Unitary Representations of Unitary Groups

Zuckerman’s derived functor module of a semisimple Lie group [Formula: see text] yields a unitary representation [Formula: see text] which may be regarded as a geometric quantization of an elliptic orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. We highlight a certain family of those irreducible unitary representations [Formula: see text] of the indefinite unitary group [Formula: see text] and a family of subgroups [Formula: see text] of [Formula: see text] such that the restriction [Formula: see text] is known to be discretely decomposable and multiplicity-free by the general theory of Kobayashi (Discrete decomposibility of the restrictions of [Formula: see text] with respect to reductive subgroups, II, Ann. of Math. 147 (1998) 1–21; Multiplicity-free representations and visible action on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005) 497–549), where [Formula: see text] is not necessarily tempered and [Formula: see text] is not necessarily compact. We prove that the corresponding moment map [Formula: see text] is proper, determine the image [Formula: see text], and compute the Corwin–Greenleaf multiplicity function explicitly.

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Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

International Mathematics Research Notices ◽

10.1093/imrn/rny119 ◽

2018 ◽

Vol 2020 (13) ◽

pp. 3902-3926

Author(s):

Réda Boumasmoud ◽

Ernest Hunter Brooks ◽

Dimitar P Jetchev

Keyword(s):

Vertical Distribution ◽

Three Dimensional ◽

Complex Multiplication ◽

Horizontal Distribution ◽

Unitary Groups ◽

Shimura Varieties ◽

Heegner Points ◽

Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

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Defect CFT techniques in the 6d $$ \mathcal{N} $$ = (2, 0) theory

Journal of High Energy Physics ◽

10.1007/jhep03(2021)261 ◽

2021 ◽

Vol 2021 (3) ◽

Author(s):

Nadav Drukker ◽

Malte Probst ◽

Maxime Trépanier

Keyword(s):

Stress Tensor ◽

Unitary Representations ◽

Point Function ◽

Displacement Operator ◽

Expectation Value ◽

Operator Expansion ◽

Bulk Stress ◽

Defect Operator ◽

Tensor Multiplet

Abstract Surface operators are among the most important observables of the 6d $$ \mathcal{N} $$ N = (2, 0) theory. Here we apply the tools of defect CFT to study local operator insertions into the 1/2-BPS plane. We first relate the 2-point function of the displacement operator to the expectation value of the bulk stress tensor and translate this relation into a constraint on the anomaly coefficients associated with the defect. Secondly, we study the defect operator expansion of the stress tensor multiplet and identify several new operators of the defect CFT. Technical results derived along the way include the explicit supersymmetry tranformations of the stress tensor multiplet and the classification of unitary representations of the superconformal algebra preserved by the defect.

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The intersection graph of a finite simple group has diameter at most 5

Archiv der Mathematik ◽

10.1007/s00013-021-01583-3 ◽

2021 ◽

Author(s):

Saul D. Freedman

Keyword(s):

Simple Group ◽

Upper Bound ◽

Finite Simple Group ◽

Unitary Groups ◽

Intersection Graph ◽

Monster Group ◽

Prime Dimension

AbstractLet G be a non-abelian finite simple group. In addition, let $$\Delta _G$$ Δ G be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of $$\Delta _G$$ Δ G has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

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The Existence of Minimal Logarithmic Signatures for Some Finite Simple Unitary Groups

Vietnam Journal of Mathematics ◽

10.1007/s10013-021-00489-5 ◽

2021 ◽

Author(s):

Ali Reza Rahimipour ◽

Ali Reza Ashrafi

Keyword(s):

Unitary Groups ◽

Logarithmic Signatures

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Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽

10.3390/sym13061060 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1060

Author(s):

Enrico Celeghini ◽

Manuel Gadella ◽

Mariano A. del del Olmo

Keyword(s):

Hilbert Space ◽

Fourier Transform ◽

Heisenberg Group ◽

Real Line ◽

Hermite Functions ◽

Unitary Representations ◽

Weyl Groups ◽

The Real ◽

Symmetry Properties ◽

The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Weyl–Heisenberg group and some of their extensions.

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