Volume 106, Issue 4
November 1996, pages 329-449
pp 329-368 November 1996
Normal subgroups ofSL_{1,D} and the classification of finite simple groups
Andrei Rapinchuk Alexander Potapchik
LetD be a division algebra of degree three over an algebraic number fieldK and let G = SL_{D}. We prove that the normal subgroup structure of G(K) is given by the Platonov-Margulis conjecture. The proof uses the classification of finite simple groups.
pp 369-377 November 1996
A theorem of the Wiener—Tauberian type forL^{1}(H^{n})
The Heisenberg motion groupHM(n), which is a semi-direct product of the Heisenberg group H^{n} and the unitary group U(n), acts on H^{n} in a natural way. Here we prove a Wiener-Tauberian theorem for L^{1} (H^{n}) with this HM(n)-action on H^{n} i.e. we give conditions on the “group theoretic” Fourier transform of a functionf in L^{1} (H^{n}) in order that the linear span of^{g}f : g∈HM(n) is dense in L^{1}(H^{n}), where^{g}f(z, t) =f(g·(z, t)), forg ∈ HM(n), (z,t)∈H^{n}.
pp 379-401 November 1996
Multiplicity formulas for finite dimensional and generalized principal series representations
The article presents two results. (1) Let a be a reductive Lie algebra over ℂ and let b be a reductive subalgebra of a. The first result gives the formula for multiplicity with which a finite dimensional irreducible representation of b appears in a given finite dimensional irreducible representation of a in a general situation. This generalizes a known theorem due to Kostant in a special case. (2) LetG be a connected real semisimple Lie group andK a maximal compact subgroup ofG. The second result is a formula for multiplicity with which an irreducible representation ofK occurs in a generalized representation ofG arising not necessarily from fundamental Cartan subgroup ofG. This generalizes a result due to Enright and Wallach in a fundamental case.
pp 403-420 November 1996
Generalized parabolic bundles and applications— II
We prove the existence of the moduli spaceM(n,d) of semistable generalised parabolic bundles (GPBs) of rankn, degreed of certain general type on a smooth curve. We study interesting cases of the moduli spacesM(n, d) and find explicit geometric descriptions for them in low ranks and genera. We define tensor products, symmetric powers etc. and the determinant of a GPB. We also define fixed determinant subvarietiesM_{L}(n, d),L being a GPB of rank 1. We apply these results to study of moduli spaces of torsionfree sheaves on a reduced irreducible curveY with nodes and ordinary cusps as singularities. We also study relations among these moduli spaces (rank 2) as polarization varies over [0, 1].
pp 421-449 November 1996
Moduli for principal bundles over algebraic curves: II
We classify principal bundles on a compact Riemann surface. A moduli space for semistable principal bundles with a reductive structure group is constructed using Mumford’s geometric invariant theory.
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