Asymptotics of operator and pseudo-differential equations by V.P. Maslov

By V.P. Maslov

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Beweis. Sei q = ehiz . Die Funktion g(q) = e- hiNz F(z) ist regulär in der Kreisscheibe Iql < e 21tÖ • Auf Grund des Maximumprinzips nimmt die Funktion in dem Bereich Iqlse 21t " ihr Maximum auf dem Rande an. Es gibt daher einen Punkt z", Imz,,= - 8 mit der Eigenschaft Ig(l)1 = W(O) 1s e- 21tN "IF(z,,)1 0 3. 14 auf die Funktion F(z) = f(Zo +Sz) an. Hierbei sei fE[T", r] eine Modulform, ZOEIHn ein fester Punkt und S = S' "2:. 0 eine semipositive ganze Matrix. Über Z 0 und S wird noch geeignet verfUgt werden.

Wir werden später sehen, daß im Falle n> 1 die Bedingung 3) schon aus 1) und 2) folgt (Koecherprinzip, s. 5). Es ist leicht zu sehen, daß die Gesamtheit aller Modulformen n-ten Grades vom Gewicht reinen Vektorraum über

2) ( D-C) A =M. M'-l= -B Die Gruppe ~ besteht also aus allen orthogonalen symplektischen Matrizen: x,. = Sp(n, IR) n 0(2n, IR). Insbesondere ist ~ eine kompakte Untergruppe von Sp(n, IR). 7 Satz. Die Abbildung p: Sp(n,IR)-> IHn' M ->M

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