Semisimple Lie Groups and Related Homogeneous Spaces

Relative BGG sequences; II. BGG machinery and invariant operators

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extensions (semi-direct product)of the nilpotent groups which allows immediately to produce the special representation of the group and then to apply the quasi-Poisson construction from the previous papers by authors in order to construct the representation of current group. The parabolic subgroup of the semisimple Lie group of rank one has such semidirect product, and special representation of it can be extended onto whole semisimple group. As result we obtain needed new model of the irreducible representation of semi-simple current groups.

Introduction Representations of $\mathbf {SU}(2)$ The general theory of linear representations of compact groups Lie theory of representations of compact connected Lie groups Induced representations of compact groups Spherical functions on compact groups Examples spherical harmonics The general theory of spherical functions Fourier and Plancherel transforms Extension of the Plancherel transform The subtleties of harmonic analysis Differential properties of spherical functions on Lie groups Spherical functions on semisimple Lie groups More on $\mathbf {SL}(2, \mathbf R)$ Automorphic functions Groups of isometries and Bessel functions Other special functions References.

The Weyl group, WG, of each exceptional simple Lie group G, is described in detail. Its structure is defined in terms of its coset decomposition with respect to the Weyl group, WH, of a classical semi-simple Lie group, H, embedded naturally in G. The concepts of G-dominance and G-equivalence are defined and used to determine, from the character formula of Weyl, the branching rule associated with the restriction of group elements from G to H. The Weyl group WG is used further to impose constraints on both the branching multiplicities for G to H and the weight multiplicities of G. These constraints are used to evaluate the weight multiplicities of F4, E6, E7 and E8 together with the branching multiplicities for E8 to SO(16).

Over the \((1, n)\)-dimensional real superspace, we consider the Lie superalgebra \(\mathcal {K}(n)\) of contact vector fields on \({\mathbb {R}}^{1|n}\). We classify \(\mathfrak {osp}(n|2)\)-invariant linear differential operators acting on the superspaces of weighted densities. This result allows us to compute the first differential cohomology of \(\mathcal {K}(n)\) with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra \(\mathfrak {osp}(n|2)\). We explicitly give 1-cocycles spanning these cohomology spaces.

The geometric realization of the irreducible square integrable representations for semisimple Lie groups (cf. [3], [6] ) and also for nilpotent Lie groups [5] suggests that, as a general phenomenon, such representations should appear as L -kernels of invariant elliptic operators. One basic problem in this respect is to decide when such a kernel is nonzero. In the compact case the basic tool for this, used in the Borel-Weil-Bott approach, is the Hirzebruch-Riemann-Roch theorem. In the noncompact case one needs an analogue of the index theorem of Atiyah-Singer [2] for noncompact manifolds. When G possesses a discrete cocompact subgroup, the L-index theorem for covering spaces of [1] and [7] provides the required analogue. Our purpose here is to give a general index theorem for homogeneous spaces of arbitrary connected unimodular Lie groups, essentially based on the index theorem for foliations [4]. So let G be a connected unimodular Lie group, and let H be a closed subgroup of G which contains the center Z of G and such that H/Z is compact. Let x be a character of Z, and let E, F be finite-dimensional unitary representations of H whose restrictions to Z are given by xDenote by E, F the corresponding (invariant) induced bundles on the homogeneous space M = G/H, and let D be an invariant elliptic differential operator from E to F. The representation of G in the kernel of D in L(M, E) is square integrable modulo the center of G (see [4]), though not necessarily irreducible. Its formal degree deg(Ker D) (as defined in [4]) is always finite, so that the analytical index of D can be defined as

