scholarly journals Decomposition of representations of SL(2, C) induced by the continuous series of E(2)

1980 ◽  
Vol 78 ◽  
pp. 95-111
Author(s):  
Hitoshi Kaneta
Keyword(s):  
Lorentz Group ◽  
Unitary Representations ◽  
Continuous Series ◽  
Induced Representations ◽  
Irreducible Unitary Representations ◽  
Inhomogeneous Lorentz Group

Since the representations of SL(2, C) induced by irreducible unitary representations of appear as the restriction to the Lorentz group of some irreducible unitary representations of the inhomogeneous Lorentz group, the decomposition of the induced representations deserves our investigation. For the representations of SL(2, C) induced by irreducible unitary representations with discrete spin of E(2), the decomposition has been obtained by Mukunda [9]. We hope that our analysis will justify the calculations by Chakrabarti [1], [2] and [3].

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.

Weak Containment and Induced Representations of Groups

1962 ◽  
Vol 14 ◽  
pp. 237-268 ◽  
Author(s):  
J. M. G. Fell
Keyword(s):  
Dual Space ◽  
Locally Compact Group ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Unitary Equivalence ◽  
Locally Compact ◽  
Induced Representations ◽  
Weak Containment ◽  
Irreducible Unitary Representations ◽  
Special Case

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

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