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Irreducibility of unitary group representations and reproducing kernels Hilbert spaces. Appendix on two point homogeneous compact ultrametric spaces in collaboration with Rostislav Grigorchuk. (English) Zbl 1037.22009
In this expository article, the authors give an introduction to Hilbert spaces with reproducing kernels and describe how they can be used in representation theory to prove irreducibility of representations. Recall that a reproducing kernel Hilbert space on a set \(X\) is a Hilbert space \({\mathcal H}\) of functions \(X\to {\mathbb C}\), such that the point evaluations \(\text{ev}_x : {\mathcal H} \to {\mathbb C}\) at elements \(x\in X\) are continuous linear functionals and hence of the form \(\langle\cdot , \phi_x\rangle\) for suitable elements \(\phi_x\in {\mathcal H}\). The map \(\Phi : X\times X\to {\mathbb C}\), \(\Phi(x,y):=\langle \phi_y,\phi_x\rangle\) is called the reproducing kernel of \(\Phi\). Given a group action \(G\times X\to X\) on \(X\), it is well known that \((\pi(g).f)(x):=f(g^{-1}.x)\) for \(g\in G\), \(f\in {\mathcal H}\), \(x\in X\) defines a unitary representation \(\pi\) of \(G\) on \({\mathcal H}\) if and only if the kernel \(\Phi\) is invariant in the sense that \(\Phi(g.x, g.y)=\Phi(x,y)\).
The paper is centered around the following irreducibility criterion for representations associated with invariant kernels: If \(G\) acts transitively on \(X\) and the subspace \({\mathcal H}^K\) of \(K\)-invariant vectors is one-dimensional for the isotropy subgroup \(K:=G_x\) of some point \(x\in X\), then \(\pi\) is irreducible (Proposition 2). Many examples are given to illustrate the usefulness of this criterion and its variants for representations defined using a cocycle (Proposition 6), resp., for representations on Hilbert spaces of vector-valued functions (Proposition 9). Further, more specialized examples are discussed in an appendix.
Reviewer’s remarks: An exposition of reproducing kernel spaces with a view towards representation theory has also been given in [K.-H. Neeb, Holomorphy and convexity in Lie theory, de Gruyter Expositions in Math. 28 (Berlin, 1999; Zbl 0936.22001)], Part A. In particular, one finds a version of Kobayashi’s Theorem for Cocycle Representations there (Theorem IV.1.10), which strengthens the authors’ method in the special case of group actions on complex manifolds [cf. S. Kobayashi, J. Math. Soc. Japan 20, 638–642 (1968; Zbl 0165.40504)].

MSC:
22D10 Unitary representations of locally compact groups
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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[1] Alpay, D., Algorithme de Schur, espaces à noyau reproduisant et théorie des systèmes, (), No 6 · Zbl 0946.93002
[2] Aronszajn, N., La théorie générale des noyaux reproduisants et ses applications, (), 133-153 · Zbl 0061.26204
[3] Aronszajn, N., Theory of reproducing kernels, Trans. amer. math. soc., 68, 337-404, (1950) · Zbl 0037.20701
[4] Bartholdi, L.; Grigorchuk, R., Sous-groupes paraboliques et représentations de groupes branchés, C.R. acad. sci. Paris, Série I, 332, 789-794, (2001) · Zbl 1011.20027
[5] Bartholdi, L.; Grigorchuk, R., On parabolic subgroups and Hecke algebras of some fractal groups, Serdica math. J., 28, 47-90, (2002) · Zbl 1011.20028
[6] Bekka, M.; Curtis, R., On Mackey’s irreducibility criterion for induced representations, (2002), Preprint, September · Zbl 1027.22010
[7] Bergman, S., Ueber die kernelfunktion eines bereiches und ihre verhalten am rande, J. reine angew. math., 199, 89-128, (1933)
[8] Bergman, S., The kernel function and conformal mapping, Math. surveys V, amer. math. soc., (1950) · Zbl 0040.19001
