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New PDF release: Algebraic Structures of Symmetric Domains

By Ichiro Satake

This ebook is a entire therapy of the overall (algebraic) idea of symmetric domains.

Originally released in 1981.

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Additional resources for Algebraic Structures of Symmetric Domains

Example text

C(a)+n is a minimal parabolic subalgebra of g. From the theory of algebraic Lie algebras (Chevalley [2], [4]), it is easy to see that all Lie subalgebras of g containing b~ are algebraic and hence of the form br for some I'cJ. It is known that any algebraic group defined over F contains an (absolute) maximal torus j defined over F. It follows that there is an F-torus Tin C(A) such that 7=TFismaximalinC(A)F. Then Tis (absolutely) maximalinG and A coincides with the F-split part of T. Let X be the (absolute) character module of j and Xo the annihilator of A in X.

The second part will be generalized in Proposition 9. ) Proposition 7. , the centralizer of ad(Iv) in Aut(@). Thus I'(V, { } ) is reductive, and Lie I'(V, { }) may be identified with the derivation algebra of the graded Lie algebra @. But, since @ is semi-simple, one has Der(@)=ad(®)=®· Therefore a derivation of@preserving the gradation must be of the form D=ad Twith Te@0 = Vo V. Thus one has Proposition 7. 3. The structure group I' ( V, { } ) of a non-degenerate ]TS is reductive, and Lie I' ( V, { } ) coincides with VD V, i.

Therefore, by Lemma 4. ;} where a'=g2 111 eCc,(G). ) q. e. d. Corollary 4. 3. Any Zariski connected reductive R-group G (and hence also g=Lie G) has a Cartan involution. Two Cartan involutions of Gare conjugate to each other by an inner automorphism of G. This follows from Theorem 4. 2 (ii), (iii) applied to the case G= {l). Corollary 4. 4 (Mostow). Let G be a Zariski connected R-group in GLn(R). Then G is reductive if and only if one has 'G= G for a suitable choice of basis and, when this is the case, the map gr-+'g-' (ge G) is a Cartan involution of G.

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Algebraic Structures of Symmetric Domains by Ichiro Satake


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