Formule de Weyl pour les groupes de Lie nilpotents. (Weyl formula for nilpotent Lie groups). (French) Zbl 0721.22004

Manchon, Dominique

J. Reine Angew. Math. 418, 77-129 (1991).

For any real nilpotent connected and simply connected Lie group G and for any unitary irreducible representation \(\pi\) of G we define the (generally unbounded) operator with Weyl symbol p acting on the Hilbert space of the representation, the symbol p being some \(C^{\infty}\) function defined on the dual of the Lie algebra of G, and we prove the so-called Weyl formula: namely if p belongs to a certain class of elliptic real-valued functions, the operator with Weyl symbol p is essentially self-adjoint, has a discrete (real) left-bounded spectrum, and the number N(t) of eigenvalues smaller than t (including multiplicities) is equivalent, as t tends to \(+\infty\), to the symplectic volume of the domain \(\Omega\cap \{p\leq t\}\) of the coadjoint orbit \(\Omega\) associated to the representation \(\pi\) by means of Kirillov’s method.

Reviewer: D.Manchon (Nancy)

MSC:

22E27	Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
47B38	Linear operators on function spaces (general)

Keywords:

real nilpotent connected and simply connected Lie group; unitary irreducible representation; Weyl symbol; Hilbert space; Lie algebra; Weyl formula; self-adjoint; left-bounded spectrum; coadjoint orbit; Kirillov’s method