IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q I: THE DISCRETE SERIES

2001 ◽  
Vol 12 (01) ◽  
pp. 1-36 ◽  
Author(s):  
RAJ WILSON ◽  
ELIZABETH TANNER
Keyword(s):  
Differential Operators ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Holomorphic Functions ◽  
Irreducible Unitary Representation ◽  
Explicit Construction ◽  
Unitary Representations ◽  
Infinitesimal Generators ◽  
Algebraic Properties ◽  
Irreducible Unitary Representations

A class of irreducible unitary representations in the discrete series of SUp,q is explicitly determined in a space of holomorphic functions of three complex matrices. The discrete series, which is the set of all square integrable representations, corresponds to a compact subgroup of SUp,q. The relevant algebraic properties of the group SUp,q are discussed in detail. For a degenerate irreducible unitary representation an explicit construction of the infinitesimal generators of the Lie algebra [Formula: see text] in terms of differential operators is given.

MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

2008 ◽  
Vol 19 (10) ◽  
pp. 1187-1201 ◽  
Author(s):  
MASAYASU MORIWAKI
Keyword(s):  
Unitary Representation ◽  
Orthogonal Group ◽  
Irreducible Unitary Representation ◽  
Unitary Representations ◽  
Minimal Representation ◽  
Infinite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Multiplicity Free ◽  
Kirillov Dimension

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q II: THE CONTINUOUS SERIES

2001 ◽  
Vol 12 (01) ◽  
pp. 37-47 ◽  
Author(s):  
RAJ WILSON ◽  
ELIZABETH TANNER
Keyword(s):  
Kronecker Product ◽  
Discrete Series ◽  
Unitary Representations ◽  
Continuous Series ◽  
Maximal Dimension ◽  
Cartan Subgroup ◽  
Integrable Functions ◽  
Irreducible Unitary Representations ◽  
Square Integrable Functions ◽  
Vector Part

A class of irreducible unitary representations belonging to the continuous series of SUp,q is explicitly determined in a space composed of the Kronecker product of three spaces of square integrable functions. The continuous series corresponds to a Cartan subgroup whose vector part has maximal dimension. These representations are distinguished by a parameter r = 1, 2, …, p for p ≤ q in SUp,q. For r = 0, one obtains the representations in the discrete series as in [5], and all representations in the continuous series, for r ≠ 0, are obtained explicitly.

scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

scholarly journals K-invariant cusp forms for reductive symmetric spaces of split rank one

2019 ◽  
Vol 31 (2) ◽  
pp. 341-349
Author(s):  
Erik P. van den Ban ◽  
Job J. Kuit ◽  
Henrik Schlichtkrull
Keyword(s):  
Symmetric Spaces ◽  
Discrete Series ◽  
Series Representation ◽  
Compact Subgroup ◽  
Cusp Forms ◽  
Series Representations ◽  
Rank One ◽  
Discrete Series Representations ◽  
Generalized Matrix ◽  
Split Rank

AbstractLet {G/H} be a reductive symmetric space of split rank one and let K be a maximal compact subgroup of G. In a previous article the first two authors introduced a notion of cusp forms for {G/H}. We show that the space of cusp forms coincides with the closure of the space of K-finite generalized matrix coefficients of discrete series representations if and only if there exist no K-spherical discrete series representations. Moreover, we prove that every K-spherical discrete series representation occurs with multiplicity one in the Plancherel decomposition of {G/H}.

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