Representations of Quantum Groups at Roots of Unity and Their Reductions mod p to Algebraic Group Representations

The dissertation consists of two main parts. One part concerns representations of quantum groups at a root of unity. We compute the dimensions of Ext-spaces between two irreducibles in singular blocks. They are expressed explicitly in terms of appropriate coefficients of the (parabolic) Kazhdan-Lusztig polynomials. A parity vanishing property is needed to derive the singular Ext formula from the regular Ext formula. For this, we assume the Kazhdan-Lusztig equivalence between the quantum group representations and certain representations for the corresponding affine Lie algebra, where we have the Koszul grading of Shan-Varagnolo-Vasserot. The other part of the dissertation is on the ``reduction mod p'' method which directly relates representations of a semisimple algebraic group in characteristic p>0 and representations of quantum groups of the same type at roots of unity having orders that are powers of p. We see that the formulas for Ext between quantum group representations obtained above give lower bounds for the dimensions for the Ext between ``reduction mod p'' modules. We then present an example where the Ext in the algebraic group case is strictly greater than the Ext in the quantum case.