In this paper, we study G-g.o. metrics on compact homogeneous spaces G/H with an intermediate subgroup K such that H C K C G. In the beginning, we prove that the restricted metrics of g on K/H and G/K are both g.o. metrics under certain conditions when g is a G-g.o. metric on G/H. Then we develop several methods to determine G-g.o. metrics on G/H by the representations of K/H and G/K. As an application, we study g.o. metrics on a class of homogeneous spaces and find that SO(11)/(Spin(7) & times; SO(2)) admits non-naturally reductive g.o. metrics.
An alternative construction of the affine root system of an isoparametric submanifold in Hilbert space to that in [GH12] is provided, without invoking Dadok's theorem.
This paper is partially an exposition of the method of the proof of the continuous part of the general distributional Whittaker Plancherel Theorem in the special case of the spherical spectrum. It is also an explanation of how this result solves the quantum non-periodic Toda lattices. Combining the ideas involved in both of these results the paper also gives a new reduction of the calculation of spherical Whittaker functions to split groups over R. It concludes with a new proof of an explicit functional equation which is used in the surjectivity result in the L2 Plancherel Theorem and an explicit isomorphism theorem for the Whittaker Schwartz Space.
Dirac cohomology of a discrete series representation is an analogue of the highest weight vectors of a highest weight representation. We employ Dirac cohomology for the induction and character lifting of discrete series as well as determining the lifting of L-packets of discrete series.
We propose a novel framework that unifies two fundamental concepts in mathematics: matrix-valued spherical functions and scalar modular forms. By extending the classical theory of modular forms to the matrix-valued setting, we introduce and study modular spherical functions. These are smooth functions defined on a connected unimodular Lie group G, a compact subgroup K and a discrete subgroup Gamma, with values in endomorphism spaces of finite-dimensional vector spaces. Modular spherical functions are characterized as eigenfunctions of the algebra D(G)K, consisting of all G-left invariant and K-right invariant differential operators on G, while satisfying specific transformation properties under the actions of G, K and Gamma. Focusing on the paradigmatic case G = SL(2, R), K = SO(2) and Gamma = SL(2, Z), we partially extend the characterization of modular spherical functions. We describe these functions in terms of polynomial eigenfunctions in two real variables for the n-Laplacian in the upper half-plane. The results include explicit bases for associated function spaces, recurrence relations for orthogonal polynomials, and analytic continuation. This work advances the study of modular spherical functions, opening new avenues in the representation theory of reductive Lie groups, orthogonal polynomials, and modular forms.
A Riemannian manifold is called weakly symmetric if any two points in M can be interchanged by an isometry. We give a complete classification of simply connected non-singular weakly symmetric nilmanifolds. Besides previously known examples, there are new families with 3-dimensional center, and a one-parameter family of dimensions 14.
Let G be a complex reductive Lie group acting on a compact K & auml;hler manifold X. Assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there exists an associated open dense subset of X, that is invariant under the action of a parabolic subgroup Q of G. We prove a Q-equivariant product decomposition for the Q-action on this subset and discuss some applications of this result. Additionaly, we establish a similar statement for real reductive subgroups of G and the restricted momentum map.
Time derivatives of pullbacks and push forwards along smooth curves of diffeomorphisms of sections of natural vector bundles are computed in terms of Lie derivatives along adapted non-autonomous vector fields by extending a key lemma of Mauhart and Michor [Arch. Math., Brno 28 (1992)]. There is also the analogous result about the first non-vanishing derivative of higher order.
The main objective of this paper is twofold. One is to classify and construct SL(3, R)-intertwining differential operators between vector bundles over the real projective space RP2. It turns out that two kinds of operators appear. We call them Cartan operators and PRV operators. The second objective is then to study the representations realized on the kernel of those operators both in the smooth and holomorphic setting. A key machinery is the BGG resolution. In particular, by exploiting some results of Davidson-Enright-Stanke and Enright-Joseph, the irreducible unitary highest weight modules of SU(1, 2) at the (first) reduction points are classified by the image of Cartan operators and kernel of PRV operators.
By a grassmannian we understand a usual complex grassmannian or possibly an orthogonal or symplectic grassmannian. We classify, with few exceptions, linear embeddings of grassmannians into larger grassmannians, where the linearity requirement is the condition that the embedding induces an isomorphism on Picard groups. This classification implies that most linear embeddings of grassmannians are equivariant. A linear ind-grassmannian is the direct limit of a chain of linear embeddings of grassmannians. We conclude the paper by classifying linear embeddings of linear ind-grassmannians.
We prove that a compact, intrinsically symmetric submanifold of an Euclidean space is extrinsically symmetric if and only if its maximal tori are Clifford tori in the ambient space. Moreover, we show that this result can be used to give a geometric proof of a result of Harish-Chandra on strongly orthogonal roots in semisimple Lie algebras.
