On the relations of general homogeneous spaces and the two-wavelet localization operators

In this note, the two-wavelet localization operator for square integrable representation of a general homogeneous space is defined. Then among other things, the boundedness properties of this operator is investigated. In particular, it is shown that it is in the Schatten p-class.

The distinction problems for Sp4 and SO3,3

This paper studies the Prasad conjecture for the special orthogonal group \mathrm{SO}_{3,3}. Then we use the local theta correspondence between \mathrm{Sp}_{4} and \mathrm{O}(V) to study the \mathrm{Sp}_{4}-distinction problems over a quadratic field extension E/F and \dim V=4 or 6. Thus we can verify the Prasad conjecture for a square-integrable representation of \mathrm{Sp}_{4}(E).

Duflo-Moore Operator for The Square-Integrable Representation of 2-Dimensional Affine Lie Group

In this paper, we study the quasi-regular and the irreducible unitary representation of affine Lie group of dimension two. First, we prove a sharpening of Fuhr’s work of Fourier transform of quasi-regular representation of . The second, in such the representation of affine Lie group is square-integrable then we compute its Duflo-Moore operator instead of using Fourier transform as in F hr’s work.

Associated symmetric pair and multiplicities of admissible restriction of discrete series

Let [Formula: see text] be a symmetric pair for a real semisimple Lie group [Formula: see text] and [Formula: see text] its associated pair. For each irreducible square integrable representation [Formula: see text] of [Formula: see text] so that its restriction to [Formula: see text] is admissible, we find an irreducible square integrable representation [Formula: see text] of [Formula: see text] which allows us to compute the Harish-Chandra parameter of each irreducible [Formula: see text]-subrepresentation of [Formula: see text] as well as its multiplicity. The computation is based on the spectral analysis of the restriction of [Formula: see text] to a maximal compact subgroup of [Formula: see text]

Square-integrable representations of non-unimodular groups

In the last three years a number of people have investigated the orthogonality relations for square integrable representations of non-unimodular groups, extending the known results for the unimodular case. The results are stated in the language of left (or generalized) Hilbert algebras. This paper is devoted to proving the orthogonality relations without recourse to left Hilbert algebra techniques. Our main technical tool is to realise the square integrable representation in question in a reproducing kernel Hilbert space.