Index above, 85 addable i-node, 76 adjoint representation, 2 aﬃne Hecke algebra, 105 aperiodic, 139 bar involution, 47 bar involution of V (λ), 74 below, 85 Bernstein presentation, 106 canonical basis, 48 Cartan matrix, 17 Cartan subalgebra, 9 Casimir element of Uv( s l2), 25 Casimir operator, 40 coproduct, 6 crystal basis, 27, 46 crystal graph, 27 cyclic quiver, 123 decomposition number, 115 definition by generators and relations, 4 dimension vector, 123 dominance ordering, 113 dominant weight, 40 enveloping algebra, 3 free Lie algebra, 15 fundamental weight, 40 G-equivariant sheaf, 134 Hecke algebra, 97 highest weight module, 40 highest weight vector, 40 ideal of a Lie algebra, 2 i-good, 87 i-induction, 100, 118 i-normal, 87 integrable Uv-module, 23 integral multisegment, 141 intersection cohomology complex, 134 i-restriction, 100, 119 Jacobi identity, 2 Kac-Moody Lie algebra, 18 Kashiwara algebra, 42 Kashiwara operator, 27, 46 Kleshchev multipartition, 94 Kostant-Lusztig form, 47 ladder, 98 length, 105 Lie algebra, 2 Lie algebra defined by generators and rela- tions, 16 Lie algebra homomorphism, 2 Lie group, 1 Lie subalgebra, 2 LLT conjecture, 97 modular system with parameters, 111 Morita equivalence, 117 multipartition, 85 multisegment, 125 node, 75 partition, 75 perverse sheaf, 134 primitive vector, 40 progenerator, 117 pullback, 127 pushforward, 127 quantum algebra of type Ar−1, 13 quantum algebra of type A (1) r−1 , 21 quantum Casimir operator, 40 realization of a Cartan matrix, 17 reduced expression, 105 removable i-node, 76 representations of a Lie group, 1 representation of a Lie algebra, 2 representation of a quiver, 123 residue of a segment, 125 Ringel-Hall algebra, 133 root, 9, 18 root space decomposition, 9, 18 root space decomposition of Uv, 39 root system, 9 r-residue, 75, 85 r-restricted, 98 157

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