# 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**

**Sample text**

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.