Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups

Ryosuke Nakahama

Symmetry, integrability and geometry: methods and applications, Tome 18 (2022)

Résumé

Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ on $D$. Its restriction to the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua–Kostant–Schmid–Kobayashi's formula in terms of the $K_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement $\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object of this article is to compute explicitly the inner product $\big\langle f(x_2),{\rm e}^{(x|\overline{z})_{\mathfrak{p}^+}}\big\rangle_\lambda$ for $f(x_2)\in\mathcal{P}(\mathfrak{p}^+_2)$, $x=(x_1,x_2)$, $z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$. For example, when $\mathfrak{p}^+$, $\mathfrak{p}^+_2$ are of tube type and $f(x_2)=\det(x_2)^k$, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials ${}_2F_1$. Also, as an application, we construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_\mu(D_1)$ from holomorphic discrete series representations of $\widetilde{G}$ to those of $\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large $\lambda$.

Détail

Keywords: weighted Bergman spaces, holomorphic discrete series representations, branching laws, intertwining operators, symmetry breaking operators, highest weight modules.

Bibliographie

[1] Arazy J., “A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains”, Multivariable Operator Theory (Seattle, WA, 1993), Contemp. Math., 185, Amer. Math. Soc., Providence, RI, 1995, 7–65 | DOI | MR | Zbl

[2] Beerends R.J., Opdam E.M., “Certain hypergeometric series related to the root system $BC$”, Trans. Amer. Math. Soc., 339 (1993), 581–609 | DOI | MR | Zbl

[3] Ben Saïd S., Clerc J.-L., Koufany K., “Conformally covariant bi-differential operators on a simple real Jordan algebra”, Int. Math. Res. Not., 2020 (2020), 2287–2351, arXiv: 1704.01817 | DOI | MR | Zbl

[4] Clerc J.-L., “Symmetry breaking differential operators, the source operator and Rodrigues formulae”, Pacific J. Math., 307 (2020), 79–107, arXiv: 1902.06073 | DOI | MR | Zbl

[5] Cohen H., “Sums involving the values at negative integers of $L$-functions of quadratic characters”, Math. Ann., 217 (1975), 271–285 | DOI | MR | Zbl

[6] Dib H., “Fonctions de Bessel sur une algèbre de Jordan”, J. Math. Pures Appl., 69 (1990), 403–448 | MR | Zbl

[7] Enright T., Howe R., Wallach N., A classification of unitary highest weight modules, Representation Theory of Reductive Groups (Park City, Utah, 1982), Progr. Math., 40, Birkhäuser Boston, Boston, MA, 1983, 97–143 | DOI | MR

[8] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, v. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981 | MR

[9] Faraut J., Kaneyuki S., Korányi A., Lu Q.-K., Roos G., Analysis and geometry on complex homogeneous domains, Progress in Mathematics, 185, Birkhäuser Boston, Inc., Boston, MA, 2000 | DOI | MR | Zbl

[10] Faraut J., Korányi A., “Function spaces and reproducing kernels on bounded symmetric domains”, J. Funct. Anal., 88 (1990), 64–89 | DOI | MR | Zbl

[11] Faraut J., Korányi A., Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994 | MR

[12] Goodman R., Wallach N.R., Symmetry, representations, and invariants, Graduate Texts in Mathematics, 255, Springer, Dordrecht, 2009 | DOI | MR | Zbl

[13] Ibukiyama T., Kuzumaki T., Ochiai H., “Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms”, J. Math. Soc. Japan, 64 (2012), 273–316 | DOI | MR | Zbl

[14] Jakobsen H.P., “Intertwining differential operators for ${\rm Mp}(n,R)$ and ${\rm SU}(n,n)$”, Trans. Amer. Math. Soc., 246 (1978), 311–337 | DOI | MR | Zbl

[15] Jakobsen H.P., “Hermitian symmetric spaces and their unitary highest weight modules”, J. Funct. Anal., 52 (1983), 385–412 | DOI | MR | Zbl

[16] Jakobsen H.P., Vergne M., “Restrictions and expansions of holomorphic representations”, J. Funct. Anal., 34 (1979), 29–53 | DOI | MR | Zbl

[17] Juhl A., Families of conformally covariant differential operators, $Q$-curvature and holography, Progress in Mathematics, 275, Birkhäuser Verlag, Basel, 2009 | DOI | MR | Zbl

[18] Kobayashi T., “Multiplicity-free representations and visible actions on complex manifolds”, Publ. Res. Inst. Math. Sci., 41 (2005), 497–549 | DOI | MR | Zbl

[19] Kobayashi T., “Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs”, Representation Theory and Automorphic Forms, Progr. Math., 255, Birkhäuser Boston, Boston, MA, 2008, 45–109, arXiv: math.RT/0607002 | DOI | MR | Zbl

