128 research outputs found
Nilpotent orbits and some small unitary representations of indefinite orthogonal groups
AbstractFor 2ā©½mā©½l/2, let G be a simply connected Lie group with g0=so(2m,2lā2m) as Lie algebra, let g=kāp be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra kā©g0, and let U(g) be the universal enveloping algebra of g. This work examines the unitarity and K spectrum of representations in the āanalytic continuationā of discrete series of G, relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g.The roots with respect to the usual compact Cartan subalgebra are all Ā±eiĀ±ej with 1ā©½i<jā©½l. In the usual positive system of roots, the simple root emāem+1 is noncompact and the other simple roots are compact. Let q=lāu be the parabolic subalgebra of g for which emāem+1 contributes to u and the other simple roots contribute to l, let L be the analytic subgroup of G with Lie algebra lā©g0, let LC=Intg(l), let 2Ī“(u) be the sum of the roots contributing to u, and let qĢ=lāuĢ be the parabolic subalgebra opposite to q.The members of uā©p are nilpotent members of g. The group LC acts on uā©p with finitely many orbits, and the topological closure of each orbit is an irreducible algebraic variety. If Y is one of these varieties, let R(Y) be the dual coordinate ring of Y; this is a quotient of the algebra of symmetric tensors on uā©p that carries a fully reducible representation of LC.For sāZ, let Ī»s=āk=1m(āl+s2)ek. Then Ī»s defines a one-dimensional (l,L) module CĪ»s. Extend this to a (qĢ,L) module by having uĢ act by 0, and define N(Ī»s+2Ī“(u))=U(g)āqĢCĪ»s+2Ī“(u). Let Nā²(Ī»s+2Ī“(u)) be the unique irreducible quotient of N(Ī»s+2Ī“(u)). The representations under study are Ļs=Ī S(N(Ī»s+2Ī“(u))) and Ļsā²=Ī S(Nā²(Ī»s+2Ī“(u))), where S=dim(uā©k) and Ī S is the Sth derived Bernstein functor.For s>2lā2, it is known that Ļs=Ļsā² and that Ļsā² is in the discrete series. Enright, Parthsarathy, Wallach, and Wolf showed for mā©½sā©½2lā2 that Ļs=Ļsā² and that Ļsā² is still unitary. The present paper shows that Ļsā² is unitary for 0ā©½sā©½mā1 even though Ļsā Ļsā², and it relates the K spectrum of the representations Ļsā² to the representation of LC on a suitable R(Y) with Y depending on s. Use of a branching formula of D. E. Littlewood allows one to obtain an explicit multiplicity formula for each K type in Ļsā²; the variety Y is indispensable in the proof. The chief tools involved are an idea of B. Gross and Wallach, a geometric interpretation of Littlewood's theorem, and some estimates of norms.It is shown further that the natural invariant Hermitian form on Ļsā² does not make Ļsā² unitary for s<0 and that the K spectrum of Ļsā² in these cases is not related in the above way to the representation of LC on any R(Y).A final section of the paper treats in similar fashion the simply connected Lie group with Lie algebra g0=so(2m,2lā2m+1), 2ā©½mā©½l/2
Invariant Differential Operators for Non-Compact Lie Groups: the Sp(n,R) Case
In the present paper we continue the project of systematic construction of
invariant differential operators on the example of the non-compact algebras
sp(n,R), in detail for n=6. Our choice of these algebras is motivated by the
fact that they belong to a narrow class of algebras, which we call 'conformal
Lie algebras', which have very similar properties to the conformal algebras of
Minkowski space-time. We give the main multiplets and the main reduced
multiplets of indecomposable elementary representations for n=6, including the
necessary data for all relevant invariant differential operators. In fact, this
gives by reduction also the cases for n<6, since the main multiplet for fixed n
coincides with one reduced case for n+1.Comment: Latex2e, 27 pages, 8 figures. arXiv admin note: substantial text
overlap with arXiv:0812.2690, arXiv:0812.265
Domains of holomorphy for irreducible unitary representations of simple Lie groups
We classify the domains of holomorphy of all Harish-Chandra modules of
irreducible unitary representations of simple non-compact Lie groups.Comment: revised version, to appear in Invent. math., 14 page
Symmetric spaces of higher rank do not admit differentiable compactifications
Any nonpositively curved symmetric space admits a topological
compactification, namely the Hadamard compactification. For rank one spaces,
this topological compactification can be endowed with a differentiable
structure such that the action of the isometry group is differentiable.
