A Linkage Atlas On Baby Verma Modules Of Subregular Nilpotent Representations Of Lie Algebras Of Type A_n

In this thesis, we focus our research interest mainly on the baby Verma mod-ules with x subregular nilpotent for the Lie algebras of type An over an algebraically closed field of characteristic p, and on the Hom spaces between these baby Verma modules. They are objects of great significance in the representation theory of Lie algebras. Jantzen has made a detailed investigation on this category. With his initia-tive notion the Uχ(g)-To module and Premet's excellent methord, he gives a delicate analysis and construction on the composition factors of the baby Verma modules on the first dominant alcove, and finally obtain a complete description of irreducible modules in this category (cf. [J2]). Based on Jantzen's work, we prove that in the Uχ(g) module category withχa subregular nilpotent p-character, the Hom spaces between any two baby Verma modules in the same given block are always nonzero, which reveals a complete linkage atlas on baby Verma modules of subregular nilpo-tent representations of Lie algebras of type An in prime characteristics. This is a new result in the modular representation theory of reductive Lie algebras.The main results in this thesis are listed as follows:1. Based on Jantzen's work (cf. [J2],2.1-2.14), we precisely list the composition series and the composition factors for the baby Verma modules Zχ(λκ) with arbitrary positive integer k.2. As Uχ(g)-To modules, we take into account the Hom spaces between the adjacent baby Verma modules in the sequence Zχ(λκ),κ=0,1,…,n. We give sufficient conditions for these Hom spaces being nonzero.3. As Uχ(g) modules, we consider the Horn spaces between Zχ(λi) and Zχ(λj) for arbitrary i,j∈{0,1,…,n}. We firstly reduce this problem to the Hom spaces between Zχ(λo) and Zχ(λi), then prove that HomUχ(g)(Zχ(λo),Zχ(λi)) and HomUχ(g)(Zχ(λi),Zχ(λo)) are not zero for the weightλo which lies in arbitrary sites in the closure of the first dominant alcove. As a consequence, we obtain the result that HomUχ(g)(Zχ(λi), Zχ(λj)) are not zero for arbitrary i,j∈{0,1,…, n}, and ultimately reveals the strong linkage relationship between any two baby Verma modules in the same block.