[9] Biggs, N., Algebraic graph theory, (1993), Cambridge Univ. Press
[10] Bourbaki, N., Intégration, chapitres 1, 2, 3 et 4, (1965), Hermann · Zbl 0136.03404
[11] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, () · Zbl 0747.05073
[12] Burger, M.; de la Harpe, P., Constructing irreducible representations of discrete groups, (), 223-235 · Zbl 0908.22005
[13] Cartan, E., Sur la détermination d’un système orthogonal complet dans un espace de Riemann symétrique clos, Rend. circ. math. Palermo, 53, 217-252, (1929), [see also Oeuvres complètes I2, Gauthier Villars (1952) 1045-1080] · JFM 55.1029.01
[14] Carey, A.L., Induced representations, reproducing kernels and the conformal group, Commun. math. phys., 52, 77-101, (1977) · Zbl 0339.43007
[15] Carey, A.I., Group representations in reproducing kernel Hilbert spaces, Report on math. physics, 14, 247-259, (1978) · Zbl 0418.22004
[16] Chatterji, S.D., Factorization of positive finite operator-valued kernels, (), 23-36 · Zbl 0516.47009
[17] Chatterji, S.D., Positive definite kernels, Bol. soc. mat. mexicana, 28, 59-65, (1983) · Zbl 0584.60050
[18] Conway, J.H.; Sloane, N.J.A., Sphere packings, lattices and groups, ()
[19] Curtis, C.W.; Reiner, I., Theory of finite groups and associative algebras, (1962), Wiley Interscience · Zbl 0131.25601
[20] Curtis, R., Hecke algebras associated with induced representations, C.R. acad. sci. Paris, 334, 31-35, (2002), Sér. I · Zbl 1014.46038
[21] Delsarte, P., Hahn polynomials, discrete harmonics, and t-designs, SIAM J. appl. math., 34, 157-166, (1978) · Zbl 0533.05009
[22] Dieudonné, J., Eléments d’analyse, 6, (1975), Bordas
[23] Dixmier, J., LES C*-algèbres et leurs représentations, (1969), Gauthier-Villars · Zbl 0174.18601
[24] Duflo, M.; Moore, C.C., On the regular representation of a nonunimodular locally compact group, J. functional analysis, 21, 209-243, (1976) · Zbl 0317.43013
[25] Dunkl, C.F., Krawtchouk polynomial addition theorem and wreath product of symmetric groups, Indian univ. math. J., 25, 335-358, (1976) · Zbl 0326.33008
[26] Evans, D.; Kawahigashi, Y., Quantum symmetries on operator algebras, (1998), Oxford Univ. Press · Zbl 0924.46054
[27] Fabrikowsky, J.; Gupta, N.D., On groups with sub-exponential growth functions, II, J. Indian math. soc. (N.S.), 56, 217-228, (1991) · Zbl 0868.20029
[28] J. Faraut, Analyse hamonique sur les espaces riemanniens symétriques de rang un (Ecole d’été “Analyse harmonique”, Université de Nancy I, 15 septembre — 3 octobre 1980)
[29] Figà-Talamanca, A., An application of Gelfand pairs to a problem of diffusion in compact ultrametric spaces, (), 51-67 · Zbl 0985.60004
[30] Gelfand, I.M.; Berezin, F.A.; Gelfand, I.M., Some remarks on the theory of spherical functions on symmetric Riemannian manifolds, Dokl. akad. nauk. SSSR, Tr. mosk. math. O.-va, 5, 311-351, (1956), [Collected papers, Vol. II, Springer (1988) 275-320]
[31] Godement, R., LES fonctions de type positif et la théorie des groupes, Trans. amer. math. soc., 63, 1-84, (1948) · Zbl 0031.35903
[32] Goodman, R.; Wallach, N.R., Representations and invariants of the classical groups, (1998), Cambridge Univ. Press · Zbl 0901.22001
[33] Grigorchuk, R.I., Bernside’s problem on periodic groups, Functional anal. appl., 14, 41-43, (1980) · Zbl 0595.20029
[34] Grigorchuk, R.I.; Nekrashevich, V.V.; Sushchanskii, V.I., Automata, dynamical systems, and groups, (), 128-203 · Zbl 1155.37311
[35] Gupta, N.D.; Sidki, S.N., On the Burnside problem for periodic groups, Math. Z., 182, 385-388, (1983) · Zbl 0513.20024
[36] Gupta, N.D.; Sidki, S.N., Some infinite p-groups, Algebra i logika, 22, 584-589, (1983) · Zbl 0546.20026
