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

SIGMA 21 (2025), 095, 37 pages      arXiv:2410.17125      https://doi.org/10.3842/SIGMA.2025.095

Construction of Irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-Modules and Discretely Decomposable Restrictions

Masatoshi Kitagawa
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Fukuoka-shi, 819-0395, Fukuoka, Japan

Received March 31, 2025, in final form October 27, 2025; Published online November 08, 2025

Abstract
In this paper, we study the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-modules using the Zuckerman derived functors. We provide criteria for the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules in the cases of generalized Verma modules, cohomologically induced modules, and discrete series representations. We treat only discrete decomposable restrictions with certain dominance conditions (quasi-abelian and in the good range). To describe the $\mathcal{U}(\mathfrak{g})^{G'}$-modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when $K'$ acts on $K/L_K$ transitively.

Key words: Lie group; representation theory; branching problem; highest weight module; holomorphic discrete series representation; Zuckerman's derived functor.

pdf (680 kb)   tex (47 kb)  

References

  1. Bernstein J.N., Gel'fand S.I., Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245-285.
  2. Borel A., Wallach N., Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Math. Surveys Monogr., Vol. 67, American Mathematical Society, Providence, RI, 2000.
  3. Duflo M., Vargas J.A., Branching laws for square integrable representations, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 49-54.
  4. Enright T.J., Lectures on representations of complex semisimple Lie groups, Tata Inst. Fund. Res. Lectures on Math. and Phys., Vol. 66, Tata Institute of Fundamental Research, Bombay, 1981.
  5. Enright T.J., Parthasarathy R., Wallach N.R., Wolf J.A., Unitary derived functor modules with small spectrum, Acta Math. 154 (1985), 105-136.
  6. Goodman R., Wallach N.R., Symmetry, representations, and invariants, Grad. Texts in Math., Vol. 255, Springer, Dordrecht, 2009.
  7. Gross B., Wallach N., Restriction of small discrete series representations to symmetric subgroups, in The Mathematical Legacy of Harish-Chandra (Baltimore, MD, 1998), Proc. Sympos. Pure Math., Vol. 68, American Mathematical Society, Providence, RI, 2000, 255-272.
  8. Hecht H., Miličić D., Schmid W., Wolf J.A., Localization and standard modules for real semisimple Lie groups. I. The duality theorem, Invent. Math. 90 (1987), 297-332.
  9. Howe R., Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570.
  10. Humphreys J.E., Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$, Grad. Stud. Math., Vol. 94, American Mathematical Society, Providence, RI, 2008.
  11. Jakobsen H.P., Vergne M., Restrictions and expansions of holomorphic representations, J. Funct. Anal. 34 (1979), 29-53.
  12. Kashiwara M., Vergne M., On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), 1-47.
  13. Kitagawa M., Uniformly bounded multiplicities, polynomial identities and coisotropic actions, arXiv:2109.05555.
  14. Kitagawa M., Family of $\mathcal{D}$-modules and representations with a boundedness property, Represent. Theory 27 (2023), 292-355, arXiv:2109.05556.
  15. Knapp A.W., Representation theory of semisimple groups. An overview based on examples, Princeton Math. Ser., Vol. 36, Princeton University Press, Princeton, NJ, 1986.
  16. Knapp A.W., Vogan Jr. D.A., Cohomological induction and unitary representations, Princeton Math. Ser., Vol. 45, Princeton University Press, Princeton, NJ, 1995.
  17. Kobayashi T., The restriction of $A_q(\lambda)$ to reductive subgroups, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 262-267.
  18. 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.
  19. 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.
  20. 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.
  21. Kobayashi T., Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal. 152 (1998), 100-135.
  22. 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, MA, 2008, 45-109, arXiv:math.RT/0607002.
  23. Kobayashi T., Branching problems of Zuckerman derived functor modules, in Representation Theory and Mathematical Physics, Contemp. Math., Vol. 557, American Mathematical Society, Providence, RI, 2011, 23-40, arXiv:1104.4399.
  24. Kobayashi T., Restrictions of generalized Verma modules to symmetric pairs, Transform. Groups 17 (2012), 523-546, arXiv:1008.4544.
  25. Kobayashi T., Recent advances in branching problems of representations, Sugaku Expositions 37 (2024), 129-177, arXiv:2112.00642.
  26. 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.
  27. 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.
  28. 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.
  29. 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.
  30. Ma J.-J., Derived functor modules, dual pairs and $\mathcal{U}(\mathfrak{g})^K$-actions, J. Algebra 450 (2016), 629-645, arXiv:1310.6378.
  31. McConnell J.C., Robson J.C., Noncommutative Noetherian rings, Grad. Stud. Math, Vol. 30, American Mathematical Society, Providence, RI, 2001.
  32. Ørsted B., Vargas J.A., Branching problems in reproducing kernel spaces, Duke Math. J. 169 (2020), 3477-3537, arXiv:1906.08119.
  33. Ørsted B., Vargas J.A., Pseudo-dual pairs and branching of discrete series, in Symmetry in Geometry and Analysis. Vol. 2. Festschrift in Honor of Toshiyuki Kobayashi, Progr. Math., Vol. 358, Birkhäuser, Singapore, 2025, 497-549, arXiv:2302.14190.
  34. Oshima Y., Discrete branching laws of Zuckerman's derived functor modules, Ph.D. Thesis, The University of Tokyo, 2013.
  35. Oshima Y., Discrete branching laws of derived functor modules, in Symmetry in Geometry and Analysis. Vol. 3. Festschrift in Honor of Toshiyuki Kobayashi, Progr. Math., Vol. 359, Birkhäuser, Singapore, 2025, 127-636.
  36. Penkov I., Serganova V., On bounded generalized Harish-Chandra modules, Ann. Inst. Fourier (Grenoble) 62 (2012), 477-496, arXiv:0710.0906.
  37. Penkov I., Zuckerman G., Generalized Harish-Chandra modules: a new direction in the structure theory of representations, Acta Appl. Math. 81 (2004), 311-326, arXiv:math.RT/0310140.
  38. Penkov I., Zuckerman G., Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type, Asian J. Math. 8 (2004), 795-811, arXiv:math.RT/0409285.
  39. Penkov I., Zuckerman G., On the structure of the fundamental series of generalized Harish-Chandra modules, Asian J. Math. 16 (2012), 489-514, arXiv:1109.1804.
  40. Sekiguchi H., Branching rules of singular unitary representations with respect to symmetric pairs $(A_{2n-1},D_n)$, Internat. J. Math. 24 (2013), 1350011, 25 pages.
  41. Vargas J.A., Associated symmetric pair and multiplicities of admissible restriction of discrete series, Internat. J. Math. 27 (2016), 1650100, 29 pages, arXiv:1311.3851.
  42. Wallach N.R., Real reductive groups. I, Pure Appl. Math., Vol. 132, Academic Press, Inc., Boston, MA, 1988.

Previous article  Next article  Contents of Volume 21 (2025)