本学位论文主要研究在单位根处的混合量子群的范畴O. 混合量子群的定义最初由Gaitsgory提出，它是在三角分解下由Lusztig量子群的正部分及De Concini–Kac量子群的负部分组成的量子代数. 其范畴O可视作BGG范畴O的量子群版本. 本文主要包含三个部分:（1）混合量子群范畴O的基本性质;（2）该范畴的中心与仿射旗簇上同调环的同构;（3）Steinberg块与本原块的凝聚层模型.我们首先建立了该范畴O的若干基本性质，包括Jantzen滤过及Jantzen求和公式、链接原理及块分解、单模与投射模的构造及BGG互反律等. 我们也研究了该范畴O的形变. 本文的主要结果之一是证明了混合量子群范畴O的块的中心与Langlands对偶群的相应仿射旗簇的奇异上同调环的代数同构. 该范畴O的中心与仿射旗簇上同调的联系最初是Bezrukavnikov，Boixeda-Alvarez，单芃和Vasserot在研究小量 子群中心的几何实现时发现并预测为同构的. 我们这一结论可视作由 Soergel 得到 的关于 BGG 范畴O的中心的经典结论的量子群版本. 我们先考察Steinberg块，利用其凝聚层模型，我们给出其到仿射Grassmann簇上可构层范畴的导出等价，并利用该范畴层面的结论，证明Steinberg块的中心与仿射Grassmann簇上同调的同构. 随后，我们对本原块证明该结论，其中，我们构造本原块中的“大投射模的自同态环”，并将之比较于本原块的中心及仿射旗簇的上同调. 最后，我们利用平移函子的迹，来比较本原块及任意块的中心，从而把结果推广到范畴?的所有块. 我 们亦建立了形变版本的结论，即形变范畴O的块的中心与相应仿射旗簇的等变上同调的代数同构. 本文的主要结果之二是构造了Steinberg块与本原块的凝聚层模型. 具体地，对Steinberg块，我们建立其与Springer消解上的等变凝聚层范畴的等价. 对本原块， 我们利用非交换Springer消解构造其凝聚层模型. 我们构造本原块及其凝聚层模型 上的“Soergel 函子”，并基于Steinberg块的等价来联系二者. 我们的结论说明了混 合量子群范畴?的本原块是一个仿射Hecke范畴.

This thesis aims to study the category O of hybrid quantum groups at roots of unity. Firstly introduced by Gaitsgory, the hybrid quantum group admits a triangular decomposition, whose positive part is given by Lusztig’s version and whose negative part is given by De Concini–Kac version. Its category O can be viewed as the quantum analogue of BGG category O. There are three main parts in this thesis: (1) basic properties of quantum category O; (2) isomorphism between center of quantum category O and cohomology ring of affine flag varieties; (3) coherent models for the Steinberg block and the principal block. We firstly developed some basic properties of quantum category O, including the Jantzen filtration and Jantzen sum formula, linkage principle and block decomposition, construction of simple modules and projective modules, and the BGG reciprocity. The deformation of quantum category O is also discussed. The first main result is an algebra isomorphism between the center of each block of quantum category O and the singular cohomology ring of certain affine flag variety associated to the Langlands dual group. The relation between center of quantum category O and cohomology of affine flag varieties was firstly observed and was predicted to be an isomorphism by Bezrukavnikov，Boixeda-Alvarez，Shan and Vasserot in their study of geometric realization of center if small quantum groups. This result can be viewed as the quantum analogue of a classical result for center of BGG category O obtained by Soergel. We firstly establish the result for Steinberg block by a categorical enhancement that it is derived equivalent to the category of certain constructible sheaves on affine Grassmannian, by making use of its coherent model. Then we prove the result for principal block, by comparing the center and cohomology to the “endomorphism algebra of big projective object”. Finally, by using the trace of translation functors to compare the centers between principal block and arbitrary block, we deduce the result for all the blocks. We also established a deformed version of this result, namely the center of deformed category O is isomorphic to the equivariant cohomology of certain affine flag varieties. Second main result of this thesis is the construction of coherent models for the Steinberg block and the principal block. More precisely, for Steinberg block, we established an equivalence to the category of equivariant coherent sheaves on Springer resolution. For principal block, we constructed its coherent model using the non-commutative Springer resolution. We established “Soergel functors” for principal block and its coherent model, and related these two categories based on the equivalence for Steinberg block. Our result shows that the principal block is a version of the affine Hecke category.