[37] de la Harpe, P., Topics in geometric group theory, (2000), University of Chicago Press · Zbl 0965.20025
[38] Hille, E., Introduction to general theory of reproducing kernels, Rocky mountain J. math., 2, 321-368, (1972) · Zbl 0266.30009
[39] Kolmogorov, A.N., Stationary sequences in Hilbert space, (), 2, 228-271, (1941)
[40] Krantz, S.G., Function theory of several complex variables, (1982), J. Wiley · Zbl 0471.32008
[41] Krein, M.G., Hermitian-positive kernels on homogenous spaces. I, Ukrain math. zurnal, Amer. math. soc. translations, 34, 2, 69-108, (1963), see also
[42] Krein, M.G., Hermitian-positive kernels on homogenous spaces. II, Ukrain math. zurnal, Amer. math. soc. translations, 34, 109-164, (1963), see also
[43] Kunze, R.A., Positive definite operator-valued kernels and unitary representations, (), 235-247
[44] Lang, S., Sl_2(ℝ), (1975), Addison-Wesley
[45] Letac, G., LES fonctions sphériques d’un couple de Gelfand symétrique et LES chaînes de Markov, Adv. appl. prob., 14, 272-294, (1982) · Zbl 0482.60011
[46] Mackey, G.W., The theory of unitary group representations, (1976), The University of Chicago Press · Zbl 0344.22002
[47] Mautner, F.I., Spherical functions over p-adic groups, II, Amer. J. math., 86, 171-200, (1964) · Zbl 0135.17204
[48] Mercer, J., Functions of positive and negative type, and their connection with the theory of integral equations, Philos. trans. royal soc. London, 209, 415-466, (1909), Ser. A · JFM 40.0408.02
[49] Meschkowski, H., Hilbertsche Räume mit kernfunktion, () · Zbl 0103.08802
[50] Moore, E.H., On the developments of Bessel’s functions, (), Bull. amer. math. soc., Bull. amer. math. soc., 23, 59-27, (1916-1917), Part II, 1939 [prepared from Moore’s notes and revised by R.W. Barnard, after Moore’s death (1932)]. See also
[51] Nehari, Z., Conformal mapping, (1952), McGraw Hill, [and Dover, 1975] · Zbl 0048.31503
[52] Neumann, P.M., Finite permutation groups, edge-coloured graphs and matrices, (), 82-118
[53] Neumann, P.M., Some questions on edjvet and pride about infinite groups, Illinois J. math., 30, 301-316, (1986) · Zbl 0598.20029
[54] Parthasarathy, K.R.; Schmidt, K., Positive definite kernels, continuous tensor products, and central limit theorems in probability theory, () · Zbl 0237.43005
[55] Poletskii, E.A.; Shabat, B.V., Invariant metrics, (), 63-111
[56] Robert, A., Introduction to the representation theory of compact and locally compact groups, ()
[57] Saxl, J., On multiplicity-free permutation representations, (), 337-353 · Zbl 0578.20006
[58] Schoenberg, I.J., Metric spaces and positive definite functions, Trans. amer. math. soc., 44, 522-536, (1938) · Zbl 0019.41502
[59] Schoenberg, I.J., Positive definite functions on spheres, Duke math. J., 9, 96-108, (1938) · Zbl 0063.06808
[60] Schur, I., Bemerkungen zur theorie der beschränkten bilinearformen mit unendlich vielen veränderlichen, (), 140, 464-491, (1911) · JFM 42.0367.01
[61] Schwartz, L., Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. d’analyse math., 13, 115-256, (1964) · Zbl 0124.06504
[62] Serre, J.-P., Arbres, amalgames, SL_2, () · Zbl 0302.20039
[63] Stein, E.M.; Weiss, G., Introduction to Fourier analysis on Euclidean spaces, (1971), Princeton University Press · Zbl 0232.42007
[64] Szegö, G., Orthogonal polynomials, () · JFM 61.0386.03
[65] Vesentini, E., Capitoli scelti Della teoria delle funzione olomorfe, Unione matematica italiana, (1984) · Zbl 0691.30003
[66] Vilenkin, N.J., Special functions and the theory of group representations, Transl. math. monographs, 22, (1968), Amer. Math. Soc.
[67] Vilenkin, N.J.; Klimyk, A.U., ()
[68] Weyl, H., Harmonics on homogeneous manifolds, (), 35, 2, 386-399, (1934) · JFM 60.0360.01
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