic continuation of the holomorphic discrete series. For a representation corresponding to a discrete point in the Wallach set, we find the decomposition under restriction to the identity component of $GL(\Omega)$. Using Riesz distributions, an explicit intertwining operator is constructed as an analytic continuation of an integral operator. The density for the Plancherel measure involves quotients of $\Gamma$-functions and the $c$-function for a symmetric cone of smaller rank., Comment: 22 pages

We prove a theorem of Dixmier and Malliavin type for analytic vectors of bounded representations of ( R , + ) : Let ( π , E ) be a bounded representation of ( R , + ) in a Banach space E and let A be the convolution algebra of analytic vectors for the left regular representation of R on L 1 ( R ) , then A ∗ A = A and π ( A ) E = E ω .

We consider the Segal-Bargmann transform for a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities., Comment: Final version. To appear in Journal of Functional Analysis. Minor revisions

We consider the generalized Segal–Bargmann transform, defined in terms of the heat operator, for a noncompact symmetric space of the complex type. For radial functions, we show that the Segal–Bargmann transform is a unitary map onto a certain L2 space of meromorphic functions. For general functions, we give an inversion formula for the Segal–Bargmann transform, involving integration against an “unwrapped” version of the heat kernel for the dual compact symmetric space. Both results involve delicate cancellations of singularities.

Let Hν be the weighted Bergman space on a bounded symmetric domain D=G/K. It has analytic continuation in the weight ν and for ν in the so-called Wallach set Hν still forms unitary irreducible (projective) representations of G. We give the irreducible decomposition of the tensor product Hν1⊗Hν2 of the representations for any two unitary weights ν and we find the highest weight vectors of the irreducible components. We find also certain bilinear differential intertwining operators realizing the decomposition, and they generalize the classical transvectants in invariant theory of SL(2,C). As applications, we find a generalization of the Bol's lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product Hν1⊗Hν2 modulo the subspace of functions vanishing of certain degree on the diagonal.

Let D=G/K be an irreducible bounded symmetric domain of dimension d and let Hν(D) be the analytic continuation of the weighted Bergman spaces of holomorphic functions on D. We consider the d-tuple M=(M1,…,Md) of multiplication operators by coordinate functions and consider its spectral properties. We find those parameters ν for which the tuple M is subnormal and we answer some open questions of Bagchi and Misra. In particular, we prove that when D=Bd is the unit ball in Cd, then Bd is a k-spectral set of M if and only if Hν(Bd) is the Hardy space or a weighted Bergman space.

Systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by Walsh and Szego, and in full generality by Bergman. In principle, these are very interesting objects, allowing application to analytic continuation that is not restricted (as Weierstrassian continuation via power series) either by circular geometry or considerations of locality. However, few explicit examples are known, and in general one does not know even gross qualitative features of such systems. The main contribution of the present paper is to prove qualitative results in a quite general situation. It is by now very well known that the phenomenon of “double orthogonality” is not restricted to Bergman spaces of analytic functions, nor even indeed has it any intrinsic relation to analyticity; its essence is an eigenvalue problem arising whenever one considers the operator of restriction on a Hilbert space of functions on some set, to a subset thereof, provided this restriction is injective and compact. However, in this paper only Hilbert spaces of analytic functions are considered, especially Bergman spaces. In the case of the Hardy spaces Fisher and Micchelli discovered remarkable qualitative features of doubly orthogonal systems, and we have shown how, based on the classical potential-theoretic notion of balayage, and its modern generalizations, one can deduce analogous results in the Bergman space set-up, but with restrictions imposed on the geometry of the considered domains and measures; these were not needed in the Fisher–Micchelli analysis, but are necessary here as shown by examples. From a more constructive point of view we study the Bergman restriction operator between the unit disk and a compactly contained quadrature domain and show that the representing kernel of this operator is rational and it is expressible (as an inversion followed by a logarithmic derivative) in terms of the polynomial equation of the boundary of the inner domain.

LetKbe a connected Lie group of compact type and letW(K) denote the set of continuous paths inK, starting at the identity and with time-interval [0, 1]. ThenW(K) forms an infinite-dimensional group under the operation of pointwise multiplication. Letρdenote the Wiener measure onW(K). We construct an analog of the Segal–Bargmann transform forW(K). LetKCbe the complexification ofK,W(KC) the set of continuous paths inKCstarting at the identity, andμthe Wiener measure onW(KC). Our transform is a unitary map ofL2(W(K),ρ) onto the “holomorphic” subspace ofL2(W(KC),μ). By analogy with the classical transform, our transform is given by convolution with the Wiener measure, followed by analytic continuation. We prove that the transform forW(K) is nicely related by means of the Itô map to the classical Segal–Bargmann transform for the path-space in the Lie algebra ofK.

A holomorphic family of differential operators of infinite order is constructed that transforms conical vectors for principal series representations of quasi-split, linear, semi-simple Lie groups into Whittaker vectors. Using this transform, it is shown that algebraic Whittaker vectors (as studied by Kostant) extend to ultradistributions of Gevrey type on principal series representations. For each element of the small Weyl group, a meromorphic family of Whittaker vectors is constructed from this transform and the Kunze-Stein intertwining integrals. An explict formula is derived for the smooth Whittaker vector (discovered by Jacquet), in terms of these families of ultradistribution Whittaker vectors. In particular, this gives new proofs of Jacquet's analytic continuation of the smooth Whittaker vector and its functional equation (Jacquet and Schiffman). Applications of the transform are also given to the theory of Verma modules.

Let G be a connected, simply-connected complex nilpotent Lie group, and Gr ⊂( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of GR to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of GR, we study the convex cone of entire functions on G whose restrictions to GR are positive-definite, and determine the minimal order of growth at infinity of such functions.