Langlands classification for real Lie groups with reductive Lie algebra

LetG be a (not necessarily connected) real Lie group with reductive Lie algebra. We consider representations ofG which some call admissible but we call them of Harish-Chandra type. We show that any nontempered irreducible Harish-Chandra type representation ofG is infinitesimally equivalent to the Langlands quotient obtained from an essentially unique triple (M, V, ) of Langlands data; while for tempered irreducible Harish-Chandra type representations we prove they are infinitesimally subrepresentations of some induced representations U^V, with imaginary and withV from the quasi-discrete series of a suitableM (perhapsG=M; we define the quasi-discrete series in Definition 4.5 of this paper.We show that irreducible continuous unitary representations of really reductive groups are of Harish-Chandra type. Then the results above yield the canonical decomposition of the unitary spectrum>G for any really reductiveG. In particular, this holds ifG/G ₀ is finite, so the center of the connected semi-simple subgroup with Lie algebra [g, g] may be infinite!Research supported, in part, by the Hungarian National Fund for Scientific Research (grant Nos. 1900 and 2648).     

The prime spectrum of the enveloping algebra of a reductive Lie algebra

Written during the author's stay at MSRI, supported by a Stipendium der Clemens Plassmann Stiftung     

New soliton hierarchies associated with the real Lie algebra

We present some new matrix spectral problems, based on the real special orthogonal Lie algebra , and construct corresponding soliton hierarchies by means of zero curvature equations associated with these spectral problems. With the aid of symbolic computation by Maple, new soliton hierarchies of Kaup–Newell type, Ablowitz–Kaup–Newell–Segur type and Wadati–Konno–Ichikawa type are obtained to illustrate the use of . Copyright © 2016 John Wiley & Sons, Ltd.     

Span of the orthogonal orbit of real matrices

We determine the real linear span of the orthogonal orbit of a real matrix. Except the trivial case when the matrix is a scalar, there are six subspaces which can be such linear spans. The situation is different from the complex case. We also pose some problems involving orthogonal matrices.     

Span of the orthogonal orbit of real matrices

We determine the real linear span of the orthogonal orbit of a real matrix. Except the trivial case when the matrix is a scalar, there are six subspaces which can be such linear spans. The situation is different from the complex case. We also pose some problems involving orthogonal matrices.     

Asymptotic behavior of the products of random matrices with values on a solvable Lie algebra

The index of exponential growth is found for the products of random matrices with values on a solvable Lie algebra. The result is expressed via the eigenvalues of the terms.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 1045–1048, July, 1993.     

On complex matrices with simple spectrum that are unitarily similar to real matrices

Suppose that one should verify whether a given complex n × n matrix can be converted into a real matrix by a unitary similarity transformation. Sufficient conditions for this property to hold were found in an earlier publication of this author. These conditions are relaxed in the following way: as before, the spectrum is required to be simple, but pairs of complex conjugate eigenvalues $ \lambda ,\bar \lambda $ \lambda ,\bar \lambda are now allowed. However, the eigenvectors corresponding to such eigenvalues must not be orthogonal.     

Stirling partitions of the symmetric group and Laplace operators for the orthogonal Lie algebra

Several constructions of Laplace operators for the canonical realization of the orthogonal Lie algebra are discussed. All of them are related with the Capelli-type determinant of a matrix formed by the generators of this Lie algebra. Combinatorial properties of the projection map used in the definition of the Capelli-type determinant are studied. It is proved that the fibers of this projection form a partition of the Bruhat order on into Boolean intervals such that the number of intervals with 2^k elements is the Stirling number of the first kind c(N − 1,k).     

A new Lie algebra along with its induced Lie algebra and associated with applications

A 3 × 3 Lie algebra H is introduced whose induced Lie algebra by decomposition and linear combinations is obtained, which may reduce to the Lie algebra given by AP Fordy and J Gibbons. By employing the induced Lie algebra and the zero curvature equation, a kind of enlarged Boussinesq soliton hierarchy is produced. Again making use of a subalgebra of the induced Lie algebra leads to the well-known KdV hierarchy whose expanding integrable system is also worked out. As an applied example of the Lie algebra H, we obtain a new integrable coupling of the well-known AKNS hierarchy.