On Mirabolic D-modules
On Mirabolic D-modules
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DOI：
10.1093/imrn/rnp216
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发表时间：
2010-01-01
影响因子：
1
通讯作者：
Ginzburg, Victor
中科院分区：
文献类型：
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作者：
Finkelberg, Michael;Ginzburg, Victor
设代数群G作用于具有有限多个轨道的连通代数流形X上。对于任意Harish-Chandra对(D，G)，其中D是X上的扭曲微分算子束，我们构造了由李代数g=LiE G生成的D的左理想DG子集，则D/DG是完整D-模，它对X中唯一的Zariski开稠密G-轨道的限制是G-等变局部系统.我们证明了D-模D/DG在一定(相当严格的)条件下同构于该局部系统到X的直接映象的一个判据。我们将这个判据应用于群G=SLN在X=B×B×Pn-1上对角作用的特殊情况，其中B表示SLN的标志流形。通过一对类似于Lusztig特征标理论的伴随函子(CH，HC)，我们进一步将BxBxPn-1上的D-模与笛卡尔积SLn×Pn-1上的D-模联系起来。本文的第二个重要结果给出了这些函子的一个显式刻画，证明了函子HC给出了Mirabolic D-模的交换范畴上的一个精确函子。
Let an algebraic group G act on X, a connected algebraic manifold, with finitely many orbits. For any Harish-Chandra pair (D, G) where D is a sheaf of twisted differential operators on X, we form a left ideal Dg subset of D generated by the Lie algebra g = Lie G. Then, D/Dg is a holonomic D-module, and its restriction to a unique Zariski open dense G-orbit in X is a G-equivariant local system. We prove a criterion saying that the D-module D/Dg is isomorphic, under certain (quite restrictive) conditions, to a direct image of that local system to X. We apply this criterion in the special case of the group G = SLn acting diagonally on X = B x B x Pn-1, where B denotes the flag manifold for SLn. We further relate D-modules on B x B x Pn-1 to D-modules on the Cartesian product SLn x Pn-1 via a pair (CH, HC), of adjoint functors analogous to those used in Lusztig's theory of character sheaves. A second important result of the paper provides an explicit description of these functors, showing that the functor HC gives an exact functor on the abelian category of mirabolic D-modules.