[20] Kobayashi T., “$F$-method for constructing equivariant differential operators”, Geometric Analysis and Integral Geometry, Contemp. Math., 598, Amer. Math. Soc., Providence, RI, 2013, 139–146, arXiv: 1212.6862 | DOI | MR | Zbl

[21] Kobayashi T., “A program for branching problems in the representation theory of real reductive groups”, Representations of Reductive Groups, Progr. Math., 312, Birkhäuser/Springer, Cham, 2015, 277–322, arXiv: 1509.08861 | DOI | MR | Zbl

[22] Kobayashi T., Kubo T., Pevzner M., Conformal symmetry breaking operators for differential forms on spheres, Lecture Notes in Math., 2170, Springer, Singapore, 2016 | DOI | MR | Zbl

[23] Kobayashi T., Ørsted B., Somberg P., Souček V., “Branching laws for Verma modules and applications in parabolic geometry. I”, Adv. Math., 285 (2015), 1796–1852, arXiv: 1305.6040 | DOI | MR | Zbl

[24] Kobayashi T., Pevzner M., “Differential symmetry breaking operators: I General theory and F-method”, Selecta Math. (N.S.), 22 (2016), 801–845, arXiv: 1301.2111 | DOI | MR | Zbl

[25] Kobayashi T., Pevzner M., “Differential symmetry breaking operators: II Rankin–Cohen operators for symmetric pairs”, Selecta Math. (N.S.), 22 (2016), 847–911, arXiv: 1301.2111 | DOI | MR | Zbl

[26] Kobayashi T., Pevzner M., “Inversion of Rankin–Cohen operators via holographic transform”, Ann. Inst. Fourier (Grenoble), 70 (2020), 2131–2190, arXiv: 1812.09733 | DOI | MR | Zbl

[27] Kobayashi T., Speh B., Symmetry breaking for representations of rank one orthogonal groups, Mem. Amer. Math. Soc., 238, 2015, v+110 pp. | DOI | MR

[28] Kobayashi T., Speh B., Symmetry breaking for representations of rank one orthogonal groups II, Lecture Notes in Math., 2234, Springer, Singapore, 2018 | DOI | MR | Zbl

[29] Loos O., Bounded symmetric domains and Jordan pairs, Mathematical Lectures, University of California, Irvine, 1977 | MR

[30] Martens S., “The characters of the holomorphic discrete series”, Proc. Nat. Acad. Sci. USA, 72 (1975), 3275–3276 | DOI | MR | Zbl

[31] Möllers J., Oshima Y., “Restriction of most degenerate representations of $O(1,N)$ with respect to symmetric pairs”, J. Math. Sci. Univ. Tokyo, 22 (2015), 279–338, arXiv: 1209.2312 | MR | Zbl

[32] Nakahama R., “Norm computation and analytic continuation of vector valued holomorphic discrete series representations”, J. Lie Theory, 26 (2016), 927–990, arXiv: 1506.05919 | MR | Zbl

[33] Nakahama R., “Construction of intertwining operators between holomorphic discrete series representations”, SIGMA, 15 (2019), 036, 101 pp., arXiv: 1804.07100 | DOI | MR | Zbl

[34] Ovsienko V., Redou P., “Generalized transvectants-Rankin–Cohen brackets”, Lett. Math. Phys., 63 (2003), 19–28 | DOI | MR | Zbl

[35] Peetre J., “Hankel forms of arbitrary weight over a symmetric domain via the transvectant”, Rocky Mountain J. Math., 24 (1994), 1065–1085 | DOI | MR | Zbl

[36] Peng L., Zhang G., “Tensor products of holomorphic representations and bilinear differential operators”, J. Funct. Anal., 210 (2004), 171–192 | DOI | MR | Zbl

[37] Rankin R.A., “The construction of automorphic forms from the derivatives of a given form”, J. Indian Math. Soc. (N.S.), 20 (1956), 103–116 | MR | Zbl

[38] Satake I., Algebraic structures of symmetric domains, Kanô Memorial Lectures, 4, Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980 | MR | Zbl

[39] Shimura G., “Arithmetic of differential operators on symmetric domains”, Duke Math. J., 48 (1981), 813–843 | DOI | MR | Zbl

[40] Yan Z.M., “A class of generalized hypergeometric functions in several variables”, Canad. J. Math., 44 (1992), 1317–1338 | DOI | MR | Zbl

[41] Yokota I., “Realizations of involutive automorphisms $\sigma$ and $G^\sigma$ of exceptional linear Lie groups $G$ I $G=G_2$, $F_4$ and $E_6$”, Tsukuba J. Math., 14 (1990), 185–223 | DOI | MR | Zbl

[42] Yokota I., Exceptional Lie groups, arXiv: 0902.0431

[43] Zhang G., “Branching coefficients of holomorphic representations and Segal–Bargmann transform”, J. Funct. Anal., 195 (2002), 306–349, arXiv: math.RT/0110212 | DOI | MR | Zbl