Unitary Duals Research Articles

Characters of infinite-dimensional quantum classical groups: BCD cases

We study the character theory of inductive limits of [Formula: see text]-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld–Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of [Formula: see text], [Formula: see text], and their inductive limits.

Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher-rank Lie groups

Let \(\mathbb{G}\) denote a higher-rank ℝ-split simple Lie group of the following type: SL(n, ℝ), SOo(m, m), E6(6), E7(7) and E8(8), where m ≥ 4 and n ≥ 3. We study the cohomological equation for discrete, abelian parabolic actions on \(\mathbb{G}\) via representation theory. Specifically, we characterize the obstructions to solving the cohomological equation and construct smooth solutions with Sobolev estimates. We prove that global estimates of the solution are generally not tame, and our non-tame estimates in the case \(\mathbb{G}\) = SL(n, ℝ) are sharp up to finite loss of regularity. Moreover, we prove that for general \(\mathbb{G}\) the estimates are tame in all but one direction, and as an application, we obtain tame estimates for the common solution of the cocycle equations. We also give a sufficient condition for which the first cohomology with coefficients in smooth vector fields is trivial. In the case that \(\mathbb{G}\) = SL(n, ℝ), we show this condition is also necessary. A new method is developed to prove tame directions involving computations within maximal unipotent subgroups of the unitary duals of SL(2, ℝ) ⋉ ℝ2 and SL(2, ℝ) ⋉ ℝ4. A new technique is also developed to prove non-tameness for solutions of the cohomological equation.

Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds

In this paper we prove new inequalities describing the relationship between the “size” of a function on a compact homogeneous manifold and the “size” of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood [17] on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in Lp(SU(2)) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the Lp-Lq boundedness of Fourier multipliers for 1<p≤2≤q<∞ on compact homogeneous manifolds as well as the Lp-Lq boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.

Amenability and representation theory of pro-Lie groups

We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups. The usual \(C^*\)-algebraic approach to group representation theory positively breaks down for infinite direct products of non-compact locally compact groups, hence the description of their unitary duals in terms of coadjoint orbits is particularly important whenever it is available, being the only description known so far.

Comparison of Unitary Duals of Drinfeld Doubles and Complex Semisimple Lie Groups

We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every $q$-deformation of a higher rank compact Lie group has central property ${\rm (T)}$. We also determine the unitary dual of $SL_q(n,\mathbb{C})$.

Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups

Let \({\mathbb{G}}\) denote a higher-rank \({\mathbb{R}}\)-split simple Lie group of the following type: \({SL(n,\mathbb{R})}\), SOo(m, m), E6(6), E7(7) and E8(8), m ≥ 4. For any unitary representation \({(\pi,\mathcal{H})}\) of \({\mathbb{G}}\) without non-trivial \({\mathbb{G}}\)-invariant vectors, we study smooth solutions of the cohomological equation \({\mathfrak{u}f=g}\), where \({\mathfrak{u}}\) is a vector in the root space of \({\mathfrak{G}}\), the Lie algebra of \({\mathbb{G}}\) and g is a given vector in \({\mathcal{H}}\). We characterize the obstructions to solving the cohomological equation, construct smooth solutions of the cohomological equation and obtain tame Sobolev estimates for f. We also study common solutions to (the infinitesimal version of) the cocycle equation \({\mathfrak{u}h=\mathfrak{v}g}\), where \({\mathfrak{u}}\) and \({\mathfrak{v}}\) are commutative vectors in different root spaces of \({\mathfrak{G}}\) and g and h are given vectors in \({\mathcal{H}}\). We give the sufficient condition under which the cocycle equation has a common solution: (∗) if \({\mathfrak{u}}\) and \({\mathfrak{v}}\) embed in \({\mathfrak{sl}(2,\mathbb{R})\times \mathbb{R}}\), then the common solution exists. In fact, condition (∗) is also necessary when \({\mathbb{G}=SL(n,\mathbb{R})}\). We show counter examples in each \({SL(n,\mathbb{R})}\), n ≥ 3 if condition (∗) is not satisfied. Especially, the cocycle rigidity always fails for \({SL(3,\mathbb{R})}\). As an application, we obtain smooth cocycle rigidity for higher rank parabolic actions over \({\mathbb{G}/\Gamma}\) if the Lie algebra of the acting parabolic subgroup contains a pair \({\mathfrak{u}}\) and \({\mathfrak{v}}\) satisfying property (∗). The main new ingredient in the proof is making use of unitary duals of \({SL(2,\mathbb{R})\ltimes\mathbb{R}^2}\) and \({(SL(2,\mathbb{R})\ltimes\mathbb{R}^2)\ltimes\mathbb{R}^3}\) obtained by Mackey theory.

