Download e-book for kindle: Algebra and analysis: problems and solutions by Lefort G.

By Lefort G.

ISBN-10: 0720420164

ISBN-13: 9780720420166

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Examples. (a) Let us consider the example (a). Let u be the antisymmetric bilinear form on 1R2n whose matrix is J. The cone C is defined by We have CC = Sp(n, e). ,,) = iu(f" 17)· Then it can be shown that r(c) = {g E Sp(n,C) I Tlf, E e 2n , f3(gf"gf,) ~ f3(f"f,)}. (b) Let us consider the example (b), and the cone C = by Cmin defined C = {iX E 9 I Tlf, E en, f3(Xf"f,) ~ o}. We have CC = SL(n, e). )}. References [Vinberg,1980j, [Olshanski,1981]' [Paneitz,1984j, [Hilgert-HofmannLawson,1989j, [Dorr,1990j, [Lawson,1994j, [Lawson,1995]' [HilgertOlafsson,1997j.

3 Invariant cones in a simple Lie algebra 27 and [Y, Zl] E p, [Y, Z2] E t, therefore [Y, Z2] = 0. If X E I, B([X, YJ, Z2) = B(X, [Y, Z2]) = 0, therefore [X, Y] E I. Now let X E I, YEp, and decompose X, Then B([XI, YJ, Z2) = -B(Y, [X}, Z2]) = 0, since [X}, Z2] = 0, and on the other side since [X2' Y] E t, Z2 E p, therefore [X, Y] E I. , Z = Zl E t. • A real Lie algebra is said to be Hermitian if it is simple and if the dimension of the center of t is equal to one. The Hermitian Lie algebras are the following classical simple Lie algebras: 5p(n, lR), 5U(p, q) (p, q ~ 1), 50"'(2n) = 50(2n, C) n M(n, IHI), 50(2, n), and the two following exceptional simple Lie algebras: e6(-14) and e7(-25)' Let g be a Hermitian Lie algebra.

Proof. p(t) = 0 for t E R. p(i) = J(gexpiX) = o. 1. For -y = 9 exp iX we put 1i"(r) = 7r(g)Exp(id7r(X)). Then 111i"(r)II ~ 1, and 1i"(-y)* = 1i"(r#). In fact -y# = expiX. g-1 = g-1 exp(iAd(g)X), 42 III. Positive Unitary Representations and 1i'(-y). = Exp(id1l"(X))1I"(g-1), 1i'(-y*) = 1I"(g-1)ExP(id1l"(Ad(g)X)) = 1I"(g-1 )11" (g )Exp( id1l"(X) )1I"(g-1) = Exp( id1l"(X))1I"(g-1). We will prove first that 1i' is weakly holomorphic, then that 1i' is a representation. 1): there exists an open neighborhood U of e in GC, and a dense subspace 'Ho C 'H such that, for u E 'Ho, the map CPu : g ~ 11" (g)u has a holomorphic extension to U, CPu: U -t 'H.

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Algebra and analysis: problems and solutions by Lefort G.


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