8,646 research outputs found
Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits
We approach the topic of Classical group nilpotent orbits from the
perspective of their moduli spaces, described in terms of Hilbert series and
generating functions. We review the established Higgs and Coulomb branch quiver
theory constructions for A series nilpotent orbits. We present systematic
constructions for BCD series nilpotent orbits on the Higgs branches of quiver
theories defined by canonical partitions; this paper collects earlier work into
a systematic framework, filling in gaps and providing a complete treatment. We
find new Coulomb branch constructions for above minimal nilpotent orbits,
including some based upon twisted affine Dynkin diagrams. We also discuss
aspects of 3d mirror symmetry between these Higgs and Coulomb branch
constructions and explore dualities and other relationships, such as
HyperKahler quotients, between quivers. We analyse all Classical group
nilpotent orbit moduli spaces up to rank 4 by giving their unrefined Hilbert
series and the Highest Weight Generating functions for their decompositions
into characters of irreducible representations and/or Hall Littlewood
polynomials.Comment: 67 pages, 11 figure
Quiver Theories and Formulae for Nilpotent Orbits of Exceptional Algebras
We treat the topic of the closures of the nilpotent orbits of the Lie
algebras of Exceptional groups through their descriptions as moduli spaces, in
terms of Hilbert series and the highest weight generating functions for their
representation content. We extend the set of known Coulomb branch quiver theory
constructions for Exceptional group minimal nilpotent orbits, or reduced single
instanton moduli spaces, to include all orbits of Characteristic Height 2,
drawing on extended Dynkin diagrams and the unitary monopole formula. We also
present a representation theoretic formula, based on localisation methods, for
the normal nilpotent orbits of the Lie algebras of any Classical or Exceptional
group. We analyse lower dimensioned Exceptional group nilpotent orbits in terms
of Hilbert series and the Highest Weight Generating functions for their
decompositions into characters of irreducible representations and/or Hall
Littlewood polynomials. We investigate the relationships between the moduli
spaces describing different nilpotent orbits and propose candidates for the
constructions of some non-normal nilpotent orbits of Exceptional algebras.Comment: 87 pages, 4 figure
Series of nilpotent orbits
We organize the nilpotent orbits in the exceptional complex Lie algebras into
series using the triality model and show that within each series the dimension
of the orbit is a linear function of the natural parameter a=1,2,4,8,
respectively for f_4,e_6,e_7,e_8. We also obtain explicit representatives in a
uniform manner. We observe similar regularities for the centralizers of
nilpotent elements in a series and graded components in the associated grading
of the ambient Lie algebra. More strikingly, for a greater than one, the
degrees of the unipotent characters of the corresponding Chevalley groups,
associated to these series through the Springer correspondance are given by
polynomials which have uniform expressions in terms of a.Comment: 20 pages, revised version with more formulas for unipotent character
Nilpotent orbits and the Coulomb branch of theories: special orthogonal vs orthogonal gauge group factors
Coulomb branches of a set of supersymmetric gauge
theories are closures of nilpotent orbits of the algebra .
From the point of view of string theory, these quantum field theories can be
understood as effective gauge theories describing the low energy dynamics of a
brane configuration with the presence of orientifold planes. The presence of
the orientifold planes raises the question to whether the orthogonal factors of
a the gauge group are indeed orthogonal or special orthogonal .
In order to investigate this problem, we compute the Hilbert series for the
Coulomb branch of theories, utilizing the monopole
formula. The results for all nilpotent orbits from to
which are special and normal are presented. A new
relationship between the choice of factors in the gauge group and the
Lusztig's Canonical Quotient of the corresponding nilpotent orbit is observed.
We also provide a new way of projecting several magnetic lattices of different
gauge group factors by the simultaneous action of a
group.Comment: 33 pages, 3 figures, 28 table
Asymptotic cone of semisimple orbits for symmetric pairs
Let G be a reductive algebraic group over the complex field and O_h be a
closed adjoint orbit through a semisimple element h. By a result of Borho and
Kraft (1979), it is known that the asymptotic cone of the orbit O_h is the
closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup
whose Levi component is the centralizer Z_G(h) in G.
In this paper, we prove an analogue on a semisimple orbit for a symmetric
pair (G, K).Comment: 14 page
Tempered Representations and Nilpotent Orbits
Given a nilpotent orbit O of a real, reductive algebraic group, a necessary
condition is given for the existence of a tempered representation pi such that
O occurs in the wave front cycle of pi. The coefficients of the wave front
cycle of a tempered representation are expressed in terms of volumes of
precompact submanifolds of an affine space.Comment: The class of nilpotent orbits studied in this paper is different from
the class of noticed nilpotent orbits studied by Noel. A previous version of
this paper erroneously stated that these two classes are the same.
Representation Theory, Volume 16, 201
Unipotent representations of real classical groups
Let be a complex orthogonal or complex symplectic group, and let
be a real form of , namely is a real orthogonal group, a
real symplectic group, a quaternionic orthogonal group, or a quaternionic
symplectic group. For a fixed parity , we
define a set of nilpotent
-orbits in (the Lie algebra of ). When
is the parity of the dimension of the standard module of , this is the set of the stably trivial special nilpotent orbits, which
includes all rigid special nilpotent orbits. For each , we construct all unipotent
representations of (or its metaplectic cover when is a real symplectic
group and is odd) attached to via the method of theta
lifting and show in particular that they are unitary
- …