Adeles and the spectrum of compact nilmanifolds
Let G be a simply-connected, connected nilpotent Lie group. Let \(\Gamma\) be a discrete cocompact subgroup of G, and let \(\lambda\) be the quasi- regular representation of \(\Gamma\), i.e. the representation of G induced by the trivial one-dimensional representation of \(\Gamma\). Then \(\lambda\) can be decomposed into a direct sum \(\lambda =\sum m(\pi,\lambda)\pi\) of irreducible representations \(\pi\) of G with multiplicities m(\(\pi\),\(\lambda)\). The problem of determining the multiplicities m(\(\pi\),\(\lambda)\) has been initiated by \textit{L. Auslander}, \textit{L. Green}, \textit{F. Hahn} [Flows on homogeneous spaces (Princeton 1963; Zbl 0106.368)] and \textit{C. Moore} [Ann. Math., II. Ser. 82, 146-182 (1965; Zbl 0139.307)]. The problem was then completely solved by \textit{R. Howe} [Am. J. Math. 93, 163-172 (1971; Zbl 0215.118)] and \textit{L. Richardson} [Am. J. Math. 93, 173-190 (1971; Zbl 0265.43012)]. Another formula for m(\(\pi\),\(\lambda)\) was given by \textit{L. Corwin}, \textit{F. Greenleaf} [J. Funct. Anal. 21, 123-154 (1976; Zbl 0319.22011)] some time later. In the present paper the author takes a different approach. He first develops a ``rational'' Kirillov theory for the adele group \(G_{{\mathbb{A}}}\) which he uses to determine the decomposition of \(L^ 2(G_{{\mathbb{A}}}/G_{{\mathbb{Q}}})\). From this he deduces the multiplicity formulas of Howe-Richardson and Corwin-Greenleaf as simple consequences.
scientific article; zbMATH DE number 4142342
scientific article