• Volume 106, Issue 4

November 1996,   pages  329-449

• Normal subgroups ofSL1,D and the classification of finite simple groups

LetD be a division algebra of degree three over an algebraic number fieldK and let G = SLD. 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.

• A theorem of the Wiener—Tauberian type forL1(Hn)

The Heisenberg motion groupHM(n), which is a semi-direct product of the Heisenberg group Hn and the unitary group U(n), acts on Hn in a natural way. Here we prove a Wiener-Tauberian theorem for L1 (Hn) with this HM(n)-action on Hn i.e. we give conditions on the “group theoretic” Fourier transform of a functionf in L1 (Hn) in order that the linear span ofgf : g∈HM(n) is dense in L1(Hn), wheregf(z, t) =f(g·(z, t)), forg ∈ HM(n), (z,t)∈Hn.

• 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.

• 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 subvarietiesML(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].

• 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.

• # Proceedings – Mathematical Sciences

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