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On the characters of exponential solvable Lie groups. (English) Zbl 0578.22007
Let $$G$$ be a connected, simply connected solvable, exponential Lie group with Lie algebra $$\mathfrak g$$. Let the Jordan-Hölder sequence $$\mathfrak g_{\mathbb C}=\mathfrak f_ m\supset \ldots \supset\mathfrak f_0=\{0\}$$ have the property that if $$\overline{\mathfrak f}_j\neq \mathfrak f_j$$, then $$\overline{\mathfrak f}_{j-1}=\mathfrak f_{j-1}$$ and $$\overline{\mathfrak f}_{j+1}=\mathfrak f_{j+1}$$, $$1\leq j\leq m-1$$. Let $$G$$ act on $$\mathfrak g'$$ via the coadjoint representation. The author makes a construction, depending only on the choice of the Jordan-Hölder sequence, of a set of polynomial functions $$Q_j\geq 0$$, $$j=1,\ldots,n$$, on $$\mathfrak g'$$, a corresponding partition of $$\mathfrak g'$$ into $$G$$-invariant subsets $\Omega_j=\{g\in\mathfrak g'\mid Q_j(g)\neq 0,\;Q_k(g)\neq 0 \text{ for }k<j\}$ and a finite set of continuous homomorphisms $$\chi_j: G\to \mathbb R_+^*$$, $$j=1,\ldots,n$$, such that $$Q_j(sg)=\chi_j(s) Q_j(g)$$ for $$s\in G$$, $$g\in \Omega_j$$ and for any $$G$$-orbit $$O$$ contained in $$\Omega_j$$ with canonical measure $$\beta_O$$ the measure $$Q_j\beta_O$$ is a non-zero, positive, tempered, relatively invariant Radon measure on $$O$$ with multiplier $$\chi_j$$.

The author also constructs positive, $$G$$-invariant analytic functions $$\alpha_1,\ldots,\alpha_n$$ on $$\mathfrak g$$ and he sets $$u_j$$ equal to the element in $$U(\mathfrak g_{\mathbb C})$$ corresponding via symmetrization to the polynomial function $$g\mapsto Q_j(ig)$$ on $$\mathfrak g'_{\mathbb C}$$. Now he proves several results about the relationship between an irreducible representation $$\pi$$ of $$G$$ associated with an orbit $$O$$ contained in $$\Omega_j$$ and the earlier constructed objects. In particular, he shows that for all $$\phi \in C_c^\infty(G)$$ the operator $$\pi (u_j*\phi)$$ is trace class and $\text{Tr}(\pi (u_j*\phi))=\int_O((\alpha_j\phi)\circ \exp){\hat{\;}}(l) Q_j(l) \,d\beta_O(l),$ where $$f\mapsto \hat f$$ stands for the ordinary Euclidean Fourier transform.
The main difference of this last formula with earlier character formulas for solvable Lie groups [in particular J.-Y. Charbonnel, J. Funct. Anal. 41, 175–203 (1981; Zbl 0471.22010)] is that, once a Jordan-Hölder basis in $$\mathfrak g_{\mathbb C}$$ has been selected, all objects in the formula are explicitly constructible. In the case where $$\mathfrak g$$ is nilpotent, the formula reduces to the Kirillov character formula.

##### MSC:
 22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
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##### References:
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