Degenerate group of type \(A\): representations and flag varieties (Q482594)

Let \(g\) be a simple Lie algebra with Borel subalgebra \(b\). The irreducible highest weight representations of such \(g\) play a fundamental role in algebraic and geometric Lie theory. In particular, generalized flag varieties for the corresponding simple Lie group \(G\) can be realized as \(G\)-orbits in the projectivizations of these modules. The author considers representations of some special solvable Lie algebras \(g^a\) (and corresponding solvable Lie groups), which are contractions (degenerations) of simple Lie algebras \(g\). The Lie algebra \(g^a\) is a semidirect sum of the subalgebra \(b\) and the abelian ideal \(g/b\) with the adjoint action of \(b\) on \(g/b\). In this paper the following question is investigated: what are the analogues of the finite-dimensional representations of \(g\) and of the flag varieties in the degenerate situation? The Lie algebras \(g^a\) have much more representations than \(g\). Some constructions (in particular, analogs of higher weight representation and flag variety) and results to answer these questions are given. It is shown that the degenerate flag varieties and their desingularizations \(R_n\) can be obtained via their construction. It is proved that the coordinate ring of \(R_n\) is isomorphic to the direct sum of duals of the highest weight representations of the degenerate group.