On the geometry of conjugacy classes in classical groups

Abstract:We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3d N = 4 superconformal gauge theories T σ ρ (G) corresponding to a set of D3 branes ending on NS5 and D5-branes, with or without O3 planes. Here G is a classical group, σ is a partition of G and ρ a partition of the dual group G ∨ . In deriving such a formula we make use of the recently discovered formula for the Hilbert series of the quantum Coulomb branch of N = 4 superconformal theories. The result can be expressed in terms of a generalization of a class of symmetric functions, the Hall-Littlewood polynomials, and can be interpreted in mathematical language in terms of localization. We mainly consider the case G = SU(N ) but some interesting results are also given for orthogonal and symplectic groups.

show abstract

“…It is known that, as an algebraic variety, the Higgs branch of T σ (SU(N )) is the closure of the nilpotent orbit of Jordan type σ T [27,37,38] …”

Section: The Localisation Formulamentioning

confidence: 99%

T ρ σ (G) theories and their Hilbert series

Cremonesi

Hanany

Mekareeya

et al. 2015

J. High Energ. Phys.

181

View full text Add to dashboard Cite

show abstract

“…In (6) choose O to be the nilpotent orbit of Jordan type (see e.g., [3]) (3, 1 2n−3 ) and let M be the simply connected double cover of O. In (7) choose O to be of Jordan type (2 4 , 1) and let M be the simply connected (double cover) of O.…”

Section: Explanation Of the Tablementioning

confidence: 99%

Nilpotent orbits, normality and Hamiltonian group actions

Brylinski¹,

Kostant²

1994

J. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let M be a G-covering of a nilpotent orbit in g where G is a complex semisimple Lie group and g = Lie(G). We prove that under Poisson bracket the space R[2] of homogeneous functions on M of degree 2 is the unique maximal semisimple Lie subalgebra of R = R(M ) containing g. The action of g ′ ≃ R[2] exponentiates to an action of the corresponding Lie groupThe theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familar of the orbits are of semisimple elements. In that case the associated representation theory is pretty much understood (Borel-Weil-Bott and noncompact analogues, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations.A totally different experience is encountered with nilpotent orbits of semisimple groups. Here the associated representation theory (unipotent representations) is poorly understood and there is a loss of reductivity of isotropy subgroups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations.The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the reoccuring experience that sophisticated aspects of semisimple group theory often leads one to these orbits (e.g., the Springer correspondence with representations of the Weyl group).In this note we announce new results concerning the symplectic and algebraic geometry of the nilpotent orbits O and the symmetry groups of that geometry. The starting point is the recognition (made also by others) that the ring R of regular functions on any G-cover M of O is not only a Poisson algebra (the case for any coadjoint orbit) but that R is also naturally graded. The key theme is that the

show abstract

“…However every spherical conjugacy class in the symplectic group has normal closure, since from the classification we know that the unipotent spherical conjugacy classes have only 2 columns (see also [17], §5, Criterion 2). For special orhogonal groups the results in [25] left open the cases of the very even unipotent classes. E. Sommers proved that these have normal closure in [39].…”

Section: Semisimple Classes In Gmentioning

confidence: 99%