By Michel Brion (auth.), Abraham Broer, A. Daigneault, Gert Sabidussi (eds.)

The 12 lectures offered in *Representation Theories and Algebraic**Geometry* specialise in the very wealthy and robust interaction among algebraic geometry and the illustration theories of varied sleek mathematical buildings, reminiscent of reductive teams, quantum teams, Hecke algebras, constrained Lie algebras, and their partners. This interaction has been generally exploited in the course of fresh years, leading to nice growth in those illustration theories. Conversely, an exceptional stimulus has been given to the advance of such geometric theories as D-modules, perverse sheafs and equivariant intersection cohomology.

the variety of themes coated is large, from equivariant Chow teams, decomposition sessions and Schubert kinds, multiplicity loose activities, convolution algebras, normal monomial concept, and canonical bases, to annihilators of quantum Verma modules, modular illustration idea of Lie algebras and combinatorics of illustration different types of Harish-Chandra modules.

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The homogeneous polynomial function Xl ... Xnex X = det(TxX)exX =: JxX of degree n - dim(X) is the Joseph polynomial introduced in [35] in relation to representation theory, see also [9]. In the example of the Hag manifold, each fixed point x is attractive, and where U- c G is the unipotent subgroup normalized by T with root set CP+. Moreover, the cone U x is the image ofthe positive Weyl chamber under x-I E W. Because Nx,w is contained in U:;, the product of all roots in cP+ n x(cp+) is adenominator for exNx,w.

As a consequence, we have an isomorphism (by Proposition 1) HT(X T )W ~ II SWj . m j=l We now describe the image of Ha(X) ~ HT(X)W in HT(XT)W under restrietion to fixed points iT. Theorem 9 Let X be a compact multiplicity-/ree space under a compact connected Lie group G. Then, with notation as above, the algebra Ha(X) is isomorphie via iT to the subalgebra 01 sm consisting 01 all m-uples (11, ... ; is in SWj, and 2) Ij == w(fk) (modAj - W(Ak)) whenever W E Wand the segment [Aj,W(Ak)] is a component 01 J-t(Xd n t*.

For equidimensional X, we have Aä(X) = A~m(x)_n(X), We now list some properties of equivariant Chow groups; let us mention first that they satisfy the functorial properties of usual Chow groups, and that each linearized vector bundle has equivariant Chern classes. Equivariant eohomology and equivariant interseetion theory 19 If Xis smooth, then the same holds for eaeh X XG U. It follows that Aa(X) is a graded ring for the intersection produet. In partieular, Aa(pt) is a graded ring. For arbitrary X, the projeetion X x U -+ U deseends to a flat map Px : X XG U -+ UjG.