A descriptive view of unitary group representations

In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if G and H are countable amenable non-type I groups, then the unitary duals of G and H are Borel isomorphic.

Irreducibility criterion for representations induced by essentially unitary ones (case of non-archimedean GL(n,𝒜))

The fundamental representations for the description of the unitary dual of general linear groups over a local field F are the Speh representations, which are denoted by u(d, m). Let A be a finite dimensional central division algebra over F. One can define in a similar way representations u(d, m) for general linear groups over A, and they play a similar fundamental role for the unitary duals of these groups. In the non-archimedean case, for two such representations u(d, m) and u(d', m'), and for real numbers a and b, we obtain a necessary and sufficient condition that the representation parabolically induced by the tensor product of |det|^a u(d, m) and |\det|^b u(d', m') is irreducible. Our motivation for working on this problem comes from our work on unitary duals of classical (symplectic, orthogonal and unitary) groups, related to constructions of complementary series for these groups.

Unitary equivalences for reductive p -adic groups

We establish a transfer of unitarity for a Bernstein component of the category of smooth representations of a reductive $p$-adic group to the associated Hecke algebra, in the framework of the theory of types, whenever the Hecke algebra is an affine Hecke algebra with geometric parameters, in the sense of Lusztig (possibly extended by a group of automorphisms of the root datum). It is known that there is a large class of such examples (detailed in the paper). As a consequence, we establish relations between the unitary duals of different groups, in the spirit of endoscopy.

The unitary dual ofp-adic Sp(2)

We investigate the composition series of the induced admissible representations of the metaplectic group Sp(2) over a p-adic field F. In this way, we determine the nonunitary and unitary duals of Sp(2) modulo cuspidal representations.

On Unitary Representations of Nilpotent Gauge Groups

Groups of smooth maps from spheres to appropriate nilpotent Lie groups exhibit some peculiar properties of the unitary duals of infinite-dimensional groups.

Wigner functions for a class of semi-direct product groups

Following a general method proposed earlier, we construct here Wigner functions defined on coadjoint orbits of a class of semidirect product groups. The groups in question are such that their unitary duals consist purely of representations from the discrete series and each unitary irreducible representation is associated with a coadjoint orbit. The set of all coadjoint orbits (hence UIRs) is finite and their union is dense in the dual of the Lie algebra. The simple structure of the groups and the orbits enables us to compute the various quantities appearing in the definition of the Wigner function explicitly. Possible use of the Wigner functions so constructed, in image analysis and quantum optical measurements, is suggested.

An external approach to unitary representations

The principal ideas of harmonic analysis on a locally compact group G which is not necessarily compact or commutative, were developed in the 1940’s and early 1950’s. In this theory, the role of the classical fundamental harmonics is played by the irreducible unitary representations of G. The set of all equivalence classes of such representations is denoted by Ĝ and is called the dual object of G or the unitary dual of G. Since the 1940’s, an intensive study of the foundations of harmonic analysis on complex and real reductive groups has been in progress (for a definition of reductive groups, the reader may consult the appendix at the end of the second section). The motivation for this development came from mathematical physics, differential equations, differential geometry, number theory, etc. Through the 1960’s, progress in the direction of the Plancherel formula for real reductive groups was great, due mainly to Harish-Chandra’s monumental work, while, at the same time, the unitary duals of only a few groups had been parametrized. With F. Mautner’s work [Ma], a study of harmonic analysis on reductive groups over other locally compact non-discrete fields was started. We shall first describe such fields. In the sequel, a locally compact non-discrete field will be called a local field. If we have a non-discrete absolute value on the field Q of rational numbers, then it is equivalent either to the standard absolute value (and the completion is the field R of real numbers), or to a p-adic absolute value for some prime number p. For r 2 Q× write r = pa/b where α, a and b are integers and neither a nor b are divisible by p. Then the p-adic absolute value of r is jrjp = p−α. A completion of Q with respect to the p-adic absolute value is denoted by Qp. It is called a field of p-adic numbers. Each finite dimensional extension F of Qp has a natural topology of a vector space over Qp. With this topology, F becomes a local field. The topology of F can be also introduced with an absolute value which is denoted by j jF (in the fifth section we shall fix a natural absolute value). The fields of real and complex numbers, together with the finite extensions of p-adic numbers, exhaust all local fields of characteristic zero up to isomorphisms ([We]). Let F be a finite field. Denote by F((X)) the field of formal power series over F. Elements of this field are series of the form f = P∞ n=k anX , an 2 F, for some integer k. Fix q > 1.

On the Unitary Duals of C(X, G)

On the Unitary Duals of <i>C(X, G)</i>