Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra (Q2723471)

Let \(\mathfrak g\) be a semisimpe complex Lie algebra with adjoint group \(G\). Let \(S(\mathfrak g^*)\) be the algebra of polynomial functions on \(\mathfrak g\) and \(D(\mathfrak g)\) be the algebra of differential operators on \(\mathfrak g\) with coefficients in \(S(\mathfrak g^*)\). Vie the adjoint action, one has the induced action of the group \(G\) on \(S(\mathfrak g^*)\), \(S(\mathfrak g)\) and \(D(\mathfrak g)\). Let \(S_+(\mathfrak g^*)^G\) be the set of invariant elements without constant term. The nilpotent cone \(N(\mathfrak g)\) of \(\mathfrak g\) is the variety of zeros of the ideal \(S_+(\mathfrak g^*)^G S(\mathfrak g^*)\). Now, let \(\mathfrak g_0\) be a real form of \(\mathfrak g\) with the adjoint group \(G_0\subset G\). Denote by \(Db(\mathfrak g_0)\) the \(D(\mathfrak g)\)-module of distributions on \(\mathfrak g_0\). Then the space of invariant distributions \(Db(\mathfrak g_0)^{G_0}\) is a \(D(\mathfrak g)^G\)-module, containing the submodule of invariant distributions supported on the nilpotent cone. The paper gives the decomposition of the semisimple \(D(\mathfrak g)^G\)-module of invariant distributions on \(\mathfrak g_0\) supported on the nilpotent cone.