Fundamental Solutions of Invariant Differential Operators on a Semisimple Lie Group Ii
نویسندگان
چکیده
Let G be a linear connected semisimple Lie group. We denote by U(g)K the algebra of left invariant differential operators on G that are also right invariant by K, and Z(U(g)K) denotes center of U(g)K . In this paper we give a sufficient condition for a differential operator P ∈ Z(U(g)K) to have a fundamental solution on G. This result extends the same one obtained previously for real rank one Lie groups and groups with only one conjugacy class of Cartan subgroups.
منابع مشابه
Fundamental Solutions of Invariant Differential Operators on a Semisimple Lie Group
Let G be a connected semisimple Lie group of real rank one. We denote by U(g)K the algebra of left invariant differential operators on G right invariant by K, and let Z(U(g)K) be its center. In this paper we give a sufficient condition for a differential operator P ∈ Z(U(g)K) to have a fundamental solution on G. We verify that this condition implies P C∞(G) = C∞(G). If G has a compact Cartan su...
متن کاملInvariant Differential Operators on a Real Semisimple Lie Algebra and Their Radial Components
Let S(g ) be the symmetric algebra over the complexification 9 of the real semisimple Lie algebra g. For u £ S(g ), d(u) is the corresponding differential operator on g. 3)(g) denotes the algebra generated by d(S(g )) and multiplication by polynomials on g . For any open set U C t, Diff (LO is the algebra of differential operators with C°°-coefficients on U. Let t be a Cartan subalgebra of g, f...
متن کاملFourier Analysis on Semisimple Symmetric Spaces
A homogeneous space X = G/H of a connected Lie group G is called a symmetric homogeneous space if there exists an involution σ of G such that H lies between the fixed point group G and its identity component Go . Example 0. For a connected Lie group G′, put G = G′×G′, σ(g1, g2) ) = (g2, g1) and H = G. Then the homogeneous space X = G/H is naturally isomorphic to G′ by the map (g1, g2) 7→ g1g−1 ...
متن کاملOrbit Reduction of Exterior Differential Systems, and group-invariant Variational Problems
For a given PDE system (or an exterior differential system) possessing a Lie group of internal symmetries the orbit reduction procedure is introduced. It is proved that the solutions of the reduced exterior differential system are in one-to-one correspondence with the moduli space of regular solutions of the prolongation of the original system. The isomorphism between the local characteristic c...
متن کاملHarmonic Analysis on Real Reductive Symmetric Spaces
Let G be a reductive group in the Harish-Chandra class e.g. a connected semisimple Lie group with finite center, or the group of real points of a connected reductive algebraic group defined over R. Let σ be an involution of the Lie group G, H an open subgroup of the subgroup of fixed points of σ. One decomposes the elements of L(G/H) with the help of joint eigenfunctions under the algebra of le...
متن کامل