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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.

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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