We introduce a Lohe group which is a new class of matrix Lie groups and present a continuous dynamical system for the synchronization of group elements in a Lohe group. The Lohe group includes classical Lie groups such as the orthogonal, unitary, and symplectic groups, and since Lohe groups need not be compact, global existence of ODEs may fail. The proposed dynamical system generalizes the Lohe model (Lohe in J Phys A 43:465301, 2010; Lohe in J Phys A 42:395101–395126, 2009) itself a nonabelian generalization of the Kuramoto model, and alongside we also generalize the analytical framework (Ha and Ryoo in J Stat Phys 163:411–439, 2016) of emergent and unique phase-locked states. For the construction of the phase-locked states, we introduce Lyapunov functions measuring the ensemble diameter and the dissimilarity between two Lohe flows, and derive Gronwall-type differential inequalities for them. The global existence of solutions then become a consequence of the boundedness of these Lyapunov functions. Our sufficient framework for the emergent dynamics is formulated in terms of coupling strength and initial states, and it leads to the global existence of solutions and the formation and uniqueness of a phase-locked asymptotic state. As a concrete example, we demonstrate how our theory can show emergent phenomenon on the Heisenberg group, where all initial configurations tend to a unique phase-locked state exponentially fast.

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

Let \(G\) be a connected and simply connected Lie group with Lie algebra \(\mathfrak g \). We say that a subset \(X\) in the set \(\mathfrak g ^\star / G\) of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in \(X\). In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup \(G^+\) and a quadratic map \(\varphi \) sending each generic orbit in \(\mathfrak g ^\star \) to a \(G^+\)-orbit in \(\mathfrak{g ^+}^\star \), in such a manner that the set \(\varphi (\mathfrak g ^\star _{gen}){/ G^+}\) is convex hull separable. We then call \(G^+\) a weak quadratic overgroup for \(G\). Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.

Weights for symmetric and general linear groups

Let G = exp ⁡ ( g ) G = \operatorname {exp}(\mathfrak {g}) be a connected and simply connected real nilpotent Lie group with Lie algebra g \mathfrak g , H = exp ⁡ ( h ) H = \operatorname {exp}(\mathfrak {h}) an analytic subgroup of G G with Lie algebra h \mathfrak h , χ \chi a unitary character of H H and τ = ind H G χ \tau = \text {ind}_H^G \chi the monomial representation of G G induced by χ \chi . Let D τ ( G / H ) D_{\tau }(G/H) be the algebra of the G G -invariant differential operators on the fiber bundle over the base space G / H G/H associated to the data ( H , χ ) (H,\chi ) . We prove the polynomial conjecture due to Corwin-Greenleaf stating that if τ \tau is of finite multiplicities, the algebra D τ ( G / H ) D_{\tau }(G/H) is isomorphic to the algebra C [ Γ τ ] H {{\mathbb C}[{\Gamma }_{\tau }]}^H of the H H -invariant polynomial functions on the affine subspace Γ τ = { ℓ ∈ g ∗ ; ℓ | h = − − 1 d χ } {\Gamma }_{\tau } = \{\ell \in {\mathfrak g}^*; {\ell }|_{\mathfrak h} = -\sqrt {-1}d\chi \} of g ∗ {\mathfrak g}^* . In this case, we show that any non-zero element of D τ ( G / H ) D_{\tau }(G/H) admits a fundamental tempered solution.

An algorithm for constructing canonical forms for any tensor representation of the classical compact Lie groups is given. This method is used to find a complete list of the symmetry breaking patterns produced by Higgs fields in the third-rank antisymmetric representations of U(n), SU(n) and SO(n) for n<or=7. A simple canonical form is also given for kth-rank symmetric tensor representations.

Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner–Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions.

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n, n). Our choice of these algebras is motivated by the fact that for n = 2 this is the conformal algebra of 4-dimensional Minkowski space-time. Furthermore for general n these algebras belong to a narrow class of algebras, which we call “conformal Lie algebras”, which have very similar properties to the conformal algebras of n 2-dimensional Minkowski space-time. We give the main multiplets of indecomposable elementary representations for n = 2, 3, 4, including the necessary data for all relevant invariant differential operators.

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.