Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 036, 101 pages      arXiv:1804.07100      https://doi.org/10.3842/SIGMA.2019.036

Construction of Intertwining Operators between Holomorphic Discrete Series Representations

Ryosuke Nakahama
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan

Received April 24, 2018, in final form April 02, 2019; Published online May 05, 2019

Abstract
In this paper we explicitly construct $G_1$-intertwining operators between holomorphic discrete series representations $\mathcal{H}$ of a Lie group $G$ and those $\mathcal{H}_1$ of a subgroup $G_1\subset G$ when $(G,G_1)$ is a symmetric pair of holomorphic type. More precisely, we construct $G_1$-intertwining projection operators from $\mathcal{H}$ onto $\mathcal{H}_1$ as differential operators, in the case $(G,G_1)=(G_0\times G_0,\Delta G_0)$ and both $\mathcal{H}$, $\mathcal{H}_1$ are of scalar type, and also construct $G_1$-intertwining embedding operators from $\mathcal{H}_1$ into $\mathcal{H}$ as infinite-order differential operators, in the case $G$ is simple, $\mathcal{H}$ is of scalar type,and $\mathcal{H}_1$ is multiplicity-free under a maximal compact subgroup $K_1\subset K$. In the actual computation we make use of series expansions of integral kernels and the result of Faraut-Korányi (1990) or the author's previous result (2016) on norm computation. As an application, we observe the behavior of residues of the intertwining operators, which define the maps from some subquotient modules, when the parameters are at poles.

Key words: branching laws; intertwining operators; symmetry breaking operators; symmetric pairs; holomorphic discrete series representations; highest weight modules.