We study open orbits of symmetric subgroups of a simple connected Lie group G on a causal flag manifold. First we show that a flag manifold M of G carries an invariant causal structure if and only if G is hermitian of tube type and M is the conformal completion of the corresponding simple euclidean Jordan algebra, resp., the Shilov boundary of the associated symmetric tube domain. We then study open orbits in M under symmetric subgroups, also called causal Makarevic spaces, from the perspective of applications in Algebraic Quantum Field Theory (AQFT). A key motivation is the geometry of corresponding modular flows. The open orbits are reductive causal symmetric spaces, which arise in two flavors: compactly causal and non-compactly causal ones. In the non-compactly causal case we determine the corresponding Euler elements and their positivity regions. For compactly causal spaces, modular flows do not always exist and we determine when this is the case. Then the positivity regions of the modular flows are not globally hyperbolic, but these spaces contain other interesting globally hyperbolic subsets that can be described in terms of the conformally flat Jordan coordinates via Cayley charts. We discuss the Lorentzian case, involving de Sitter and anti-de Sitter space in some detail.
We present a proof of a generalization of the theorem of H. Matsumoto on Coxeter groups. Our generalized version is applicable to "graphs admitting geometric realization". The original version of the theorem for Coxeter groups is a special case when applied to the Cayley graph and the geometric representation of a Coxeter group. Our version of Matsumoto theorem is also applicable to skeleta, graphs that were defined in the recent paper by the authors on root Lie superalgebras.
For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the Discrete Series found by Harish-Chandra. In this paper, we continue our study of the branching laws for Discrete Series when restricted to a subgroup H of the same type by use of integral and differential operators in combination with our previous duality principle. Many results are presented in generality, others are shown in detail for Holomorphic Discrete Series.
Let Gamma be a discrete subgroup of a unimodular locally compact group G. M. Caspers et al. [Local and multilinear noncommutative de Leeuw theorems, Math. Ann. 388 (2024) 4251-4305] showed that the L-p-norm of a Fourier multiplier m: G -> C on Gamma can be bounded locally by its L-p-norm on G, modulo a constant c(A) which depends on the support A of m|(Gamma). In the context where G is a connected Lie group with Lie algebra g, we develop tools to find explicit bounds on c(A). We show that the problem reduces to: (1) The adjoint representation of the semisimple quotient s = g/r of g by the radical r subset of g (which was handled in the paper of M. Caspers et al. cited above). (2) The action of s on a set of real irreducible representations that arise from quotients of the commutator series of r. In particular, we show that c(G) = 1 for unimodular connected solvable Lie groups.
For negatively curved symmetric spaces it is known from S. Hansen, J. Hilgert, and A. Parthasarathy [Resonances and scattering poles in symmetric spaces of rank one, Int. Math. Res. Notices 20 (2019) 6362-6389] that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-co compact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.
We investigate local properties of the Schr & ouml;dinger algebra in (n + 1)-dimensional space-time of Schr & ouml;dinger Lie groups. Specifically, for any positive integer n, it initiates the study of 2-local derivations of this Lie algebra, denoted by S-n. The main result establishes that every 2-local derivation on Sn is actually a derivation.
The flag type of a semigroup S of a noncompact semisimple Lie group is an algebraic tool related to the geometry of the invariant control set determined by S on the flag manifolds of G. In the present paper, we show that it is possible to recover the flag type by studying the existence of lower bounds for co cycles on the maximal flag manifold.
Let (K, N) be a nilpotent Gelfand pair and let G := K (Sic) N be the semidirect product associated with (K, N). Let pi E G(<^>) be a generic representation of G and let tau is an element of K-<^>. The Kirillov-Lipsman's orbit method suggests that the multiplicity m(pi)(tau) of an irreducible K-module tau occurring in the restriction of pi|K can be linked to (the number of K-orbits) the Corwin-Greenleaf multiplicity function (C.G.M.F for short). Under some assumptions on the pair (K, N), in this work we focus on the connection between the geometric number C.G.M.F and the multiplicity (m(pi)(.)). In the geometric counterpart we give a necessary and sufficient conditions associated with the C.G.M.F. Moreover, we prove that this function is bounded for a special class of subgroups of G.
We study AV-modules, as in the work of Billig and collaborators, from a more geometric perspective. We show that if the underlying sheaf is a vector bundle, then the covariant derivative by a vector field depends almost O-linearly on the vector field. More precisely, we will show that a certain Lie map is a differential operator. This strengthens a theorem of the author and Rocha, in the sense that the bound on the order of a certain differential operator is improved upon quadratically.
