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Malliavin calculus for stable processes on homogeneous groups

Piotr Graczyk (1991)

Studia Mathematica

Let μ t t > 0 be a symmetric semigroup of stable measures on a homogeneous group, with smooth Lévy measure. Applying Malliavin calculus for jump processes we prove that the measures μ t have smooth densities.

Maslov indices on the metaplectic group M p ( n )

Maurice De Gosson (1990)

Annales de l'institut Fourier

We use the properties of M p ( n ) to construct functions μ : M p ( n ) Z 8 associated with the elements of the lagrangian grassmannian Λ (n) which generalize the Maslov index on Mp(n) defined by J. Leray in his “Lagrangian Analysis”. We deduce from these constructions the identity between M p ( n ) and a subset of S p ( n ) × Z 8 , equipped with appropriate algebraic and topological structures.

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 { T H G } of compact subsets, there exists η > 0 such that # [ e ] Γ T = ( T ) + O ( ( T ) 1 - η ) for an explicit measure 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.

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