Moreover, the restriction of the action on the boundary leads to a flat model
for some geometry (conformal, CR or quaternionic CR depending of the space).
One can ask whether such a differentiable compactification exists for higher
rank spaces, hopefully leading to some knew geometry to explore. In this paper
we answer negatively.Comment: 13 pages, to appear in Mathematische Annale
An interacting particle model and a Pieri-type formula for the orthogonal group
We introduce a new interacting particles model with blocking and pushing
interactions. Particles evolve on the positive line jumping on their own
volition rightwards or leftwards according to geometric jumps with parameter q.
We show that the model involves a Pieri-type formula for the orthogonal group.
We prove that the two extreme cases - q=0 and q=1 - lead respectively to a
random tiling model studied by Borodin and Kuan and to a random matrix model.Comment: 1
The projective translation equation and unramified 2-dimensional flows with rational vector fields
Let X=(x,y). Previously we have found all rational solutions of the
2-dimensional projective translation equation, or PrTE,
(1-z)f(X)=f(f(Xz)(1-z)/z); here f(X)=(u(x,y),v(x,y)) is a pair of two (real or
complex) functions. Solutions of this functional equation are called projective
flows. A vector field of a rational flow is a pair of 2-homogenic rational
functions. On the other hand, only special pairs of 2-homogenic rational
functions give rise to rational flows. In this paper we are interested in all
non-singular (satisfying the boundary condition) and unramified (without
branching points, i.e. single-valued functions in C^2\{union of curves})
projective flows whose vector field is still rational. We prove that, up to
conjugation with 1-homogenic birational plane transformation, these are of 6
types: 1) the identity flow; 2) one flow for each non-negative integer N -
these flows are rational of level N; 3) the level 1 exponential flow, which is
also conjugate to the level 1 tangent flow; 4) the level 3 flow expressable in
terms of Dixonian (equianharmonic) elliptic functions; 5) the level 4 flow
expressable in terms of lemniscatic elliptic functions; 6) the level 6 flow
expressable in terms of Dixonian elliptic functions again. This reveals another
aspect of the PrTE: in the latter four cases this equation is equivalent and
provides a uniform framework to addition formulas for exponential, tangent, or
special elliptic functions (also addition formulas for polynomials and the
logarithm, though the latter appears only in branched flows). Moreover, the
PrTE turns out to have a connection with Polya-Eggenberger urn models. Another
purpose of this study is expository, and we provide the list of open problems
and directions in the theory of PrTE; for example, we define the notion of
quasi-rational projective flows which includes curves of arbitrary genus.Comment: 34 pages, 2 figure
Quantum differential forms
Formalism of differential forms is developed for a variety of Quantum and
noncommutative situations
Fusion and singular vectors in A1{(1)} highest weight cyclic modules
We show how the interplay between the fusion formalism of conformal field
theory and the Knizhnik--Zamolodchikov equation leads to explicit formulae for
the singular vectors in the highest weight representations of A1{(1)}.Comment: 42 page
On the Strong Coupling Limit of the Faddeev-Hopf Model
The variational calculus for the Faddeev-Hopf model on a general Riemannian
domain, with general Kaehler target space, is studied in the strong coupling
limit. In this limit, the model has key similarities with pure Yang-Mills
theory, namely conformal invariance in dimension 4 and an infinite dimensional
symmetry group. The first and second variation formulae are calculated and
several examples of stable solutions are obtained. In particular, it is proved
that all immersive solutions are stable. Topological lower energy bounds are
found in dimensions 2 and 4. An explicit description of the spectral behaviour
of the Hopf map S^3 -> S^2 is given, and a conjecture of Ward concerning the
stability of this map in the full Faddeev-Hopf model is proved.Comment: 21 pages, 0 figure
Special Reduced Multiplets and Minimal Representations for SO(p,q)
Using our previous results on the systematic construction of invariant
differential operators for non-compact semisimple Lie groups we classify the
special reduced multiplets and minimal representations in the case of SO(p,q).Comment: 26 pages, 11 figures, to appear in the Proceedings of the X
International Workshop "Lie Theory and Its Applications in Physics}, (Varna,
Bulgaria, June 2013), "Springer Proceedings in Mathematics and Statistics",
Vol. 11
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