By Bent Orsted

ISBN-10: 0126254400

ISBN-13: 9780126254402

This e-book is a compilation of numerous works from well-recognized figures within the box of illustration conception. The presentation of the subject is exclusive in supplying numerous diverse issues of view, which should still makethe ebook very precious to scholars and specialists alike. provides numerous assorted issues of view on key subject matters in illustration idea, from the world over recognized specialists within the box

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**Extra resources for Algebraic and Analytic Methods in Representation Theory **

**Example text**

N E Mn(A) I det(aij) 6 A ×} for any R-algebra A. For this algebraic group, we have R[GLn] = R[Xij, T]/(T det(Xij) - 1). 3 is consistent with the conventions just given). Likewise, the various definitions and constructions found in the previous sections also generalize. Let us give the details in the case of the construction of induced modules: Let G be an algebraic group over k, and suppose V is a vector space over k. Then a representation of G on V is a morphism of functors between the two algebraic groups G and GL(V).

H. 5 Let A e X(T). Then Q~(A) has a Z~-filtration, and for each # E X(T) we have [Q~(~). 2~(~)1- [2~(~). L~(~)] (the latter symbol denoting composition factor multiplicity). " Note that IndTG~T -- 2r o Ind B~T. 4. , B' = woBwo, where w0 is (a representative in G of) the longest element in W. Set 2 ' - I n d ~ T T. Then, clearly, Z~ has properties completely analogous to Z~. 2) for all A E X(T). We now have Let A,# E X(T). Then k, i f # - A + 2 ( p r - 1 ) p , i) Homa~T(2'~(A),2~(tt)) ~ O, otherwise.

1 directly from the results of the previous sections. 3 we obtain the complete reducibility of all finite dimensional G-modules in characteristic zero. 2 g E G , v E E f E V*. ~ k. 2). [:] Let E and F be two B-modules. 3 Hi(E) ® H i (F) ~ H i+j (E ® F) for all i, j > O. Proof: We go by induction on i + j . For i - j - 0, we have the map g ° ( E ) ® H°(F) ~ H ° ( E ® F) given by fl Q f2 ~-+ (g ~-+ f l ( g ) Q f2(g), g e G), fl e H°(E), f2 e H°(F). Suppose j > 0, and embed F ~ I with I an injective B-module.

### Algebraic and Analytic Methods in Representation Theory by Bent Orsted

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