...of t = 0. This is, F(t) being ^ 0 for 1 1 ^ Rl} impossible unless Gr(-} vanishes identically: which proves the first part of the theorem. To prove the second part we denote by p the root (or one of the roots) satisfying the condition P =?. Then putting in (14) #! =...

...0 for [ í ' á R¡, impossible unless G(-j vanishes identically: .»•, = 0, x.¿ = 0, ... which proves the first part of the theorem. To prove the second part we denote by p the root (or one of the roots) satisfying the condition \P\-l Then putting in (14) #1=1»...

...l/o^O^I- Here» equality holds only for a function of the form f(z) = €/0(z), where |e | = 1. This proves the first part of the theorem. To prove the second part, we note that the extremal function f$(z) can be characterized by the following considerations: l)/0(z)...

...- / is of Hilbert-Schmidt class. By virtue of Theorem 3.2 and result 1, this completes the proof of the first part of the theorem. To prove the second part, we introduce the matrix IP- '.2. ..-..» (>,*= 1,2, ...). This matrix generates some operator A jk which...

...a(x)i + a(x2), xlt x2 e X. Comparison of dimensions now shows that Ì = A + A is a direct sum. This proves the first part of the theorem. To prove the second part, it suffices to consider the case of a simply connected group ®. Let H and H be simply connected Lie...

...</'(t 0 )y, d«^ 1 > = <y,rftl n /2>=0; hence y = 0. Since n 2 l /2=f*£ i we have (21.4.2), which proves the first part of the theorem. To prove the second part we choose, using Theorem C.4.4, C° ° functions q j , p k in a neighborhood of f(t 0 ) such that f* qj...

...most two irreducible components in V(Z'). On the other hand, it obviously preserves / 2 -grading; this proves the first part of the theorem. To prove the second part of the theorem one has to verify (1.1.12) and compare the highest weight with (1.1.14). Note that A,(0)...

...Therefore n(T(H,v)\v,x) n(T(H,v),x) and so ^ n(T(H,v)\v,x) ' , u) \ u, x) - Eue „ M(^(G, u) \ w, x) This proves the first part of the theorem. To prove the second part, we note that since T(G\u,v) is isomorphic to a component of T(G,u)\u, it follows that p,(T(G,u) \u,x)...

...inequalities (1.8.3), (1.8.5) we obtain E oj«?; < EE «oí» •»; < E «>i'fí = ^'. j=l ;=1 i=li=l which proves the first part of the theorem. To prove the second part of the theorem (assertion 4), it suffices to note that in the case of strict dominance of the mth row...

...greater than (1 — £)\/27r(7ro(#*) — <5) for sufficiently large n, which concludes the proof of the first part of the theorem. To prove the second part, we assume that q £ Q have bounded densities with bounded derivatives. Thus, we get from (10) - e) nl...