×

A Kirillov theory for divisible nilpotent groups. (English) Zbl 0832.20054

Let \(N\) be a connected and simply connected nilpotent Lie group with Lie algebra \(\mathfrak n\). \(N\) acts on the linear dual \({\mathfrak n}^*\) of \(\mathfrak n\) by the coadjoint action and A. A. Kirillov [Usp. Mat. Nauk 17, 57-110 (1962; Zbl 0106.250)] constructed a bijection between the orbit space \({\mathfrak n}^*/N\) and the unitary dual \(\widehat {N}\) of \(N\). It turned out [cf. I. Brown, Ann. Sci. Ec. Norm. Supér., IV. Sér. 6(1973), 407-411 (1974; Zbl 0284.57026)] that this bijection is a homeomorphism between quotient topology and hull-kernel topology. Attempts to extend this Kirillov theory to non-Lie groups encounter two main problems: non type I-ness and finding a suitable substitute for the Lie algebra.
Here the authors present a Kirillov theory for divisible nilpotent groups. A group \(G\) is divisible if \(x^n = a\) has a solution for all \(n\) and all \(a \in G\). \(G\) is complete if the solution is unique. It is known that nilpotent groups are complete if and only if they are torsion- free and divisible. The results are first shown for complete groups and then for divisible groups by using the fact that every divisible nilpotent group \(G\) is a quotient of a complete nilpotent group \(H\) by a central group \(K\).
The main results are as follows. There is a Kirillov theory for \(G\) in the sense that \(\text{Prim }G\), the primitive ideal space of the \(C^*\)-algebra of \(G\), is homeomorphic to the space of \(K\)-integral \(G\) quasi-orbits in the dual \({\mathfrak h}^*\) of the suitably defined Lie algebra \(\mathfrak h\) of \(H\). Plancherel measure for \(G\) is canonically identified with Haar measure \(\mu\) on \((Z(H)/K)^\wedge\). In fact, with respect to \(\mu\) almost all characters of \(G\) are zero off \(Z(H)/K\) and are faithful circle-valued homomorphisms on \(Z(H)/K\).

MSC:

20F18 Nilpotent groups
22E25 Nilpotent and solvable Lie groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] L. Asimow, A.J. Ellis, Convexity theory and its applications in functional analysis. Academic Press 1980 · Zbl 0453.46013
[2] G. Baumslag, Lecture notes on nilpotent groups. CBMS Reg. Conf. Ser., vol. 2, 1971 · Zbl 0241.20001
[3] I. Brown, Dual topology of a nilpotent Lie group. Ann. Sci. Ec. Norm. Supér (4)6 (1973), 407-411 · Zbl 0284.57026
[4] A.L. Carey, W. Moran, Characters of nilpotent groups. Math. Proc. Camb. Philos. Soc.96 (1984), 123-137 · Zbl 0549.43004 · doi:10.1017/S0305004100062009
[5] L. Corwin, C.Pfeffer, On factor representations of discrete rational nilpotent groups and the Plancherel formula (to appear) · Zbl 0789.22014
[6] J. Dixmier, C*-algebras. North-Holland 1982
[7] E.G. Effros, F. Hahn, Locally compact transformation groups andC *-algebras. Mem. Am. Math. Soc.75 (1967) · Zbl 0166.11802
[8] J.M.G. Fell, The structure of algebras of operator fields. Acta Math.106 (1961), 233-280 · Zbl 0101.09301 · doi:10.1007/BF02545788
[9] J. Glimm, Type IC *-algebras. Ann. Math.73 (1961), 572-612 · Zbl 0152.33002 · doi:10.2307/1970319
[10] R.E. Howe, On representations of discrete, finitely generated, torsion-free, nilpotent groups. Pac. J. Math.73 (1977), 281-305 · Zbl 0387.22005
[11] R.E. Howe, The Fourier transform for nilpotent locally compact groups: I, Pac. J. Math.73 (1977), 307-327 · Zbl 0396.43013
[12] R.E. Howe, Kirillov theory for compactp-adic groups. Pac. J. Math.73 (1977), 365-381 · Zbl 0385.22007
[13] K.I. Joy, A description of the topology on the dual space of a nilpotent Lie group. Pac. J. Math.112 (1984), 135-139 · Zbl 0535.22008
[14] A.A. Kirillov, Unitary representations of nilpotent Lie groups. Russ. Math. Surv.17 (1962), 53-104 · Zbl 0106.25001 · doi:10.1070/RM1962v017n04ABEH004118
[15] L.G. Kovács, Groups with uniform automorphisms. Rend. Circ. Mat. Palermo (2) Suppl.19 (1988), 125-133
[16] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. Ec. Norm. Supér (3)71 (1954), 101-190 · Zbl 0055.25103
[17] G.K. Pedersen,C *-algebras and their automorphism groups. Academic Press 1979 · Zbl 0416.46043
[18] C. Pfeffer Johnston, Primitive ideal spaces of discrete rational nilpotent groups (to appear) · Zbl 0857.22005
[19] D. Poguntke, Discrete nilpotent groups haveT 1 primitive ideal space. Stud. Math.71 (1981-82), 271-275
[20] E. Thoma, Über unitäre Darstellungen abzählbarer, diskreter Gruppen. Math. Ann.153 (1964), 111-138 · Zbl 0136.11603 · doi:10.1007/BF01361180
[21] R.B. Warfield, Nilpotent groups. Lect. Notes Math., vol 513. Springer 1976. · Zbl 0347.20018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.