×

Sums of adjoint orbits and \(L^2\)-singular dichotomy for \(SU(m)\). (English) Zbl 1223.22007

An old result of D. L. Ragozin [J. Funct. Anal. 10, 212–229 (1972; Zbl 0286.43002)] implies that if \(G\) is any compact, connected, simple Lie group with Lie algebra \(\mathfrak{g}\), then any convolution product of dimension \(G\), \(G\)-invariant, continuous measures on \(\mathfrak{g}\) is absolute continuous with respect to Haar measure on \(\mathfrak{g}\), and any sum of dimension \(G\), adjoint orbits has non-empty interior. In a series of papers, S. K. Gupta et al. (cf. S. K. Gupta, K. E.Hare and S. Seyfaddini [Math. Z. 262, No. 1, 91–124 (2009; Zbl 1168.43003)]) found the minimum integer \(k(G)\) such that any convolution product of \(k(G)\), \(G\)-invariant, continuous measures is absolutely continuous, and any sum of \(k(G)\) adjoint orbits has non-empty interior, the number \(k(G)\) being roughly the rank of \(G\). They also proved that the convolution of \(k(G)\) orbital measures, the \(G\)-invariant measures supported on orbits, belongs to the smaller space \(L^2\cap L^1\).
This investigation is continued in the interesting paper under review. The author considers the special case \(G= SU(n)\) and finds necessary and sufficient conditions for a sum of adjoint orbits to have non-empty interior or, equivalently, for a product of orbital measures to belong to \(L^2 \cap L^1\). Indeed, if \(O_X = \{\text{Ad}(g) X:g \in SU (n)\}\) is an adjoint orbit in the Lie algebra \(su(n)\), then \(\sum^k_{j=1} O_{X_j}\) has non-empty interior if and only if \(\sum^k_{j=1} d_j/n\leq k-1\), where \(d_j\) = maximum dimension of an eigenspace of \(X_j\), except if \(k=2\) and both \(X_1\) and \(X_2\) have precisely two eigenvalues with equal multiplicities. A similar statement holds for products of conjugacy classes in \(SU(n)\) and convolution products of orbital measures in \(SU(n)\).

MSC:

22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alekseev, Anton; Malkin, Anton; Meinrenken, Eckhard, Lie group valued moment maps, J. Differential Geom., 48, 3, 445-495 (1998) · Zbl 0948.53045
[2] Dooley, A. H.; Repka, J.; Wildberger, N. J., Sums of adjoint orbits, Linear Multilinear Algebra, 36, 2, 79-101 (1993) · Zbl 0797.15010
[3] Duli, A. Kh.; Vildberger, N. Dzh., Harmonic analysis and global exponential mapping for compact Lie groups, Funktsional. Anal. i Prilozhen., 27, 1, 25-32 (1993)
[4] Frumkin, Avital; Goldberger, Assaf, On the distribution of the spectrum of the sum of two Hermitian or real symmetric matrices, Adv. in Appl. Math., 37, 2, 268-286 (2006) · Zbl 1115.22010
[5] Guillemin, V.; Sternberg, S., Convexity properties of the moment mapping, Invent. Math., 67, 3, 491-513 (1982) · Zbl 0503.58017
[6] Guillemin, V.; Sternberg, S., Convexity properties of the moment mapping. II, Invent. Math., 77, 3, 533-546 (1984) · Zbl 0561.58015
[7] Gupta, Sanjiv Kumar; Hare, Kathryn E., \(L^2\)-singular dichotomy for orbital measures of classical compact Lie groups, Adv. Math., 222, 5, 1521-1573 (2009) · Zbl 1179.43006
[8] Gupta, Sanjiv Kumar; Hare, Kathryn E.; Seyfaddini, Sobhan, \(L^2\)-singular dichotomy for orbital measures of classical simple Lie algebras, Math. Z., 262, 1, 91-124 (2009) · Zbl 1168.43003
[9] Kirwan, Frances, Convexity properties of the moment mapping. III, Invent. Math., 77, 3, 547-552 (1984) · Zbl 0561.58016
[10] Knutson, Allen; Tao, Terence, Honeycombs and sums of Hermitian matrices, Notices Amer. Math. Soc., 48, 2, 175-186 (2001) · Zbl 1047.15006
[11] Ragozin, David L., Rotation invariant measure algebras on Euclidean space, Indiana Univ. Math. J., 23, 1139-1154 (1973/1974) · Zbl 0302.43015
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.