No Access

Corwin–Greenleaf multiplicity function of a class of Lie groups

Université de Sfax, Faculté des Sciences Sfax, BP 1171, 3038 Sfax, Tunisie

https://doi.org/10.1142/S0129167X24500745Cited by:0 (Source: Crossref)

Abstract

Let $N$ be a simply connected nilpotent Lie group, and let $K$ be a connected compact subgroup of the automorphism group, $A u t (N),$ of $N .$ Let $G : = K ⋉ N$ be the semidirect product (of $K$ and $N$ ). Let $𝔤 \supset 𝔨$ be the respective Lie algebras of $G$ and $K$ and $q : 𝔤^{*} \to 𝔨^{*}$ be the natural projection. It was pointed out by Lipsman, that the unitary dual $\hat{G}$ of $G$ is in one-to-one correspondence with the space of admissible coadjoint orbits $𝔤^{‡} / G$ (see [R. L. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures Appl.59 (1980) 337–374]). Let $π \in \hat{G}$ be a generic representation of $G$ and let $τ \in \hat{K} .$ To these representations we associate, respectively, the admissible coadjoint orbit $𝒪^{G} \subset 𝔤^{*}$ and $𝒪^{K} \subset 𝔨^{*}$ (via the Lipsman’s correspondence). We denote by $χ (𝒪^{G}, 𝒪^{K})$ the number of $K$ -orbits in $𝒪^{G} \cap q^{- 1} (𝒪^{K}),$ which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman’s orbit method suggests that the multiplicity $m_{π} (τ)$ of an irreducible $K$ -module $τ$ occurring in the restriction $π |_{K}$ could be read from the coadjoint action of $K$ on $𝒪^{G} \cap q^{- 1} (𝒪^{K}) .$ Under some assumptions on the pair $(K, N),$ we prove that for a class of generic representations $π \in \hat{G},$ one has

m π (τ) \neq 0 \Rightarrow χ (𝒪 G, 𝒪 K) \neq 0 . <math display="block" altimg="eq-00031.gif"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>τ</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\neq</mo><mn>0</mn><mo class="MathClass-rel">\Rightarrow</mo><mi>χ</mi><mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>G</mi></mrow></msup><mo class="MathClass-punc">,</mo><msup><mrow><mi mathvariant="cal">𝒪</mi></mrow><mrow><mi>K</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\neq</mo><mn>0</mn><mo class="MathClass-punc">.</mo></mrow></math>

Moreover, we show that the Corwin–Greenleaf multiplicity function is bounded (

$\leq 1$ ) for a special class of subgroups of

$G .$ Finally, we give a necessary and sufficient conditions to obtain a nonzero multiplicity (

$m_{π} (τ_{λ}) \neq 0$ ).

Communicated by Toshiyuki Kobayashi

To the memory of M. Ben Halima

Keywords:

