Irreducible Representations Of SL Research Articles

Adequate subgroups and indecomposable modules

The notion of adequate subgroups was introduced by Jack Thorne [59]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all absolutely irreducible modules in characteristic p of dimension p. This relies on a modified definition of adequacy, provided by Thorne in [60], which allows p to divide the dimension of the module. We prove adequacy for almost all irreducible representations of SL_2(p^a) in the natural characteristic and for finite groups of Lie type as long as the field of definition is sufficiently large. We also essentially classify indecomposable modules in characteristic p of dimension less than 2p-2 and answer a question of Serre concerning complete reducibility of subgroups in classical groups of low dimension.

Unramified representations of the universal central extension of SL2(𝔽 q ((z)))

Abstract In this paper we are interested in investigating the principal series representations of a certain rank 2 Kac–Moody group. This group (denoted by SL ˜ 2 ( 𝔽 q ( ( z ) ) ) ${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$ ) is introduced as a central extension by 𝔽 q * ${\mathbb {F}_q^*}$ of the group of two-by-two unimodular matrices over the non-archimedean local field, 𝔽 q ( ( z ) ) ${\mathbb {F}_q((z))}$ . We begin by determining a formula for a cocycle that defines this central extension. This is used to classify all irreducible representations of the centrally extended torus inside SL ˜ 2 ( 𝔽 q ( ( z ) ) ) ${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$ and, as a consequence of this classification, we are able to determine the irreducible unramified principal series representations of SL ˜ 2 ( 𝔽 q ( ( z ) ) ) ${\protect \widetilde{{\rm SL}}_2(\mathbb {F}_q((z)))}$ .

Embeddings of the Euclidean algebra $\mathfrak {e}(3)$e(3) into $\mathfrak {sl}(4,\mathbb {C})$sl(4,C) and restrictions of irreducible representations of $\mathfrak {sl}(4,\mathbb {C})$sl(4,C)

The Euclidean group E(3) is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. It is the noncompact, semidirect product group \documentclass[12pt]{minimal}\begin{document}$E(3) \cong SO(3) \ltimes \mathbb {R}^{3}$\end{document}E(3)≅SO(3)⋉R3. The Euclidean algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) is the complexification of the Lie algebra of E(3). We classify the embeddings of the Euclidean algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) into the simple Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C) and, as an application of this classification, discuss the restriction to various embeddings of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) of certain irreducible representations of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C). In particular, we consider which of these restrictions are \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3)-indecomposable.

COLORED JONES POLYNOMIALS WITH POLYNOMIAL GROWTH

The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;ℂ) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N if one fixes a complex number c near [Formula: see text]. On the other hand if the absolute value of c is small enough, it converges to the inverse of the Alexander polynomial evaluated at exp c. In this paper we study cases where it grows polynomially.

Invariant theory of GL(n) and intersection theory of Grassmannians

We give an algebro-geometric relation between intersection theory of Grassmannians and invariant theory of the general linear group. From points of intersection of Schubert varieties in Gr(r,n), we produce “all” invariants in tensor products of (associated) irreducible representations of SL(r).

Aspects of hidden and manifest [formula omitted] symmetry in 2D near-horizon black-hole backgrounds

The invariance under unitary representations of the conformal group SL(2, R) of a quantum particle is rigorously investigated in two-dimensional spacetimes containing Killing horizons using de Alfaro–Fubini–Furlan's model. The limit of the near-horizon approximation is considered. If the Killing horizon is bifurcate (Schwarzschild-like near horizon limit), the conformal symmetry is hidden, i.e., it does not arise from geometrical spacetime isometries, but the whole Hilbert space turns out to be an irreducible unitary representation of SL(2, R) and the time evolution is embodied in the unitary representation. In this case the symmetry does not depend on the mass of the particle and, if the representation is faithful, the conformal observable K shows thermal properties. If the Killing horizon is nonbifurcate (extreme Reisner–Nördstrom-like near horizon limit), the conformal symmetry is manifest, i.e., it arises from geometrical spacetime isometries. The SL(2, R) representation which arises from the geometry selects a hidden conformal representation. Also in that case the Hilbert space is an irreducible representation of SL(2, R) and the group conformal symmetries embodies the time evolution with respect to the local Killing time. However no thermal properties are involved, at least considering the representations induced by the geometry. The conformal observable K gives rise to Killing time evolution of the quantum state with respect to another global Killing time present in the manifold. Mathematical proofs about the developed machinery are supplied and features of the operator H g=− d 2 dx 2 + g x 2 , with g=−1/4 are discussed. It is proven that a statement, used in the recent literature, about the spectrum of self-adjoint extensions of H g is incorrect.

Fuchsian Affine Actions of Surface Groups

Let λq be the irreducible representation of SL(2, ℝ) in SL(q, ℝ). Define a Fuchsian subgroup of SL(q, ℝ) to be a subgroup conjugate to a discrete subgroup of λq(SL(2, ℝ)). We prove in this paper that the fundamental group of a compact surface does not act properly on the affine space by affine tranformations if its linear part is Fuchsian.

Zero weight spaces and the Springer correspondence

It is known that an irreducible representation of SL n( C) whose highest weight is a partition of n affords in its zero weight space the irreducible representation of S n corresponding to the dual partition. We generalize this to groups of type D and E.

Anosov automorphisms for nilmanifolds and rigidity of group actions

AbstractWe obtain the density of Lyapunov exponents for maximal abelian ℝ-split group in kth tensor product representation of a subgroup Γ ⊂ SL(n, ℤ) of finite index under certain conditions. Anosov and Cartan actions of such groups associated with irreducible representations of SL(n, ℝ) are also classified. Examples of rigidity of actions on nilmanifolds are discussed.

