Invariant differential operators on a real semisimple Lie algebra and their radial components

Abstract

Let S ( g C ) S({\mathfrak {g}_C}) be the symmetric algebra over the complexification g C {\mathfrak {g}_C} of the real semisimple Lie algebra g \mathfrak {g} . For u ϵ S ( g C ) , ∂ ( u ) u\;\epsilon \;S({\mathfrak {g}_C}),\partial (u) is the corresponding differential operator on g \mathfrak {g} . D ( g ) \mathcal {D}(\mathfrak {g}) denotes the algebra generated by ∂ ( S ( g C ) ) \partial (S({\mathfrak {g}_C})) and multiplication by polynomials on g C {\mathfrak {g}_C} . For any open set U ⊂ g , Diff ( U ) U \subset \mathfrak {g},{\text {Diff}}(U) is the algebra of differential operators with C ∞ {C^\infty } -coefficients on U. Let h \mathfrak {h} be a Cartan subalgebra of g , h ′ \mathfrak {g},\mathfrak {h}’ the set of its regular points and π = Π α ϵ P α \pi = {\Pi _{\alpha \epsilon P}}\alpha , P some positive system of roots. Let W = ( h ′ ) G W = {(\mathfrak {h}’)^G} , G the connected adjoint group of g \mathfrak {g} . Harish-Chandra showed that, for each D ϵ Diff ( W ) D\;\epsilon \;{\text {Diff}}(W) , there is a unique differential operator δ h ′ ( D ) \delta {’_\mathfrak {h}}(D) on h ′ \mathfrak {h}’ such that ( D f ) | h ′ = δ h ′ ( D ) ( f | h ) (Df){\left | {_\mathfrak {h}’ = \delta {’_\mathfrak {h}}(D)(f} \right |_\mathfrak {h}}) for all G-invariant f ϵ C ∞ ( W ) f\epsilon \;{C^\infty }(W) , and that if D ϵ D ( h ) D\;\epsilon \mathcal {D}(\mathfrak {h}) , then δ h ′ ( D ) = π − 1 ∘ D ¯ ∘ π \delta {’_\mathfrak {h}}(D) = {\pi ^{ - 1}} \circ \bar D \circ \pi for some D ¯ ϵ D ( g ) \bar D\epsilon \mathcal {D}(\mathfrak {g}) . In particular ∂ ( u ) ¯ = ∂ ( u | h ) , u ϵ S ( g C ) \overline {\partial (u)} = \partial (u{|_\mathfrak {h}}),u\;\epsilon \;S({\mathfrak {g}_C}) and invariant. We prove these results by different, yet simpler methods. We reduce evaluation of δ h ′ ( ∂ ( u ) ) ( u ϵ S ( g C ) \delta {’_\mathfrak {h}}(\partial (u))\;(u\;\epsilon \;S({\mathfrak {g}_C}) , invariant) via Weyl’s unitarian trick, to the case of compact G. This case is proved using an evaluation of a family of G-invariant eigenfunctions on: \[ π ( H ) π ( H ′ ) ∫ G exp ⁡ B ( H x , H ′ ) d x = c ∑ S ϵ W ( g C , h C ) ϵ ( s ) exp ⁡ B ( s H , H ′ ) , H , H ′ ϵ g , c > 0. \pi (H)\pi (H’)\int _G {\exp B({H^x},H’)dx = c\sum \limits _{S\epsilon W({\mathfrak {g}_C},{\mathfrak {h}_C})} {\epsilon (s)\exp B(sH,H’),H,H’\epsilon \;\mathfrak {g},c > 0.} } \] For G-invariant D ϵ D ( g ) D\;\epsilon \;\mathcal {D}(\mathfrak {g}) , we prove π − 1 ∘ δ ′ ( D ) ∘ π ϵ D ( h ) {\pi ^{ - 1}} \circ \delta ’(D) \circ \pi \;\epsilon \;\mathcal {D}(\mathfrak {h}) using properties of derivations E → [ ∂ ( u ) , E ] E \to \left [ {\partial (u),E} \right ] of D ( g ) \mathcal {D}(\mathfrak {g}) induced by ∂ ( u ) ( u ϵ S ( g C ) ) \partial (u)\;(u\;\epsilon \;S({\mathfrak {g}_C})) and of the algebra of polynomials on h C {\mathfrak {h}_C} invariant under the Weyl group.