Szegő maps and highest weight representations

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E7(7), they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE 7(7) in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE 7(7) leads to reducible unitary representations.

We describe the embedding structure of the Verma modules of the W3 algebra using the free field representation (FFR) in the lattice approach. This structure is expressed by a set of intertwining diagrams. In particular, we show how these diagrams can be used to achieve the Bernstein-Gel’fand-Gel’fand (BGG) resolution of the irreducible highest weight representation (IHWR) in terms of the Verma modules. As an application of prime importance, we show how the character of the W3 IHWR can be readily derived using the Rocha-Caridi procedure.

Our object of study is the ring of invariant differential operators on a hermitian symmetric space G/K of classical and noncompact type. Here, as usual, G is a connected noncompact semisimple Lie group with finite center and K is a maximal compact subgroup of G. It is well-known that the ring, denoted by ?(G/K), is isomorphic to a polynomial ring of 1 variables, 1 being the rank of G/K. This is true even in the nonhermitian case. If 1 = 1, it is generated by the Laplace-Beltrami operator z/ which is essentially self-adjoint; moreover -Iz2 is nonnegative. In the general case, an easily posable problem is to find an explicitly defined set of generators. We can go one step further by focusing our attention on the nonnegativity, which necessarily limits the range of eigenvalues under a suitable integrability condition on functions. It is thus natural to ask whether there exist some canonically defined operators which generate -9(G/K) and have the property of nonnegativity. The main purpose of the present paper is to give an affirmative answer to this question. In fact, our answer applies to a somewhat more general type of rings of differential operators which includes 9(G/K) as a special case. To be explicit, we take an irreducible representation p: K -* GL(V) with a complex vector space V of finite dimension, and consider the set C'(p) of all V-valued Cx functions f on G such that f(xk -1) = p(k)f(x) for every k e K. We then denote by ?(p) the ring of left-invariant difrrential operators on G which map C'(p) into itself. Now the complexification g of the Lie algebra of G has abelian subalgebras p+ and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on G/K at the origin. For any complex vector space W let Sr(W) denote the ring of all complex-valued homogeneous polynomial functions on W of degree r, and let S(W) = Er=0Sr(W). Through the adjoint representation of G on g, K acts naturally on Sr( ?+). It is a known fact, due to Hua and Schmid, that each irreducible constituent of this representation of K has multiplicity one. Now our principal

We determine precisely for which irreducible unitary highest weight representation of the group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the countable direct limit of the finite-dimensional unitary groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the corresponding 1-cohomology space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> does not vanish. This occurs in particular if a highest weight, viewed as an integer-valued function on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℕ</mml:mi> </mml:math> , is finitely supported. In a second step, we extend the finitely supported highest weight representations to norm-continuous unitary representations of the Banach-completions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of the direct limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> th Schatten norm for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , the corresponding 1-cohomology spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> do not vanish either, except in three cases. We conclude that these groups do not have Kazhdan’s Property (T). On the other hand, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , the first cohomology spaces all vanish because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has property (FH) as a bounded topological group.

We prove that the restriction to (2, R) of the underlying HarishChandra module of a Discrete Series representation of a noncompact simple Lie group has no sl(2)-irreducible subrepresentations unless it is holomorphic. We also give a characterization of the holomorphic Discrete Series. Let G be a connected real simple Lie group. Fix a maximal compact subgroup K of G. Let g (t) denote the complex Lie algebra of G (respectively, K). Hence we have the Cartan decomposition g = t + p. Thus, p is Kinvariant under the restriction of the adjoint representation of G to K and p . Let T C K be a maximal torus. Henceforth we assume that T is a Cartan subgroup of G. Let t be the complex Lie algebra of T. Under these hypotheses Harish-Chandra in [HC2] has given a classification of the square-integrable representations of G in terms of the regular characters of T. Let (D(g, t) denote the root system of the pair (g, t) . A root of the pair (g, t) is called compact (noncompact) if its root space is contained in t (respectively, p). We now fix a noncompact root y . Then the root vectors of y span a copy by of sl(2, C) . Moreover by is invariant under the conjugation of g with respect to the Lie algebra of G. Let s,, be the real form of by associated to the conjugation of g with respect the Lie algebra of G restricted to by . Let Hy be the connected subgroup of G corresponding to sy I Let (7r, V) be the Harish-Chandra module underlying a square-integrable, irreducible representation of G. We then have Proposition 1. If the restriction of (7r, V) to by contains an irreducible bysubmodule, then G/K is a Hermitian symmetric space and (7r, V) corresponds to a holomorphic square-integrable representation. A corollary to Proposition 1 is Corollary. If G is as above, G/K is not Hermitian, and H is a semisimple subgroup of G such that its symmetric space is Hermitian, then (7r, V) restricted to [ has no [-holomorphic irreducible subrepresentations. Received by the editors April 29, 1991 and, in revised form, July 11, 1991. 1991 Mathematics Subject Classification. Primary 22E47.

