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

ISBN-10: 9048150752

ISBN-13: 9789048150755

ISBN-10: 9401591318

ISBN-13: 9789401591317

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

the diversity of themes coated is large, from equivariant Chow teams, decomposition periods and Schubert kinds, multiplicity unfastened activities, convolution algebras, regular monomial thought, 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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**Sample text**

32 M. Brion Because lP(X) and alllP(X T ') are rational cohomology projective spaces, we have dim(X) = X(lP(X)) = X(lP(X)T) = L X(IP'(X T')) T' Ldim(XT') = Ldim(VT') = dim(V). T' T' Thus, 'Fr is surjective. Let d be its degree, and let dT, be the degree of 'FrT'. Then we have by Lemma 16 exX = deoV = d d d II ex(X). II eo(V) = -TI T' T' T' T' T' T' If, moreover, each X T' is smooth, then we can take each 'FrT' to be the identity. So each dT , is 1, and c = d is an integer. (ii)~(iii) Because exX is homogeneous of degree - dim(X), we obtain dim(X) = L dim(X T ').

Graham, Characteristie classes in the Chow ring, J. Algebraic Geom. 6 (1997), 431-443. [22] D. Edidin and W. Graham, Equivariant interseetion theory, Invent. , to appear. [23] D. Edidin and W. Graham, Loealization in equivariant interseetion theory and Bott's residue formula, Amer. J. , to appear. [24] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer, New York 1994. [25] W. Fulton, Intersection Theory, Springer, New York 1984. [26] W. Fulton, R. MacPherson, F. Sottile and B.

Brion We claim that e x X( w) _ exX(T) - s"(es,,xX(T)) , a which determines inductively equivariant multiplicities of Schubert varieties. Indeed, the Tfixed points in the fiber 7[-1 (x) are the classes (1, x)B and (s", s"x)B, where the first (resp. second) point occurs if and only if x::; T (resp. s"x ::; T). Fina11y, (1, x)B has aT-invariant neighborhood isomorphie to IC(a) x X(T) where IC(a) is the one-dimensional T-module with weight a. It follows that exX(T) e(l,x)BP" XB X(T) = - - a - ' and that a similar equality holds for e(s",s"x)BP" XB X(T).

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

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