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

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 smooth mathematical constructions, similar 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, 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 lined is extensive, from equivariant Chow teams, decomposition sessions and Schubert types, multiplicity unfastened activities, convolution algebras, commonplace monomial idea, and canonical bases, to annihilators of quantum Verma modules, modular illustration conception of Lie algebras and combinatorics of illustration different types of Harish-Chandra modules.

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XjG) over Q, whenever G aets on X with finite isotropy groups [22]. Let X be a G-seheme and let H C G be a closed subgroup. Then, for U as above, the quotient U -+ UjH exists and is a principal H-bundle. Thus, there is a smooth map X XH U -+ X XG U with fiber GjH, whieh induees a map A~(X) -+ A;r (X) of degree dim( G j H). If moreover H is a Levi subgroup of G, then G j H is the unipotent radieal of G and henee is isomorphie to affine spaee. 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.

Question 2 Notation being as above, when is A*(G/H) finite in positive degree? Equivalently, when is the restrietion map SW -+ SjiH surjeetive? 3]: when is the restrietion map Clg]G -+ Cl~]H surjeetive? g. when H is isomorphie to SL(2) or to PSL(2): indeed, the Killing form of 9 restricts then to a non-zero quadratic element ofCl~]H, which generates this algebra. ;: W. It turns out that this does not hold for eertain pairs of semisimple groups (G, H) as weIl. The following example was pointed out by Bram Broer: let G be semisimple of type E 6 and H c G be a maximal semisimple subgroup of a parabolie subgroup of type D5.

Then X(w) = PaX(T) and the map 7r: Pa XB X(T) --+ X(w) is birational. Iterating this construction, we obtain a B-equivariant birational map where w = sa! sa2 ... San is a reduced decomposition. Moreover, the variety on the left is projective, smooth and contains only finitely many T-fixed points: the c1asses of sequences (SI, ... , Sn) where each Sj is either Saj or 1. 26 M. Brion We claim that e x X( w) _ exX(T) - s"(es,,xX(T)) , a which determines inductively equivariant multiplicities of Schubert varieties.