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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:
[1] P. BERNAT , Sur les représentations unitaires des groupes de Lie résolubles (Ann. Sci. École Norm. Sup., Vol. 82, 1965 , pp. 37-99). Numdam | MR 33 #2763 | Zbl 0138.07302 · Zbl 0138.07302 · numdam:ASENS_1965_3_82_1_37_0 · eudml:81805
[2] P. BERNAT et al., Représentations des groupes de Lie résolubles , Dunod, Paris, 1972 . MR 56 #3183 | Zbl 0248.22012 · Zbl 0248.22012
[3] J.-Y. CHARBONNEL , Sur les semi-caractères des groupes de Lie résolubles connexes (J. Funct. Anal., Vol. 41, 1981 , pp. 175-203). MR 83c:22011 | Zbl 0471.22010 · Zbl 0471.22010 · doi:10.1016/0022-1236(81)90086-0
[4] M. DUFLO , Caractères des représentations des groupes résolubles associées à une orbite entière , Chap. IX in [2].
[5] M. DUFLO , M. RAÏS , Sur l’analyse harmonique sur les groupes de Lie résolubles (Ann. Sci. École Norm. Sup., Vol. 9, 1976 , pp. 107-144). Numdam | MR 55 #8254 | Zbl 0324.43011 · Zbl 0324.43011 · numdam:ASENS_1976_4_9_1_107_0 · eudml:81972
[6] N. V. PEDERSEN , Semicharacters and solvable Lie groups (Math. Ann., Vol. 247, 1980 , pp. 191-244). MR 81j:22015 | Zbl 0406.22008 · Zbl 0406.22008 · doi:10.1007/BF01348956 · eudml:163360
[7] L. PUKANSZKY , Leçons sur les représentations des groupes , Dunod, Paris, 1967 . MR 36 #311 | Zbl 0152.01201 · Zbl 0152.01201
[8] L. PUKANSZKY , On the characters and the Plancherel formula of nilpotent groups (J. Funct. Anal., Vol. 1, 1967 , pp. 255-280). MR 37 #4236 | Zbl 0165.48603 · Zbl 0165.48603 · doi:10.1016/0022-1236(67)90015-8
[9] L. PUKANSZKY , On the unitary representations of exponential groups (J. Funct. Anal., Vol. 2, 1968 , pp. 73-112). MR 37 #4205 | Zbl 0172.18502 · Zbl 0172.18502 · doi:10.1016/0022-1236(68)90026-8
[10] L. PUKANSZKY , Unitary representations of solvable Lie groups (Ann. Sci. École Norm. Sup., Vol. 4, 1971 , pp. 457-608). Numdam | MR 55 #12866 | Zbl 0238.22010 · Zbl 0238.22010 · numdam:ASENS_1971_4_4_4_457_0 · eudml:81887
[11] L. PUKANSZKY , Characters of connected Lie groups (Acta Math., Vol. 133, 1974 , pp. 81-137). MR 53 #13480 | Zbl 0323.22011 · Zbl 0323.22011 · doi:10.1007/BF02392143
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