ABSTRACTThe unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.

Let (X, ω) be an integral symplectic manifold and let (L, ∇) be a quantum line bundle, with connection, over X having ω as curvature. With this data one can define an induced symplectic manifold \( (\tilde X,\omega _{\tilde X} ) \), where dim \( \tilde X = 2 + \dim X \). It is then shown that prequantization on X becomes classical Poisson bracket on \( \tilde X \). We consider the possibility that if X is the coadjoint orbit of a Lie group K, then \( \tilde X \) is the coadjoint orbit of some larger Lie group G. We show that this is the case if G is a noncompact simple Lie group with a finite center and K is the maximal compact subgroup of G. The coadjoint orbit X arises (Borel-Weil) from the action of K on \( \mathfrak{p} \), where \( \mathfrak{g} = \mathfrak{k} + \mathfrak{p} \) is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism \( (\tilde X,\omega _{\tilde X} ) \cong (Z,\omega _Z ) \), where Z is a nonzero minimal "nilpotent" coadjoint orbit of G. This is applied to show that the split forms of the five exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.

On unitary representations of the exceptional non-linear N=7 and N=8 superconformal algebras in terms of free fields

This third chapter presents a brief and somewhat sketchy introduction to the theory of unitary representations of reductive and semisimple Lie groups G. The basic fact for an irreducible unitary representation π of G on a Hilbert space ℋ, is that every irreducible representation к of a maximal compact subgroup K ⊂ G has multiplicity m(к, π| к) ≤ dim к. This yields up the infinitesimal character χπ: L(g)→ ℂ and the distribution character θπ: C c ∞ (G) → ℂ, and consequently the differential equations $$z({\theta _\pi }) = \,{\chi _\pi }(z){\theta _\pi }$$ for $$z \in L{\text{(g)}}$$ which are the starting point for serious harmonic analysis on G.

A new concept of “generalized enveloping algebra” is introduced by means of the generalized Heisenberg commutation relations of non-Abelian quantum kinematics. This concept is examined within the quantum-kinematic formalism of some noncompact Lie groups of a special kind. The well known Gel'fand theorem (which relates the center of the traditional enveloping algebra with the adjoint representation) is then extended to the generalized enveloping algebra of the group. In this way, the isomorphism of the “generalized left-center” and the “traditional right-center” of the corresponding enveloping algebras is proved within the left regular representation of noncompact Lie groups of the chosen kind. As an interesting application of generalized enveloping algebras, this paper contains a brief discussion of quantum-kinematic (boson) ladder operators for non-Abelian noncompact finite Lie groups and of their corresponding coherent states.

Affine Toda solitons and vertex operators

Chapter 3 - Lie Groups

PROOF. Replacing G by a finite covering we may and will assume that G is a direct product of simple Lie groups, none of which is compact. The group In of isometries of the Euclidean n-space is known to be the semidirect product of the orthogonal group and R , RI being the normal subgroup. Therefore the projection of In onto the orthogonal group is a differentiable group morphism. Let f be a differentiable morphism from G into In. Then f followed by the projection of In onto the orthogonal group is a differentiable morphism. If this composition were nontrivial it would be nontrivial on some simple factor of G. Therefore, there would be a noncompact simple Lie group contained in the orthogonal group. This is impossible because a semisimple Lie subgroup of a compact Lie group is closed, hence compact (cf. [2, p. 128D. Thus, the image of f is contained in RW. This implies that G has a nontrivial abelian factor, unless f is the trivial morphism.

The oscillator method for the construction of unitary highest (or lowest) weight representations of noncompact groups and supergroups is generalized. Within this generalization, the method yields unitary highest weight representations of all simple supergroups whose even subgroups are in the form of a direct product of a compact group with a simple noncompact group. The method is illustrated by studying in detail the unitary highest weight representations of the supergroup OSp (2n+1/2m,R). The generalized supercoherent states associated with these unitary representations are also defined.