TENSOR PRODUCTS OF SOME IRREDUCIBLE UNITARY REPRESENTATIONS OF THE UNIVERSAL COVERING GROUP OF SU(1, 1) THAT DEFINE REPRESENTATIONS OF SU(1, 1)
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We consider tensor products of irreducible unitary representations of the universal covering group G of SU(1, 1) belonging to the so-called positive discrete series of representations of G in the terminology used by Sally in his well-known book [P. J. Sally, Analytic Continuation of the Irreducible Unitary Representations of the Universal Covering Group of SL(2, R), Memoirs Amer. Math. Soc. No. 69, American Mathematical Society, Providence, RI, 1967]. We show that, for the special values of the parameters that are indicated in the paper, these products take the kernel of the natural homomorphism of G onto SU(1, 1) to the identity operator, and hence define representations of the group SU(1, 1). We then study the decomposition of this representation into the direct sum of irreducible representations and study some tensor products of unitary pseudorepresentations of SU(1, 1).
Abstract
We consider tensor products of irreducible unitary representations of the universal covering group G of SU(1, 1) belonging to the so-called positive discrete series of representations of G in the terminology used by Sally in his well-known book [P. J. Sally, Analytic Continuation of the Irreducible Unitary Representations of the Universal Covering Group of SL(2, R), Memoirs Amer. Math. Soc. No. 69, American Mathematical Society, Providence, RI, 1967]. We show that, for the special values of the parameters that are indicated in the paper, these products take the kernel of the natural homomorphism of G onto SU(1, 1) to the identity operator, and hence define representations of the group SU(1, 1). We then study the decomposition of this representation into the direct sum of irreducible representations and study some tensor products of unitary pseudorepresentations of SU(1, 1).
참고문헌 (5)
- A. I. Shtern, "Deformations of irreducible unitary representations of a discrete series of Hermitian-symmetric simple Lie groups in the class of pure pseudorepresentations", J. Math. Sci. (N.Y.), 123(4), pp.4324~4339 (2004)[DOI]
- A. I. Shtern, "Quasisymmetry. I", Russ. J. Math. Phys., 2(3), pp.353~382 (1994)
- A. I. Shtern, "Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko's conjecture", Fundam. Prikl. Mat., 13(7), pp.85~225 (2007)
- A. I. Shtern, "On the tensor product of unbounded quasi-representations of groups", Advanced Studies in Contemporary Mathematics, 27(4), pp.641~645 (2017), 장전수학회[DOI]
- TEX>), "Analytic Continuation of the Irreducible Unitary Representations of the Universal Covering Group of SL(2, <TEX>$ {\mathbb{R}}$<", pp.1967, P. J. Sally