Short Time Behavior of Hermite Functions on Compact Lie Groups

We study the heat kernel on a compact Lie group. In this work, we will use structure theory of semisimple Lie algebra to find equivalent forms of the heat kernel on a general compact Lie group.

Over the \((1, n)\)-dimensional real superspace, we consider the Lie superalgebra \(\mathcal {K}(n)\) of contact vector fields on \({\mathbb {R}}^{1|n}\). We classify \(\mathfrak {osp}(n|2)\)-invariant linear differential operators acting on the superspaces of weighted densities. This result allows us to compute the first differential cohomology of \(\mathcal {K}(n)\) with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra \(\mathfrak {osp}(n|2)\). We explicitly give 1-cocycles spanning these cohomology spaces.

Relative BGG sequences; II. BGG machinery and invariant operators

A new systematic method to find the relative invariant differentiation operators is developed. We incorporate this new approach with Lie's infinitesimal method to study the general class y′′′ = f(x, y, y′, y′′) under general point equivalence transformations in the generic case.As a result, all third‐order differential invariants, relative and absolute invariant differentiation operators are determined for third‐order ODEs y′′′ = f(x, y, y′, y′′), which are not quadratic in the second‐order derivative. These relative invariant differentiation operators are used to determine the fourth‐order differential invariants and absolute invariant differentiation operators in a degenerate case of interest. As an application, invariant descriptions of all the canonical forms in the complex planewith four infinitesimal symmetries for third‐order ODEs y′′′ = f(x, y, y′, y′′), which are not quadratic in the second‐order derivative, are provided. Copyright © 2015 John Wiley & Sons, Ltd.

Des que estava cursant els primers anys de la Llicenciatura en Matemtiques he tingut la illusi de fer una tesi doctoral en aquesta disciplina, com deia Gauss, la reina de les cincies. Segurament aquesta illusi ha sorgit de la curiositat que les Matemtiques i la cincia

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n, n). Our choice of these algebras is motivated by the fact that for n = 2 this is the conformal algebra of 4-dimensional Minkowski space-time. Furthermore for general n these algebras belong to a narrow class of algebras, which we call “conformal Lie algebras”, which have very similar properties to the conformal algebras of n 2-dimensional Minkowski space-time. We give the main multiplets of indecomposable elementary representations for n = 2, 3, 4, including the necessary data for all relevant invariant differential operators.

In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact group SO(p, 9 − p) (for p = 5, 6) we give the classification multiplets of indecomposable elementary representations induced from a new (relative to earlier considerations) choice of parabolic subgroup MAN so that the factor-group G/M is an Einstein manifold. This classification includes the data for the relevant invariant differential operators.

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so∗(10). We use the maximal Heisenberg parabolic subalgebra P=M⊕A⊕N with M=su(3,1)⊕su(2)≅so∗(6)⊕so(3). We give the main and the reduced multiplets of indecomposable elementary representations. This includes the explicit parametrization of the intertwining differential operators between the ERS. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras so(p,q) (with p+q=10, p≥q≥2) with maximal Heisenberg parabolic subalgebra: P′=M′⊕A′⊕N′, M′=so(p−2,q−2)⊕sl(2,IR), M′C≅MC.

Over the (1,n)-dimensional real superspace, n>1, we classify K(n)-invariant binary differential operators acting on the superspaces of weighted densities, where K(n) is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of K(n) with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities—a superization of a result by Feigin and Fuchs [“Homology of the Lie algebras of vector fields on the line,” Funct. Anal. Appl. 14, 201 (1980)]. We explicitly give 1-cocycles spanning these cohomology spaces.

Ricci-corrected derivatives and invariant differential operators

Spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces

Evolution of curvature invariants and lifting integrability

Let G = exp ⁡ ( g ) G = \operatorname {exp}(\mathfrak {g}) be a connected and simply connected real nilpotent Lie group with Lie algebra g \mathfrak g , H = exp ⁡ ( h ) H = \operatorname {exp}(\mathfrak {h}) an analytic subgroup of G G with Lie algebra h \mathfrak h , χ \chi a unitary character of H H and τ = ind H G χ \tau = \text {ind}_H^G \chi the monomial representation of G G induced by χ \chi . Let D τ ( G / H ) D_{\tau }(G/H) be the algebra of the G G -invariant differential operators on the fiber bundle over the base space G / H G/H associated to the data ( H , χ ) (H,\chi ) . We prove the polynomial conjecture due to Corwin-Greenleaf stating that if τ \tau is of finite multiplicities, the algebra D τ ( G / H ) D_{\tau }(G/H) is isomorphic to the algebra C [ Γ τ ] H {{\mathbb C}[{\Gamma }_{\tau }]}^H of the H H -invariant polynomial functions on the affine subspace Γ τ = { ℓ ∈ g ∗ ; ℓ | h = − − 1 d χ } {\Gamma }_{\tau } = \{\ell \in {\mathfrak g}^*; {\ell }|_{\mathfrak h} = -\sqrt {-1}d\chi \} of g ∗ {\mathfrak g}^* . In this case, we show that any non-zero element of D τ ( G / H ) D_{\tau }(G/H) admits a fundamental tempered solution.

Let G 0 = K p be the Cartan motion group associated with a noncompact semisimple Riemannian symmetric pair (G, K) .Let a be a maximal abelian subspace of p and let p = a + q be the corresponding orthogonal decomposition.A flat horocycle in p is a G 0 -translate of q .A conical distribution on the space 0 of flat horocycles is an eigendistribution of the algebra D( 0 ) of G 0 -invariant differential operators on 0 which is invariant under the left action of the isotropy subgroup of G 0 fixing q .We prove that the space of conical distributions belonging to each generic eigenspace of D( 0 ) is onedimensional, and we classify the set of all conical distributions on 0 when G/K has rank one.We also consider the question of the irreducibility of the natural representation of G 0 on the eigenspaces of D( 0 ) .

Asymptotic Behavior of Spherical Functions on Semisimple Symmetric Spaces