Tensor Square of the Minimal Representation of O(p, q)

2007 ◽  
Vol 50 (1) ◽  
pp. 48-55 ◽  
Author(s):  
Alexander Dvorsky
Keyword(s):  
Tensor Product ◽  
Hyperbolic Space ◽  
Irreducible Representations ◽  
Plancherel Measure ◽  
Minimal Representation ◽  
Direct Integral ◽  
Tensor Square ◽  
Real Hyperbolic Space

AbstractIn this paper, we study the tensor product π = σmin ⊗ σmin of the minimal representation σmin of O(p, q) with itself, and decompose π into a direct integral of irreducible representations. The decomposition is given in terms of the Plancherel measure on a certain real hyperbolic space.

scholarly journals HIGHER DEFORMATIONS OF LIE ALGEBRA REPRESENTATIONS II

2020 ◽  
pp. 1-24
Author(s):  
MATTHEW WESTAWAY
Keyword(s):  
Tensor Product ◽  
Algebraic Groups ◽  
Preceding Paper ◽  
Irreducible Representations ◽  
Product Theorem ◽  
Enveloping Algebras ◽  
Frobenius Kernel ◽  
Frobenius Kernels ◽  
Lie Algebra Representations ◽  
Semisimple Algebraic Groups

Steinberg’s tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

TEST MAP AND DISCRETENESS IN SL(2, ℍ)

2018 ◽  
Vol 61 (03) ◽  
pp. 523-533 ◽  
Author(s):  
KRISHNENDU GONGOPADHYAY ◽  
ABHISHEK MUKHERJEE ◽  
SUJIT KUMAR SARDAR
Keyword(s):  
Hyperbolic Space ◽  
Division Ring ◽  
Discreteness Criteria ◽  
Dense Subgroups ◽  
Real Quaternions ◽  
Real Hyperbolic Space

AbstractLet ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).

scholarly journals Isometric immersions in the hyperbolic space with their image contained in a horoball

2001 ◽  
Vol 43 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Ana Lluch
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Riemannian Manifold ◽  
Isometric Immersions ◽  
Special Groups ◽  
The Mean ◽  
Complex Hyperbolic Spaces ◽  
Real Hyperbolic Space ◽  
Sharp Lower Bound

We give a sharp lower bound for the supremum of the norm of the mean curvature of an isometric immersion of a complete Riemannian manifold with scalar curvature bounded from below into a horoball of a complex or real hyperbolic space. We also characterize the horospheres of the real or complex hyperbolic spaces as the only isometrically immersed hypersurfaces which are between two parallel horospheres, have the norm of the mean curvature vector bounded by the above sharp bound and have some special groups of symmetries.

On Springer's correspondence for simple groups of type En (n = 6, 7, 8)

1982 ◽  
Vol 92 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Dean Alvis ◽  
George Lusztig
Keyword(s):  
Irreducible Representation ◽  
Tensor Product ◽  
Algebraic Group ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Complex Numbers ◽  
Unipotent Element ◽  
Reductive Algebraic Group ◽  
Dimension 2 ◽  
Injective Map

Let G be a connected reductive algebraic group over complex numbers. To each unipotent element u ε G (up to conjugacy) and to the unit representation of the group of components of the centralizer of u, Springer (11), (12) associates an irreducible representation of the Weyl group W of G. The tensor product of that representation with the sign representation will be denoted ρu. (This agrees with the notation of (5).) This representation may be realized as a subspace of the cohomology in dimension 2β(u) of the variety of Borel subgroups containing u, where β(u) = dim . For example, when u = 1, ρu is the sign representation of W. The map u → ρu defines an injective map from the set of unipotent conjugacy classes in G to the set of irreducible representations of W (up to isomorphism). Our purpose is to describe this map in the case where G is simple of type Eu (n = 6, 7, 8). (When G is classical or of type F4, this map is described by Shoji (9), (10); the case where G is of type G2 is contained in (11).

scholarly journals Jørgensen’s Inequality and Algebraic Convergence Theorem in Quaternionic Hyperbolic Isometry Groups

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Huani Qin ◽  
Yueping Jiang ◽  
Wensheng Cao
Keyword(s):  
Hyperbolic Space ◽  
Convergence Theorem ◽  
Kleinian Group ◽  
Isometry Groups ◽  
Quaternionic Hyperbolic Space ◽  
Algebraic Convergence ◽  
Real Hyperbolic Space ◽  
Jørgensen’S Inequality ◽  
Algebraic Limit

We obtain an analogue of Jørgensen's inequality in quaternionic hyperbolic space. As an application, we prove that if ther-generator quaternionic Kleinian group satisfies I-condition, then its algebraic limit is also a quaternionic Kleinian group. Our results are generalizations of the counterparts in then-dimensional real hyperbolic space.

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