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• W2247234543 abstract "The idea of describing the eigenvalues of bosonic and fermionic fields in quantum fieldtheory by commuting and anticommuting variables led to the development of supermanifolds in the 1970s. Of particular interest to condensed matter physicists studyingdisordered systems are symmetric super spaces, i.e. quotients G/K of Lie super groups,because those arise for example as the target spaces of non-linear σ -models, whichare the effective low energy theories for disordered fermionic systems with quadraticHamiltonians in the thermodynamic limit. See [Zir98] for an elaborate discussion ofan application. An interesting development is [LSZ07] which proves that an importantstep in the development of these physical models is indeed mathematically rigorous.A recent example using this new technique can be found in [SZ10].Recently, active research has been focused on the development of harmonic analysison symmetric super spaces ([AHZ08, All10, AHL11]). One goal here is to prove a superFourier inversion formula and hence obtain a tool to solve linear partial differentialequations involving G-invariant differential operators on G/K as appear in the aforementioned examples from physics. To this end one needs to understand sphericalfunctions, i.e. the K-biinvariant joint eigenfunctions of such differential operators andin particular their asymptotics. We expect that, as in the classical case, those functionscan be characterised as matrix coeffcients of the spherical representations of G, i.e.those containing a K -invariant vector. The goal of this thesis is therefore to determinewhich finite dimensional irreducible G representations are spherical. For concretenessand to avoid problems stemming from non-compact real forms we restrict our attentionto the Lie super algebra level and study the symmetric pair g = glq|r+s with sub Liesuper algebra k = glq|r ⊕ gl0|s .Our main result, Theorem 3.60 on page 53, is a generalisation of a classical theoremdue to Helgason, [Hel84], Chapter V, Theorem 4.1, which states that a representationis spherical if and only if the highest weight vector is M -invariant, where Q = M ANis a minimal parabolic subgroup of G. It turns out that this is exactly the same in thesuper case. For the proof we use methods developed by Schlichtkrul, [Sch84], to reducethe use of integration, which in the super world holds much more pitfalls. Similar tothe ordinary case we can then classify the spherical representations in terms of theirhighest weights in Lemma 3.61 and the subsequent corollaries. At least for glq|r+1 withr > q this classiffcation is complete, but see Chapter 4 for some immediate as well asconjectured generalisations." @default.
• W2247234543 created "2016-06-24" @default.
• W2247234543 creator A5064789877 @default.
• W2247234543 date "2011-10-01" @default.
• W2247234543 modified "2023-09-27" @default.
• W2247234543 title "Spherical representations of reductive Lie super algebras" @default.
• W2247234543 hasPublicationYear "2011" @default.
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