117 research outputs found
Virasoro algebra and wreath product convolution
We present a group theoretic construction of the Virasoro algebra in the
framework of wreath products. This can be regarded as a counterpart of a
geometric construction of Lehn in the theory of Hilbert schemes of points on a
surface.Comment: Corrections of typos including signs in Lemma 1 and Theorem 4, and
other minor changes, to appear in Journal of Al
Split Quaternionic Analysis and Separation of the Series for SL(2,R) and SL(2,C)/SL(2,R)
We extend our previous study of quaternionic analysis based on representation
theory to the case of split quaternions H_R. The special role of the unit
sphere in the classical quaternions H identified with the group SU(2) is now
played by the group SL(2,R) realized by the unit quaternions in H_R. As in the
previous work, we use an analogue of the Cayley transform to relate the
analysis on SL(2,R) to the analysis on the imaginary Lobachevski space
SL(2,C)/SL(2,R) identified with the one-sheeted hyperboloid in the Minkowski
space M. We study the counterparts of Cauchy-Fueter and Poisson formulas on H_R
and M and show that they solve the problem of separation of the discrete and
continuous series. The continuous series component on H_R gives rise to the
minimal representation of the conformal group SL(4,R), while the discrete
series on M provides its K-types realized in a natural polynomial basis. We
also obtain a surprising formula for the Plancherel measure on SL(2,R) in terms
of the Poisson integral on the split quaternions H_R. Finally, we show that the
massless singular functions of four-dimensional quantum field theory are
nothing but the kernels of projectors onto the discrete and continuous series
on the imaginary Lobachevski space SL(2,C)/SL(2,R). Our results once again
reveal the central role of the Minkowski space in quaternionic and split
quaternionic analysis as well as a deep connection between split quaternionic
analysis and the four-dimensional quantum field theory.Comment: corrected, 70 pages, no figures, to appear in Advances in Mathematic
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