一般线性李超代数及其超群的无穷小子群的表示

在索特征代数闭域上考虑一般线性李超代数gl(m |n)的限制表示与超群GL(m | n)的有理表示以及它们的关系.主要结果为: (1)对gl(m | n)的不可约限制表示进行分类,其中某些单模恰是Kac-模.类似于复数域情形,给出了Kac-模不可约的充要条件; (2)当m≠n(mod p)以及p≥2h-2(h=max{m,n})时,gl(m | n)的限制投射模可以被提升为有理GL(m | n)-模,并且证明了不可约表示的投射覆盖具有Z-滤过,即滤过中的每个子商同构于"baby Vlerma模";(3)得到了一般线性超群G=GL(m | n)的r阶nobenius核的反转公式,它反映了单Gγ-模的投射覆盖的Z-滤过重数与广义baby Verma模的合成因子效之间的关系.

On Representations of gl(m | n) and Infinitesimal Subgroups of GL(m | n)

Let k be an algebraically closed field of characteristic p > 2. In this paper, the authors study the restricted representations of the general linear Lie superalgebras gl(m | n),and take the connection into account with rational representations of the corresponding algebraic supergroup GL(m | n). The main results include (1) The iso-classes of restricted irreducible modules for gl(m | n) are easily parameterized. Some of those modules are just Kac-modules, for judgement of which a necessary and sufficient condition is given,parallel to the case in the field of complex number; (2) The restricted projective modules of gl(m | n)-module can be lifted to a rational GL(m | n)-module when m ≠ n (modp) and p ≥ 2h - 2 (h = max{m, n}). Furthermore, such projective modules have Z-filtrations,i.e., each subquotient of such a filtration is isomorphic to some "baby Verma modules";(3) A reciprocity formula for the rth Frobenius kernels Gγ of G is obtained, which reflects the relation between the multiplicities in such a filtration of the projective cover of simple Gγ-modules and the composite numbers of baby Verma modules.