No abstract

This is a sequel to [25] and [30]. Associated with G := GL n and its rational representation (ρ, M ) over an algebraically closed filed k, we define an enhanced algebraic group G := G ⋉ ρ M which is a product variety GL n × M , endowed with an enhanced cross product. In this paper, we first show that the nilpotent cone N := N(g) of the enhanced Lie algebra g := Lie(G) has finite nilpotent orbits under adjoint G-action if and only if up to tensors with one-dimensional modules, M is isomorphic to one of the three kinds of modules: (i) a one-dimensional module, (ii) the natural module k n , (iii) the linear dual of k n when n > 2; and M is an irreducible module of dimension not bigger than 3 when n = 2. We then investigate the geometry of enhanced nilpotent orbits when the finiteness occurs. Our focus is on the enhanced group G = GL(V ) ⋉ η V with the natural representation (η, V ) of GL(V ), for which we give a precise classification of finite nilpotent orbits via a finite set e P n of so-called enhanced partitions of n = dim V , then give a precise description of the closures of enhanced nilpotent orbits via constructing so-called enhanced flag varieties. Finally, the G-equivariant intersection cohomology decomposition on the nilpotent cone of g along the closures of nilpotent orbits is established.

<abstract><p>Associated with a reductive algebraic group $ G $ and its rational representation $ (\rho, M) $ over an algebraically closed filed $ {\bf{k}} $, the authors define the enhanced reductive algebraic group $ {\underline{G}}: = G\ltimes_\rho M $, which is a product variety $ G\times M $ and endowed with an enhanced cross product in ^{[<xref ref-type="bibr" rid="b5">5</xref>]}. If $ {\underline{G}} = GL(V)\ltimes_{\eta} V $ with the natural representation $ (\eta, V) $ of $ {\text{GL}}(V) $, it is called enhanced general linear algebraic group. And the authors give a precise classification of finite nilpotent orbits via a finite set of so-called enhanced partitions of $ n = \dim V $ for the enhanced group $ {\underline{G}} = {\text{GL}}(V)\ltimes_{\eta}V $ in [<xref ref-type="bibr" rid="b6">6</xref>, Theorem 3.5], We will give another way to prove this classification theorem in this paper. Then we focus on the support variety of the Weyl module for $ {\underline{G}} = {\text{GL}}(V)\ltimes_{\eta}V $, and obtain that it coinsides with the closure of an enhanced nilpotent orbit under some mild condition.</p></abstract>