Glasnik Matematicki, Vol. 60, No. 2 (2025), 243-265. \( \)

INDUCTION FROM TWO LINKED SEGMENTS WITH ONE HALF BORDER AND CUSPIDAL REDUCIBILITY

Igor Ciganović

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail:igor.ciganovic@math.hr

Abstract.   In this paper, we determine the composition series of the induced representation \(\delta([\nu^{\frac{1}{2}}\rho,\nu^c\rho])\times \delta([\nu^{-a}\rho,\nu^b\rho]) \rtimes \sigma\) where \(a, b, c \in \mathbb{Z}+\frac{1}{2}\) such that \(\frac{1}{2}\leq a < b < c\), \(\rho\) is an irreducible cuspidal unitary representation of a general linear group and \(\sigma\) is an irreducible cuspidal representation of a classical group such that \(\nu^\frac{1}{2}\rho\rtimes \sigma\) reduces.

2020 Mathematics Subject Classification.   22D30, 22E50, 22D12, 11F85

Key words and phrases.   Classical group, composition series, induced representations, p-adic field, Jacquet module

https://doi.org/10.3336/gm.60.2.05

References:

1. I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive \(p\)-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), 441–472.
   MathSciNet    Link

2. B. Bošnjak, Representations induced from cuspidal and ladder representations of classical \(p\)-adic groups, Proc. Amer. Math. Soc. 149 (2021), 5081–5091.
   MathSciNet    CrossRef

3. I. Ciganović, Composition series of a class of induced representations, a case of one half cuspidal reducibility, Pacific J. Math. 296 (2018), 21–30.
   MathSciNet    CrossRef

4. I. Ciganović, Composition series of a class of induced representations built on discrete series, Manuscripta Math. 170 (2023), 1–18.
   MathSciNet    CrossRef

5. I. Ciganović, Parabolic induction from two segments, linked under contragredient, with a one half cuspidal reducibility, a special case, Glas. Mat. Ser. III 59(79) (2024), 77–105.
   MathSciNet    CrossRef

6. D. Goldberg, Reducibility of induced representations for \(Sp(2n)\) and \(SO(2n)\), Amer. J. Math, 116 (1994), 1101–1151.
   MathSciNet    CrossRef

7. Y. Kim, B. Liu and I. Matić, Degenerate principal series for classical and odd \(G\)Spin groups in the general case, Represent. Theory 24 (2020), 403–434.
   MathSciNet    CrossRef

8. I. Matić, On discrete series subrepresentations of the generalized principal series, Glas. Mat. Ser. III 51(71) (2016), 125–152.
   MathSciNet    CrossRef

9. I. Matić, Representations induced from the Zelevinsky segment and discrete series in the half-integral case, Forum Math. 33 (2021), 193–212.
   MathSciNet    CrossRef

10. I. Matić and M. Tadić, On Jacquet modules of representations of segment type, Manuscripta Math. 147 (2015), 437–476.
    MathSciNet    CrossRef

11. I. Matić and M. Tadić, On Jacquet modules of representations of segment type.
    Link

12. C. Mœglin, Sur la classification des séries discrètes des groupes classiques \(p\)-adiques: paramètres de Langlands et exhaustivité, J. Eur. Math. Soc. (JEMS) 4 (2002), 143–200.
    MathSciNet    CrossRef

13. C. Mœglin and M. Tadić, Construction of discrete series for classical \(p\)-adic groups, J. Amer. Math. Soc. 15 (2002), 715–786.
    MathSciNet    CrossRef

14. G. Muić, Composition series of generalized principal series; the case of strongly positive discrete series, Israel J. Math. 140 (2004), 157–202.
    MathSciNet    CrossRef

15. G. Muić, Reducibility of generalized principal series, Canad. J. Math. 57 (2005), 616–647.
    MathSciNet    CrossRef

16. G. Muić, Reducibility of standard representations, Pacific J. Math. 222 (2005), 133–168.
    MathSciNet    CrossRef

17. M. Tadić, Structure arising from induction and Jacquet modules of representations of classical \(p\)-adic groups, J. Algebra 177 (1995), 1–33.
    MathSciNet    CrossRef

18. M. Tadić, On reducibility of parabolic induction, Israel J. Math. 107 (1998), 29–91.
    MathSciNet    CrossRef

19. M. Tadić, On regular square integrable representations of \(p\)-adic groups, Amer. J. Math. 120 (1998), 159–210.
    MathSciNet    Link

20. M. Tadić, On tempered and square integrable representations of classical \(p\)-adic groups, Sci. China Math. 56 (2013), 2273–2313.
    MathSciNet    CrossRef

21. A. V. Zelevinsky, Induced representations of reductive \(p\)-adic groups. II. On irreducible representations of \(GL(n)\), Ann. Sci. École Norm. Sup. (4) 13 (1980), 165–210.
    MathSciNet     Link