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A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

  • H. KUMUDINI DHARMADASA (a1) and WILLIAM MORAN (a2)

Abstract

Let $G$ be a locally compact group and $K$ a closed subgroup of $G$ . Let $\unicode[STIX]{x1D6FE},$ $\unicode[STIX]{x1D70B}$ be representations of $K$ and $G$ respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space $G/K$ possesses a right-invariant measure and the representation space $H(\unicode[STIX]{x1D6FE})$ of the representation $\unicode[STIX]{x1D6FE}$ of $K$ is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on $G/K$ and that the representation spaces $\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with $\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.

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References

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[1] Dharmadasa, H. K., The Theory of  $A_{p}^{q}$  Spaces, PhD Thesis, University of Adelaide, South Australia, 1995.
[2] Dunford, N. and Schwartz, J., Linear Operators, Vols I & 2 (Interscience, New York, 1958).
[3] Fontenot, R. A. and Schochetman, I., ‘Induced representations of groups on Banach spaces’, Rocky Mountain J. Math. 7(1) (1977), 5382.
[4] Gaal, S. A., Linear Analysis and Representation Theory (Springer, Berlin, 1973).
[5] Jaming, P. and Moran, W., ‘Tensor products and p-induction of representations on Banach spaces’, Collect. Math. 51(1) (2000), 83109.
[6] Kleppner, A., ‘Intertwining forms for summable induced representations’, Trans. Amer. Math. Soc. 112 (1964), 164183.
[7] Mackey, G. W., ‘On induced representations of groups’, Amer. J. Math. 73 (1951), 576592.
[8] Mackey, G. W., ‘Induced representations of locally compact groups I’, Ann. Math. (2) 55(1) (1952), 101140.
[9] Mautner, F. I., ‘A generalization of the Frobenius reciprocity theorem’, Proc. Natl. Acad. Sci. USA 37 (1951), 431435.
[10] Moore, C. C., ‘On the Frobenius reciprocity theorem for locally compact groups’, Pacific J. Math. 12 (1962), 359365.
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A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

  • H. KUMUDINI DHARMADASA (a1) and WILLIAM MORAN (a2)

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