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References

  1. Ben Saïd S., Clerc J.L., Koufany K., Conformally covariant bi-differential operators on a simple real Jordan algebra, arXiv:1704.01817.
  2. Clerc J.L., Kobayashi T., Ørsted B., Pevzner M., Generalized Bernstein-Reznikov integrals, Math. Ann. 349 (2011), 395-431, arXiv:0906.2874.
  3. Cohen H., Sums involving the values at negative integers of $L$-functions of quadratic characters, Math. Ann. 217 (1975), 271-285.
  4. Dib H., Fonctions de Bessel sur une algèbre de Jordan, J. Math. Pures Appl. 69 (1990), 403-448.
  5. Enright T., Howe R., Wallach N., A classification of unitary highest weight modules, in Representation Theory of Reductive Groups (Park City, Utah, 1982), Progr. Math., Vol. 40, Birkhäuser Boston, Boston, MA, 1983, 97-143.
  6. Faraut J., Kaneyuki S., Korányi A., Lu Q.k., Roos G., Analysis and geometry on complex homogeneous domains, Progress in Mathematics, Vol. 185, Birkhäuser Boston, Inc., Boston, MA, 2000.
  7. Faraut J., Korányi A., Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 64-89.
  8. Faraut J., Korányi A., Analysis on symmetric cones, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.
  9. 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.
  10. Jakobsen H.P., Hermitian symmetric spaces and their unitary highest weight modules, J. Funct. Anal. 52 (1983), 385-412.
  11. Jakobsen H.P., Vergne M., Restrictions and expansions of holomorphic representations, J. Funct. Anal. 34 (1979), 29-53.
  12. Juhl A., Families of conformally covariant differential operators, $Q$-curvature and holography, Progress in Mathematics, Vol. 275, Birkhäuser Verlag, Basel, 2009.
  13. Kobayashi T., Discrete decomposability of the restriction of $A_{\mathfrak q}(\lambda)$ with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), 181-205.
  14. Kobayashi T., Multiplicity free theorem in branching problems of unitary highest weight modules, in Proceedings of the Symposium on Representation Theory (1997, Saga, Kyushu), Editor K. Mimachi, Tokyo University of Science, 1997, 9-17.
  15. Kobayashi T., Discrete decomposability of the restriction of $A_{\mathfrak q}(\lambda)$ with respect to reductive subgroups. II. Micro-local analysis and asymptotic $K$-support, Ann. of Math. 147 (1998), 709-729.
  16. Kobayashi T., Discrete decomposability of the restriction of $A_{\mathfrak q}(\lambda)$ with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), 229-256.
  17. Kobayashi T., Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005), 497-549.
  18. Kobayashi T., Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, in Representation Theory and Automorphic Forms, Progr. Math., Vol. 255, Birkhäuser Boston, Boston, MA, 2008, 45-109, arXiv:math.RT/0607002.
  19. Kobayashi T., Restrictions of generalized Verma modules to symmetric pairs, Transform. Groups 17 (2012), 523-546, arXiv:1008.4544.
  20. Kobayashi T., $F$-method for constructing equivariant differential operators, in Geometric Analysis and Integral Geometry, Contemp. Math., Vol. 598, Amer. Math. Soc., Providence, RI, 2013, 139-146, arXiv:1212.6862.
  21. Kobayashi T., Propagation of multiplicity-freeness property for holomorphic vector bundles, in Lie Groups: Structure, Actions, and Representations, Progr. Math., Vol. 306, Birkhäuser/Springer, New York, 2013, 113-140, arXiv:math.RT/0607004.
  22. Kobayashi T., Shintani functions, real spherical manifolds, and symmetry breaking operators, in Developments and Retrospectives in Lie Theory, Dev. Math., Vol. 37, Springer, Cham, 2014, 127-159, arXiv:1401.0117.
  23. Kobayashi T., Symmetric pairs with finite-multiplicity property for branching laws of admissible representations, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), 79-83.
  24. Kobayashi T., A program for branching problems in the representation theory of real reductive groups, in Representations of Reductive Groups, Progr. Math., Vol. 312, Birkhäuser/Springer, Cham, 2015, 277-322, arXiv:1509.08861.
  25. Kobayashi T., Kubo T., Pevzner M., Conformal symmetry breaking operators for differential forms on spheres, Lecture Notes in Math., Vol. 2170, Springer, Singapore, 2016, arXiv:1605.09272.
  26. Kobayashi T., Matsuki T., Classification of finite-multiplicity symmetric pairs, Transform. Groups 19 (2014), 457-493, arXiv:1312.4246.
  27. 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.
  28. Kobayashi T., Oshima T., Finite multiplicity theorems for induction and restriction, Adv. Math. 248 (2013), 921-944, arXiv:1108.3477.
  29. Kobayashi T., Oshima Y., Classification of discretely decomposable $A_{\mathfrak q}(\lambda)$ with respect to reductive symmetric pairs, Adv. Math. 231 (2012), 2013-2047, arXiv:1104.4400.
  30. Kobayashi T., Oshima Y., Classification of symmetric pairs with discretely decomposable restrictions of $(\mathfrak{g},K)$-modules, J. Reine Angew. Math. 703 (2015), 201-223, arXiv:1202.5743.
  31. 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.
  32. 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.
  33. Kobayashi T., Speh B., Intertwining operators and the restriction of representations of rank-one orthogonal groups, C. R. Math. Acad. Sci. Paris 352 (2014), 89-94.
  34. Kobayashi T., Speh B., Symmetry breaking for representations of rank one orthogonal groups, Mem. Amer. Math. Soc. 238 (2015), v+110 pages, arXiv:1310.3213.
  35. Kobayashi T., Speh B., Symmetry breaking for representations of rank one orthogonal groups II, Lecture Notes in Math., Vol. 2234, Springer, Singapore, 2018.
  36. Loos O., Bounded symmetric domains and Jordan pairs, Math. Lectures, University of California, Irvine, 1977.
  37. Martens S., The characters of the holomorphic discrete series, Proc. Nat. Acad. Sci. USA 72 (1975), 3275-3276.
  38. Möllers J., Ørsted B., Oshima Y., Knapp-Stein type intertwining operators for symmetric pairs, Adv. Math. 294 (2016), 256-306, arXiv:1309.3904.
  39. 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.
  40. Muirhead R.J., Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982.
  41. Nakahama R., Integral formula and upper estimate of I and J-Bessel functions on Jordan algebras, J. Lie Theory 24 (2014), 421-438, arXiv:1211.4702.
  42. Nakahama R., Norm computation and analytic continuation of vector valued holomorphic discrete series representations, J. Lie Theory 26 (2016), 927-990, arXiv:1506.05919.
  43. Ovsienko V., Redou P., Generalized transvectants-Rankin-Cohen brackets, Lett. Math. Phys. 63 (2003), 19-28.
  44. Peetre J., Hankel forms of arbitrary weight over a symmetric domain via the transvectant, Rocky Mountain J. Math. 24 (1994), 1065-1085.
  45. Peng L., Zhang G., Tensor products of holomorphic representations and bilinear differential operators, J. Funct. Anal. 210 (2004), 171-192.
  46. Satake I., Algebraic structures of symmetric domains, Kanô Memorial Lectures, Vol. 4, Iwanami Shoten, Tokyo, Princeton University Press, Princeton, N.J., 1980.
  47. Stembridge J.R., Multiplicity-free products and restrictions of Weyl characters, Represent. Theory 7 (2003), 404-439.
  48. Tsukamoto C., Spectra of Laplace-Beltrami operators on ${\rm SO}(n+2)/{\rm SO}(2)\times {\rm SO}(n)$ and ${\rm Sp}(n+1)/{\rm Sp}(1)\times {\rm Sp}(n)$, Osaka J. Math. 18 (1981), 407-426.
  49. 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.
  50. Yokota I., Exceptional Lie groups, arXiv:0902.0431.
  51. Zhang G., Branching coefficients of holomorphic representations and Segal-Bargmann transform, J. Funct. Anal. 195 (2002), 306-349, arXiv:math.RT/0110212.

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