AMSC: 22D10, 22E27, 22E45

References

1. P. Baguis , Semidirect product and the Pukanszky condition, J. Geom. Phys. 25 (1998) 245–270. Crossref, Google Scholar
2. A. Baklouti , On the cortex of connected simply connected nilpotent Lie groups, Russian J. Math. Phys. 5–3 (1998) 281–293. Google Scholar
3. M. E. B. Bekka and E. Kaniuth , Irreducible representations of locally compact group that cannot be Hausdorff separated from the identity representation, J. Reine Angew. Math. 385 (1988) 203–220. Web of Science, Google Scholar
4. M. Ben Halima and A. Messaoud , Corwin–Greenleaf multiplicity function for compact extensions of $ℝ^{n}$ , Int. J. Math. 26 (2015) 146–158. Link, Web of Science, Google Scholar
5. M. Ben Halima and A. Messaoud , Corwin–Greenleaf multiplicity function for compact extensions of the Heisenberg group, Int. J. Math. 29(09) (2018) 1850056. Link, Google Scholar
6. M. Ben Halima and A. Rahali , On the dual topology of a class of Cartan motion groups, J. Lie Theory 22 (2012) 491–503. Web of Science, Google Scholar
7. M. Ben Halima and A. Rahali , Dual topology of the Heisenberg motion groups, Indian J. Pure Appl. Math. 45(4) (2014) 513–530. Crossref, Web of Science, Google Scholar
8. C. Benson, J. Jenkins and G. Ratcliff , On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321(1) (1990) 85–116. Web of Science, Google Scholar
9. C. Benson, J. Jenkins and G. Ratcliff , Bounded K-spherical functions on Heisenberg groups, J. Funct. Anal. 105 (1992) 409–443. Crossref, Web of Science, Google Scholar
10. C. Benson, J. Jenkins and G. Ratcliff , The orbit method and Gelfand pairs associated with nilpotent Lie groups, J. Geom. Anal. 9 (1999) 569–582. Crossref, Google Scholar
11. C. Benson, J. Jenkins, G. Ratcliff and T. Worku , Spectra for Gelfand pairs associated with the Heisenberg group, Colloq. Math. 71 (1996) 305–328. Crossref, Google Scholar
12. C. Benson and G. Ratcliff , The space of bounded spherical functions on the free two step nilpotent Lie group, Transform. Groups 13 (2008) 243–281. Crossref, Google Scholar
13. L. Corwin and F. Greenleaf , Spectrum and multiplicities for unitary representations in nilpotent Lie groups, Pacific J. Math. 135 (1988) 233–267. Crossref, Web of Science, Google Scholar
14. J. Dixmier, Les $C^{*}$ -Algèbres et leurs Représentations (Gauthier-Villars, 1969). Google Scholar
15. M. Elloumi, Espaces duaux de certains produits semi-directs et noyaux associés aux orbites plates, Ph.D. Thesis, University of Lorraine (2009). Google Scholar
16. M. Elloumi, J.-K. Günther and J. Ludwig , On the dual topology of the groups $U (n) ⋉ ℍ_{n}$ , in Proc. Fourth Tunisian-Japanese Conference (Springer, 2017), pp. 9–68, https://arxiv.org/abs/1702.03747. Google Scholar
17. M. Elloumi and J. Ludwig , Dual topology of the motion groups $S O (n) ⋉ ℝ^{n}$ , Forum Math. 22 (2008) 397–410. Web of Science, Google Scholar
18. J. M. G. Fell , Weak containment and induced representations of groups, Can. J. Math. 14 (1962) 237–268. Crossref, Web of Science, Google Scholar
19. J. M. G. Fell , Weak containment and induced representations of groups (II), Trans. Amer. Math. Soc. 110 (1964) 424–447. Google Scholar
20. V. Fischer, F. Ricci and O. Yakimova , Nilpotent Gelfand pairs and spherical transforms of Schwartz functions I: Rank-one actions on the centre, Math. Z. 271(1) (2012) 221–255. Crossref, Web of Science, Google Scholar
21. H. Friedlander et al., An orbit model for the spectra of nilpotent Gelfand pairs, Transform. Groups 25 (2020) 859–886, https://doi.org/10.1007/s00031-019-09541-8. Crossref, Web of Science, Google Scholar
22. V. Guillemin and S. Sternberg , Convexity properties of the moment mapping, Invent. math. 67 (1982) 491–513. Crossref, Web of Science, Google Scholar
23. V. Guillemin and S. Sternberg , Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982) 515–538. Crossref, Web of Science, Google Scholar
24. A. A. Kirillov , Lectures on the Orbit Method (American Mathematical Society, Providence, RI, 2004). Crossref, Google Scholar
25. T. Kobayashi , Bounded multiplicity theorems for induction and restriction, J. Lie Thoery 32 (2022) 197–238. Web of Science, Google Scholar
26. T. Kobayashi and S. Nasrin , Multiplicity one theorem in the orbit method, in Lie Groups and Symmetric Spaces, American Mathematical Society Translations: Series 2, Vol. 210 (American Mathematical Society, Providence, RI, 2003), pp. 161–169. Crossref, Google Scholar
27. T. Kobayashi and S. Nasrin , Geometry of coadjoint orbits and multiplicity-one branching laws for symmetric pairs, Algebr. Represent. Theor. 21 (2018) 1023–1036. Crossref, Web of Science, Google Scholar
28. H. Leptin and J. Ludwig , Unitary Representation Theory of Exponential Lie Groups (de Gruyter, Berlin, 1994). Crossref, Google Scholar
29. R. L. Lipsman , Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures Appl. 59 (1980) 337–374. Web of Science, Google Scholar
30. G. W. Mackey , The Theory of Unitary Group Representations (Chicago University Press, 1976). Google Scholar
31. G. W. Mackey , Unitary Group Representations in Physics, Probability and Number Theory (Benjamin-Cummings, 1978). Google Scholar
32. A. Rahali , Dual topology of generalized motion groups, Math. Rep. 20(70) (2018) 233–243. Google Scholar
33. A. Rahali , Cartan motion groups and dual topology, Pure Appl. Math. Q. 14(3–4) (2018) 617–628. Crossref, Web of Science, Google Scholar
34. A. Rahali , Lipsman mapping and dual topology of semidirect products, Bull. Belg. Math. Soc. Simon Stevin. 26 (2019) 149–160. Crossref, Web of Science, Google Scholar
35. A. Rahali , Heisenberg motion groups and their cortex, J. Pure. Appl. Math. 53 (2022) 570–577, https://doi.org/10.1007/s13226-021-00134-4. Google Scholar
36. A. Rahali , Branching rule and coadjoint orbit for Heisenberg Gelfand pairs, Int. J. Math. 34(2) (2023) 2350004, https://doi.org/10.1142/S0129167X23500040. Link, Google Scholar
37. A. Rahali, I. B. Chenni and Z. Selmi , Orbit method and dual topology for certain Lie groups, Banach J. Math. Anal. 17 (2023) 8, https://doi.org/10.1007/s43037-022-00231-4. Crossref, Web of Science, Google Scholar
38. H. Regeiba, I. Chenni and A. Rahali Fell topology and its application for some semidirect products, Ann. Funct. Anal. 13 (2022) 28, https://doi.org/10.1007/s43034-022-00174-9. Crossref, Web of Science, Google Scholar