Graded contractions of representations of special linear Lie algebras with respect to their maximal parabolic subalgebras

Parabolic gradings of the classical simple Lie algebras sl(N,C) (N>or=3) are described for all maximal parabolic subalgebras. All contractions are found which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading (parabolic contractions of Lie algebras). Contractions of the irreducible representations of sl(N,C) for each parabolic contraction of the Lie algebra are the main results of the article.

A relation between instantons of grassmann ?-models and Toda equations: II

We construct special solutions of Toda equations and show that these are deeply related with irreducible representations of sl(2;ℂ). We also show that after taking a continuum limit of Toda equations, these become the well-known solution of the continuum Toda equation.

Notes on representations of non-Archimedean SL(n)

Let F be a non-archimedean local field. In this paper the relation between irreducible representations of GL(n, F) and SL(«, F) is studied. Using the results on GL(n, F) a parametrization of (var- ious classes of) irreducible representations of SL(n, F) by parame- ters expressed in terms of cuspidal representations of GL(n, F) is obtained. Introduction. Before we give a more detailed description of the con- tent of this paper, a few historical remarks on SL(n, F) are needed. Gelfand and Naimark gave in (8) proof of the irreducibility of unitary principal series representations of SL(n, C). The same proof gives the irreducibility of unitary principal series for GL(n) over any local field. Using the fact that the unitary principal series have non-trivial Whittaker models for GL(n), and the uniqueness of the model proved by Rodier ((18)), Howe and Silberger proved in (10) that the unitary principal series of GL(n, F) restricted to SL(n, F) are multiplicity free. The same idea appears in Labesse and Langlands paper (14). In this way, Howe and Silberger obtained that unitary principal series representations of SL(n, F) are multiplicity free. Shahidi observed in (20) that one can prove, using the same idea of Whittaker models, that any irreducible tempered representation of GL(n, F) restricted to SL(n , F) is multiplicity free. In this way one obtains that the parabolically induced representation of SL(«, F) by irreducible tem- pered representation is multiplicity free. A general approach to the reducibility and the multiplicities was done by Keys. The structure of the commuting algebras of unitary principal series representations for Chevalley groups was described by him in (11) and it turned out the multiplicities are not always one. This was also shown earlier by Knapp and Zuckerman in (12). Gelbart and Knapp gave in (5) a de- scription of irreducible constituents of the restriction to SL(w, F) of the unitary principal series representations of GL(n, F). Their pa- per (6) is based on two working hypotheses, the second of them is the multiplicity one of the restriction to SL(«,/ Γ) of irreducible represen- tations of GL(/ι, F). Bernstein showed in (1) that any parabolically

The MIC-Kepler problem and its symmetry group for positive energies both in classical and quantum mechanics

The symmetry of the Kepler problem has been well known in classical as well as quantum mechanics on the level of Lie algebra, while little is known of global symmetry. In previous papers, the MIC-Kepler problem was introduced, which is the Kepler problem along with a centrifugal potential and Dirac's monopole field. This system was shown, in the negative- (respectively, zero-) energy cases, to admit the same symmetry groupSO(4) (respectively, the Euclidean motion groupE(3)) as the Kepler problem does in classical theory, and to carry all the irreducible representations ofSU(2)×SU(2) (respectively,R3blSU(2)), the double cover ofSO(4) (respectively,E(3)), in quantum theory. This paper is a continuation of the previous ones and intended for the study of the symmetry group in the positive-energy case. In classical theory, the symmetry group of the MIC-Kepler problem of positive energy proves to be the proper Lorentz groupSO+ (1,3) acting on the positive-energy manifold diffeomorphic toR3×S2. In quantum theory, the quantised MIC-Kepler problem with positive energy, assigned by a positive numbers and an integerm, turns out to carry a unitary irreducible representation ofSL(2,C), the double cover ofSO+(1,3), in a Hilbert space, which is isomorphic with the space ofL2-cross-sections in the complex line bundleLm associated with the principalS1 bundleS3→S2. These representations ofSL(2,C) are equivalent to the principal series representations ofSL(2,C).

Matrix element expansion of a spin wavefunction

An expansion of the wavefunction for a free, massive particle with spin is obtained in terms of the matrix elements for the unitary, irreducible representations of SL(2,c) by a method based on the theory of induced representations. It is further shown that this expansion is equivalent to the Shapiro integral transformation for the wavefunction.

Representations of sl(3, IR) with half-integral spin bands

A series of irreducible representations of sl(3,IR) is constructed with half-integral spin bands which parallels the principal series of SL(3,IR). While the principal series for SL(3,IR) was shown by Weaver and Biedenharn to be applicable to even–even deformed nuclei, the series constructed here applies to odd mass nuclei. It is shown how parity is included in an SL(3,IR) algebraic model. A second (cuspidal parabolic) series of representations of SL(3,IR) is constructed, which is required for the harmonic analysis of wave functions on SL(3,IR), a necessary preliminary to the application of the results of the SL(3,IR) model in a full microscopic treatment of deformed nuclei. It is shown that the cuspidal parabolic series has undesirable properties for the description of collective states in rare-earth nuclei.

An Irreducible Representation of sl(2)

In a recent paper [1] MM. Arnal and Pinczon have classified all complex irreducible representations (ρ, V) of sl(2) having the property (P) that there exists a non-zero element x∈sl(2) such that ρ(x) admits an eigenvalue. It is the purpose of this note to demonstrate, by example, that there exist irreducible representations of sl(2) which do not have property (P). As usual, we consider sl(2) embedded in its universal enveloping algebra U and identify the representations of sl(2) and U.