Noyaux d'opérateurs sur les groupes de Lie exponentiels

Abstract : Let G be a connected, simply conncted exponential Lie group with Lie algebra g. Let n be a nilpotent ideal with g'=(g,g) c n, l [epsilon] g* and h a polarization for l in g. Let H=exph and [pi] =indGHxl,h the unitary irreductible representation induced on G by the abelian character xl,h of the subgroup H. The quotient space G/H is diffeomorphic with some Rm and the representation space of [pi] can naturally be identified with L2(Rm). Then for any f [epsilon]L1(G) the operator [pi](f) = ?g f(x)pi(x)dx is a kernel operator, i.e. there exists a function fpi on Rm x Rm such that (pi(f)e)(x) = ?Rm fpi(x, y)e(y)dy. Problem : which kernels occur in this fashion? For nilpotent G, Howe had shown in 1977 that for every Schwartz function F e S(l,h) c S(RmxRm) there exists f e L1(G) with F=Fpi. For exponential groups such a result cannot be expected. In fact, however even f e C?c(G), i.e. f is C? and has compact support, pi(f) need no longer ba a compact operator ; this happens already for the infinite dimensional representations of the ax+b group. In 1983 Ludwig had shown that for a fixed polarization h in l and a coexponential basis B for h in g, the characterization of Hawe remains valid when the space S (l,h) is change to a suitable space ES(l,h,B) (S for Sschwartz, E for exponential decay) of functions on GxG, which are Schwartz functions in some directions of G/H and which have exponential decay in the other directions on G/H with also their partial Fourier transform in these directions. In 1993 Leptin and Ludwig had extended this result to the case of Vergne polarizations h=h(l,S) through n. In this paper, i extend the last characterization to the so-called strongly-adapted polarizations to the nilpotent ideal n, which is the family of polarizations h for l [epsilon] g* such that h[pont] polarization for lln and h is g(lln) - invariant. Remark that this family strictly contains the Vergne polarizations h=h(l,S) through n and every strongly-adapted polarization satisfies the Pukansky's condition. We also treat some examples
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Joseph Andele. Noyaux d'opérateurs sur les groupes de Lie exponentiels. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1997. Français. ⟨NNT : 1997METZ015S⟩. ⟨tel-01777201⟩

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