Representation Theories and Algebraic Geometry by Michel Brion (auth.), Abraham Broer, A. Daigneault, Gert

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

The 12 lectures offered in Representation Theories and AlgebraicGeometry specialize in the very wealthy and strong interaction among algebraic geometry and the illustration theories of varied sleek mathematical buildings, comparable to reductive teams, quantum teams, Hecke algebras, limited Lie algebras, and their partners. This interaction has been generally exploited in the course of contemporary years, leading to nice development in those illustration theories. Conversely, a very good stimulus has been given to the advance of such geometric theories as D-modules, perverse sheafs and equivariant intersection cohomology.
the diversity of issues lined is broad, from equivariant Chow teams, decomposition periods and Schubert types, multiplicity loose activities, convolution algebras, ordinary monomial idea, and canonical bases, to annihilators of quantum Verma modules, modular illustration thought of Lie algebras and combinatorics of illustration different types of Harish-Chandra modules.

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Moreover, the number of subsets X T ' is finite (this can be seen by linearizing the action of Taround fixed points). Thus, it is enough to check that each J-t(GX T') n t* is a finite union of segments with ends in J-t(X T ). But and J-t(X T') is contained in (g*)T' = GT't*, whence So we can replace (G, X) by (GT ' /T' , Y) where Y is a component of X T ' , and thus we can assume that the rank of G is one. Then J-t(X) n t* is a segment, or a union of two segments with ends in J-t(X T ) [34]. 0 Observe that W acts on X T and that both J-t(X T ) and J-t(Xd nt* are W -invariant subsets of t*.

Thus, A;;(X) is isomorphie to A;r (X). Our latter remark reduees many questions on equivariant Chow groups to the ease of reduetive groups. From now on, we assurne that G is reduetive and eonneeted; we denote by T c Gamaximal torus, by W its Weyl group and by B a Borel subgroup of G eontaining T. Let Sz (resp. S) be the symmetrie algebra over the integers (resp. over the rationals) of the eharaeter group B(T) ~ B(B). Then we have the following analogue of Proposition 1, due to Edidin and Graham [21], [22].

Then we have the following analogue of Proposition 1, due to Edidin and Graham [21], [22]. 20 M. Brion Theorem 10 Notation being as above, the graded ring AT(pt) is isomorphie to Sz. Moreover, the map Aa(pt) -+ Ar(pt) is injeetive over Q and identifies Aa{pt)

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