Download Algebraic and Analytic Methods in Representation Theory by Bent Ørsted and Henrik Schlichtkrull (Eds.) PDF

Download Algebraic and Analytic Methods in Representation Theory by Bent Ørsted and Henrik Schlichtkrull (Eds.) PDF

By Bent Ørsted and Henrik Schlichtkrull (Eds.)

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Extra info for Algebraic and Analytic Methods in Representation Theory

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Form a complete set of inequivalent indecomposable tilting modules. ) We give a proof which only works for p not too small, namely we assume that H°(a;) C G for all fundamental weights a~. 2, because then all fundamental weights ~ belong to the fundamental alcove C). (i) Write A - - E i=1 n a i, ~ , where ~Zl,.. : a;,. QHO(~dn)®an belongs to G, and since this module has highest weight A and this occurs with multiplicity 1, we find D(A) as an indecomposable summand. (ii) Let M E £t, and pick a maximal weight A of M.

So, as in the previous two sections, we shall assume char(k) - p > 0. H. H. 2 Let V be a representation of G. Then we define V (r) to be the representation of G on the same vector space but with G-action given by g v - F~(g)v, g E G, v E V. 1 from Let E be a B-module. 3 Hi(E(r) ® ( p r _ 1 ) p ) ~ Hi(E)(r) ® Str. 11(iii)), we have H ~ - H~(G/GrB, 2 r ( - ) ) . 8. Hence, the tensor identity for H i ( G / G r B , - ) gives Hi(E(r) @ ( p r _ 1 ) p ) ~ Hi(G/GrB, E (r)) ® Str, and we are done if we show that H i ( G / G r B , E (r)) ~ Hi(E) (r).

It is called the associated variety V(M) of M and is trivially contained in V(Ann M). Moreover, by a deep result of O. Gabber [G], it is an involutive subvariety of V ( A n n M ) , and this remembers to a large extent the U(0) module structure of M. Here, being involutive can be expressed by saying that the ideal I(M) of definition of V(M), namely v/grAnnU(0)m, satisfies { I ( M ) , I ( M ) } C I(M). This is equivalent to (dxa, dxb) := {a, b}(x) = 0 for all a, b E I(M), x E V(M). Thus, the subspace {dxa • a C I(M)} of T*x,V(Ann M) is isotropic and so can have at most half the dimension.

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