Restricting cuspidal representations of the group of automorphisms of a homogeneous tree
Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 353-379.

Voir la notice de l'article dans Biblioteca Digitale Italiana di Matematica

Sia $\mathfrak{X}$ un albero omogeneo dove a ogni vertice si incontrano $q+1$$(q\geq 2)$ spigoli. Sia $\mathfrak{A}=Aut(\mathfrak{X})$ il gruppo di automorfismi di $\mathfrak{X}$ e $H$ un sottogruppo chiuso isomorfo a $PGL(2, F)$ ($F$ campo locale il cui campo residuo ha ordine $q$). Sia $\pi$ una rappresentazione continua unitaria e irriducibile di $\mathfrak{A}$ e si consideri $\pi_{H}$, la sua restrizione ad $H$. È noto che se $\pi$ è una rappresentazione sferica o speciale $\pi_{H}$ rimane irriducibile. In questo lavoro si mostra che quando $\pi$ è cuspidale la situazione è molto più complessa. Si studia in dettagli o il caso in cui il sotto albero minimale associato a $\pi$ sia il più piccolo possibile, ottenendo una esplicita decomposizione di $\pi_{H}$. 
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Cartwright, Donald I.; Kuhn, Gabriella. Restricting cuspidal representations of the group of automorphisms of a homogeneous tree. Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 2, pp. 353-379. https://geodesic-test.mathdoc.fr/item/BUMI_2003_8_6B_2_a5/

[1] D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics: 55, Cambridge University Press, 1997. | MR | Zbl

[2] J. Dixmier, Les $C^*$-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964. | MR | Zbl

[3] J. Faraut, Analyse harmonique sur les paires de Gelfand et les spaces hyperboliques, Analyse Harmonique Nancy 1980, CIMPA, Nice 1983. | Zbl

[4] A. Figà-Talamanca-C. Nebbia, Harmonic analysis and representation theory for groups acting on homogeneous trees, London Mathematical Society Lecture Note Series 162, Cambridge University Press, Cambridge, 1991. | MR | Zbl

[5] G. B. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton, 1995. | MR | Zbl

[6] R. Howe, On the principal series of $Gl_n$ over $p$-adic fields, Trans. Amer. Math Soc., 177 (1973), 275-296. | MR | Zbl

[7] R. Howe, Harish-Chandra Homomorphisms for $\mathfrak{p}$-adic groups, CBMS Regional Conference Series in Mathematics, No. 59, American Mathematical Society, 1985. | Zbl

[8] G. James-A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its applications, volume 16, Addison-Wesley, 1981. | MR | Zbl

[9] P. C. Kutzko, On the supercuspidal representations of $Gl_2$, Amer. J. Math., 100 (1978), 43-60. | MR | Zbl

[10] G. W. Mackey, The theory of unitary group representations, University of Chicago Press, Chicago, 1976. | MR | Zbl

[11] G. I. Ol'Shanskii, Representations of groups of automorphisms of trees, Usp. Mat. Nauk, 303 (1975), 169-170.

[12] G. I. Ol'Shanskii, Classification of irreducible representations of groups of automorphisms of Bruhat-Tits trees, Functional Anal. Appl., 11 (1977), 26-34. | MR | Zbl

[13] G. K. Pedersen, Analysis now, Graduate Texts in Mathematics 118, Springer Verlag (1989). | MR | Zbl

[14] I. Piatetski-Shapiro, Complex representations of $GL(2, K)$ for finite fields $K$, Contemporary Mathematics, Vol. 16, American Mathematical Society, Providence, 1983. | MR

[15] W. R. Scott, Group theory, Prentice-Hall, 1964. | MR | Zbl

[16] J. P. Serre, Trees, Springer Verlag, Berlin Heidelberg New York, 1980. | MR | Zbl