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Matrix coefficients, counting and primes for orbits of geometrically finite groups

Amir Mohammadi, Hee Oh (2015)

Journal of the European Mathematical Society

Let G : = SO ( n , 1 ) ∘ and Γ ( n - 1 ) / 2 for n = 2 , 3 and when δ > n - 2 for n ≥ 4 , we obtain an effective archimedean counting result for a discrete orbit of Γ in a homogeneous space H ∖ G where H is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family { ℬ T ⊂ H ∖ G } of compact subsets, there exists η > 0 such that # [ e ] Γ ∩ ℬ T = ℳ ( ℬ T ) + O ( ℳ ( ℬ T ) 1 - η ) for an explicit measure ℳ on H ∖ G which depends on Γ . We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ in arithmetic settings....

Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory

F. Alberto Grünbaum, Inés Pacharoni, Juan Alfredo Tirao (2005)

Annales de l’institut Fourier

The main purpose of this paper is to present new families of Jacobi type matrix valued orthogonal polynomials obtained from the underlying group S U ( n ) and its representations. These polynomials are eigenfunctions of some symmetric second order hypergeometric differential operator with matrix coefficients. The final result holds for arbitrary values of the parameters α , β > - 1 , but it is derived only for those values that come from the group theoretical setup.

Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group

Ewa Damek, Andrzej Hulanicki (1991)

Studia Mathematica

On the domain S_a = {(x,e^b): x ∈ N, b ∈ ℝ, b > a} where N is a simply connected nilpotent Lie group, a certain N-left-invariant, second order, degenerate elliptic operator L is considered. N × {e^a} is the Poisson boundary for L-harmonic functions F, i.e. F is the Poisson integral F(xe^b) = ʃ_N f(xy)dμ^b_a(x), for an f in L^∞(N). The main theorem of the paper asserts that the maximal function M^a f(x) = sup{|ʃf(xy)dμ_a^b(y)| : b > a} is of weak type (1,1).

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Igor Moiseev, Yuri L. Sachkov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.

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