The Gel’fand realization and the exceptional representations of SL(2,R)

Basu, Debabrata; Bhattacharya, T.

doi:doi:10.1063/1.526799

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Triality principle and G₂ group in spinor language

The presented paper is aimed at providing a systematic study of a relation between octonions and spinors corresponding to S7 7‐sphere, starting from a natural point of view, enabling us to endow spinor space S+(8,0) with octonion algebra structures. As a result we arrive at formulations of triality principle in its finite form in terms of vector and fundamental representations of Spin (8) group—both for spinors and vectors. The group of automorphisms of octonion algebra, as well as its Lie algeb...

A note on the multiplicity‐free unitary irreps of SL(3,R)

Unitary irreps of ∼(SL(3,R)) which contain a representation D j of SU(2) at most once are known to exist with SU(2) content j=k+2n (n=0,1,2,.. and k=0, 1/2 ,1). Güler, in a recent paper, claims that there also exist multiplicity‐free unitary representations with SU(2) content j=k0, k0+1, k0+2,..., for k0>3. We show that such representations do not exist for k0>1.

It is shown that the canonical representation space of Gel’fand and co‐workers is particularly appropriate for problems requiring explicit reduction under the noncompact SO(1,1) and E(1) bases for both the principal and exceptional series of representations of SL(2,R). We use this realization to set up complete orthonormal sets of eigendistributions corresponding to the three subgroup reductions, namely, SL(2,R)⊇SO(1,1), SL(2,R)⊇E(1), and SL(2,R)⊇SO(2), and evaluate the unitary transformations connecting these reductions. These overlap matrix elements appear as the applications of these distributions to a set of well‐defined test functions. Using the rigorous theory of analytic continuation we show that the results for the exceptional representations have the same analytic forms as the corresponding results for the principal series. Some of these results are essential prerequisites for the solution of the Clebsch–Gordan problem (series and coefficients) of SL(2,R) in the SO(1,1) basis.