The Study Of Nilpotent Orbits For A Class Of Enhanced Reductive Lie Algebras And Their Related Intersection Cohomology Theory

Enhanced reductive algebraic group and enhanced reductive Lie algebra are the extension of the reductive group and reductive Lie algebra,which are no longer the reductive case.Their representation theory is more complicated but close to that of the reductive case.They play an important role in the representation theory,especially in prime field.In addition,the authors give the preliminary study for the general case of semi-reductive algebraic group and semi-reductive Lie algebra.Based on this,it is significant to carry out systematic and in-depth research in this field.In this paper,we mainly give the study of the nilpotent orbits theory and the corresponding geometric properties of a special and classical semi-reductive Lie algebra arised from the linear general algebraic group.The main results are as follows.(1)Suppose V be a vector space of dimension n over the algebraically closed field k,G=GL(V)and g=gl(V).We define the corresponding enhanced reductive algebraic group G=G × V and enhanced reductive Lie algebra g=Lie(G)=g⊕ V.At the same time,denote the set of bipartitions of n by Qn.We constuct the one-one correspondence between the set of nilpotent orbits on enhanced reductive Lie algebra g=gl(V)⊕ V and the set Qn/～={(λ-(1)k;(1)k)E Qn|k∈Z≥0｝.(2)We present a resolution of sigularities of the G-orbit closures and obtain a different result on the codimension of G-orbits in its Zariski topology closure from the condition on the redutive Lie algebras.(3)The third result is the G-module stuture of functions on the first class of G-orbits(theorem 4.5.1)and extend the theory of redutive Lie algebras.(4)We obtain the intersection cohomology G-decomposition theroem on the nilpotent cone of enhanced reductive Lie algebra g.(5)The last result is the Jantzen conjecture on g about the support varieties of Weyl modules.We prove its correctness for g on the good characteristic of algebraically closed field k.The content of this theorem is that the cohomology support variety of Weyl module HG0(λ)agrees on the closure of some G-orbit:Vh((HG0(λ)))=O(π(λ)t;(0)).This is a generalization of the related results on the general linear groups of Jantzen.