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.
