## Lowest $\mathfrak {sl}(2)$-types in $\mathfrak {sl}(n)$-representations

HTML articles powered by AMS MathViewer

- by Hassan Lhou and Jeb F. Willenbring
- Represent. Theory
**21**(2017), 20-34 - DOI: https://doi.org/10.1090/ert/492
- Published electronically: March 13, 2017
- PDF | Request permission

## Abstract:

Fix $n \geq 3$. Let $\mathfrak {s}$ be a principally embedded $\mathfrak {sl}_2$-subalgebra in $\mathfrak {sl}_n$. A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer $b(n)$ such that for any finite dimensional irreducible $\mathfrak {sl}_n$-representation, $V$, there exists an irreducible $\mathfrak {s}$-representation embedding in $V$ with dimension at most $b(n)$. We prove that $b(n)=n$ is the sharpest possible bound. We also address embeddings other than the principal one.

The exposition involves an application of the CartanāHelgason theorem, Pieri rules, Hermite reciprocity, and a calculation in the ābranching algebraā introduced by Roger Howe, Eng-Chye Tan, and the second author.

## References

- I. N. BernÅ”teÄn, I. M. Gelā²fand, and S. I. Gelā²fand,
*Models of representations of Lie groups*, Trudy Sem. Petrovsk.**Vyp. 2**(1976), 3ā21 (Russian). MR**0453927** - Ranee Kathryn Brylinski,
*Limits of weight spaces, Lusztigās $q$-analogs, and fiberings of adjoint orbits*, J. Amer. Math. Soc.**2**(1989), no.Ā 3, 517ā533. MR**984511**, DOI 10.1090/S0894-0347-1989-0984511-X - DavidĀ H. Collingwood and WilliamĀ M. McGovern,
*Nilpotent orbits in semisimple Lie algebras*, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. - E.Ā B. Dynkin,
*Semisimple subalgebras of semisimple Lie algebras*, Mat. Sbornik N.S.**30(72)**(1952), 349ā462 (3 plates). - Roe Goodman and Nolan R. Wallach,
*Symmetry, representations, and invariants*, Graduate Texts in Mathematics, vol. 255, Springer, Dordrecht, 2009. MR**2522486**, DOI 10.1007/978-0-387-79852-3 - Ć. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik,
*Structure of Lie groups and Lie algebras*, Current problems in mathematics. Fundamental directions, Vol. 41 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp.Ā 5ā259 (Russian). - Roger Howe,
, The mathematical heritage of Hermann Weyl (Durham, NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp.Ā 133ā166. MR*The classical groups*and invariants of binary forms**974333**, DOI 10.1090/pspum/048/974333 - Roger E. Howe, Eng-Chye Tan, and Jeb F. Willenbring,
*Reciprocity algebras and branching for classical symmetric pairs*, Groups and analysis, London Math. Soc. Lecture Note Ser., vol. 354, Cambridge Univ. Press, Cambridge, 2008, pp.Ā 191ā231. MR**2528468**, DOI 10.1017/CBO9780511721410.011 - Bertram Kostant,
*The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group*, Amer. J. Math.**81**(1959), 973ā1032., DOI 10.2307/2372999 - Ivan Penkov and Vera Serganova,
*Bounded simple $({\mathfrak {g}},\textrm {sl}(2))$-modules for $\textrm {rk}\,{\mathfrak {g}}=2$*, J. Lie Theory**20**(2010), no.Ā 3, 581ā615. MR**2743105** - Ivan Penkov and Gregg Zuckerman,
*A construction of generalized Harish-Chandra modules for locally reductive Lie algebras*, Transform. Groups**13**(2008), no.Ā 3-4, 799ā817. MR**2452616**, DOI 10.1007/s00031-008-9034-9 - Michel Brion,
*On the representation theory of $\textrm {SL}(2)$*, Indag. Math. (N.S.)**5**(1994), no.Ā 1, 29ā36. MR**1268726**, DOI 10.1016/0019-3577(94)90030-2 - Anthony van Groningen and Jeb F. Willenbring,
*The cubic, the quartic, and the exceptional group $\rm G_2$*, Developments and retrospectives in Lie theory, Dev. Math., vol. 38, Springer, Cham, 2014, pp.Ā 385ā397. MR**3308792**, DOI 10.1007/978-3-319-09804-3_{1}7 - Ben Webster,
*Cramped subgroups and generalized Harish-Chandra modules*, Proc. Amer. Math. Soc.**136**(2008), no.Ā 11, 3809ā3814. MR**2425719**, DOI 10.1090/S0002-9939-08-09421-5 - Jeb F. Willenbring and Gregg J. Zuckerman,
*Small semisimple subalgebras of semisimple Lie algebras*, Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 12, World Sci. Publ., Hackensack, NJ, 2007, pp.Ā 403ā429. MR**2401818**, DOI 10.1142/9789812770790_{0}011

## Bibliographic Information

**Hassan Lhou**- Affiliation: Department of Mathematical Sciences, University of Wisconsin - Milwaukee, 3200 North Cramer Street, Milwaukee, Wisconsin 53211
- Email: hlhou@uwm.edu
**Jeb F. Willenbring**- Affiliation: Department of Mathematical Sciences, University of Wisconsin - Milwaukee, 3200 North Cramer Street, Milwaukee, Wisconsin 53211
- MR Author ID: 662347
- Email: jw@uwm.edu
- Received by editor(s): September 12, 2016
- Received by editor(s) in revised form: October 23, 2016
- Published electronically: March 13, 2017
- Additional Notes: The second author was supported by the National Security Agency grant # H98230-09-0054.
- © Copyright 2017 American Mathematical Society
- Journal: Represent. Theory
**21**(2017), 20-34 - MSC (2010): Primary 17B10; Secondary 05E10, 22E46
- DOI: https://doi.org/10.1090/ert/492
- MathSciNet review: 3622114