Uniformly bounded representations of the Lorentz groups

Thesis/Dissertation · Fri Jan 01 04:00:00 EST 1982

OSTI ID:5003368

Brega, A O

For the Lorentz group G = SO/sub e/(n + 1, 1)(ngreater than or equal to 2) the author constructs a family of uniformly bounded representations by means of analytically continuing a certain normalization of the unitary principal series. The method the author uses relies on an analysis of various operators under a Mellin transform and extends earlier work of E.N. Wilson. In a series of papers Kunze and Stein initiated the theory of uniformly bounded representations of semisimple Lie groups; the starting point is the unitary principal series T(sigma,s) obtained in a certain subgroup M of G and a purely imaginary number s. From there Kunze and Stein constructed families of representations R(sigma,s) depending analytically on a parameter s in a domain D of C containing the imaginary axis which are unitarily equilvalent to T(sigma,s) for s contained in the set of imaginary numbers and whose operator norms are uniformly bounded for each s in D. In the case of the Lorentz groups SO/sub e/(n + 1, 1)(ngreater than or equal to2) and the trivial representation 1 of M, E.N. Wilson obtained such a family R(1,s) for the domain D = (s contained in the set of C: absolute value Re(s) Vertical Bar2). For this domain D and for any representation sigma of M the author provides a family R(sigma,s) of uniformly bounded representations analytically continuing T(sigma,s), thereby generalizing Wilson's work. The author has also investigated certain symmetry properties of the representations R(sigma,s) under the action of the Weyl group. The trivial representation is Weyl group invariant and the family R(1,s) obtained by Wilson satisfies R(1,s) = R(1,-s) reflecting this. Obtained was the analogous result R(sigma,s) = R(sigma,-s) for some well known representations sigma that are Weyl group invariant. This involves the explicit computation of certain constants arising in the Fourier transforms of intertwining operators.

• Research Organization: Washington Univ., St. Louis, MO (USA). Edward Mallinckrodt Inst. of Radiology
• OSTI ID: 5003368
• Country of Publication: United States
• Language: English

Related Subjects

658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
IRREDUCIBLE REPRESENTATIONS
LIE GROUPS
MATHEMATICAL OPERATORS
SO GROUPS
SYMMETRY
SYMMETRY GROUPS
TRANSFORMATIONS
UNITARITY