We define a partial order on the set No,c̄ of pairs (O, C), where O is a nilpotent orbit and C is a conjugacy class in Ā(O), Lusztig’s canonical quotient of A(O). We then show that there is a unique order-reversing duality map No,c̄ → LNo,c̄ that has certain properties analogous to those of the original Lusztig-Spaltenstein duality map. This generalizes work of E. Sommers.

The integration of the exponential of the square of the moment map of the circle action is studied by a direct stationary phase computation and by applying the DuistermaatHeckman formula. Both methods yield two distinct formulas expressing the integral in terms of contributions from the critical set of the square of the moment map. The cohomological pairings on the symplectic quotient, includin...

Let K be a connected compact Lie group and M a Hamiltonian K-manifold, i.e., a symplectic K-manifold equipped with a moment map μ : M → k := (LieK). A theorem of Kirwan (implicitly in [Ki]) asserts: if M is connected and compact then the level sets of μ are connected. The purpose of this note is to prove such a statement in the category of algebraic varieties. First, we reformulate Kirwan’s the...

BACKGROUND AND PURPOSE A number of studies have associated the adult intelligence quotient with the structure and function of the bilateral parieto-frontal networks, whereas the relationship between intelligence quotient and parieto-frontal network function has been found to be relatively weak in early childhood. Because both human intelligence and brain function undergo protracted development ...

In the previous lecture we proved the Kronecker-Weber theorem: every abelian extension L of Q lies in a cyclotomic extension Q(ζm)/Q. The isomorphism Gal(Q(ζm)/Q) ' (Z/mZ)× allows us to view Gal(L/Q) as a quotient of (Z/mZ)×. Conversely, for each quotient H of (Z/mZ)×, there is a subfield L of Q(ζm) for which H ' Gal(L/Q). We now want make the correspondence between H and L explicit, and then g...

Our main subject is to study the action of the Hecke operators Tl on the Jacobian J of the modular curve X0(N). We consider the Hecke algebra T, i.e. the subring of End(J) generated by the Tl and the involution ω and define the Eisenstein ideal I ⊂ T as the ideal generated by 1 +ω and 1 + l−Tl, l prime 6= N . The Eisenstein quotient of J is the quotient J̃ := J/aJ where a denotes the kernel of t...

In the previous lecture we proved the Kronecker-Weber theorem: every abelian extension L of Q lies in a cyclotomic extension Q(ζm)/Q. The isomorphism Gal(Q(ζm)/Q) ' (Z/mZ)× allows us to view Gal(L/Q) as a quotient of (Z/mZ)×. Conversely, for each quotient H of (Z/mZ)×, there is a subfield L of Q(ζm) for which H ' Gal(L/Q). We now want make the correspondence between H and L explicit, and then g...

In the Lecture 20 we proved the Kronecker-Weber theorem: every abelian extension L of Q lies in a cyclotomic extension Q(ζm)/Q. The isomorphism Gal(Q(ζm)/Q) ' (Z/mZ)× allows us to view Gal(L/Q) as a quotient of (Z/mZ)×. Conversely, for each quotient H of (Z/mZ)×, there is a subfield L of Q(ζm) for which H ' Gal(L/Q). We would like to make the correspondence between H and L explicit, and then ge...

We calculate the character table of the maximal subgroup of the Monster N(3B) ∼= 3 + .2.Suz:2, and also of the group 31+12:6.Suz:2, which has the former as a quotient. The strategy is to induce characters from the inertia groups in 31+12:6.Suz:2 of characters of 3. We obtain the quotient map to N(3B) computationally, and our careful concrete approach allows us to produce class fusions between o...

A brief comment about items (3) and (4). If H is a subgroup of G the inclusion map is an injective homomorphism from H to G . On the other hand, the image of an injective homomorphism α : H −→ G is a subgroup that is isomorphic to H . So the study of injective homomorphisms is (up to isomorphism) the study of subgroups. After studying subgroups, we defined normal subgroup and showed several equ...