Lowest 𝔰𝔩(2)-types in 𝔰𝔩(𝔫)-representations

Lhou, Hassan; Willenbring, Jeb

doi:10.1090/ert/492

Lowest $\mathfrak {sl}(2)$-types in $\mathfrak {sl}(n)$-representations
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by Hassan Lhou and Jeb F. Willenbring PDF

Represent. Theory 21 (2017), 20-34 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 Cartan–Helgason theorem, Pieri rules, Hermite reciprocity, and a calculation in the “branching algebra” introduced by Roger Howe, Eng-Chye Tan, and the second author.

References

I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Models of representations of Lie groups, Trudy Sem. Petrovsk. Vyp. 2 (1976), 3–21 (Russian). MR 0453927
Ranee Kathryn Brylinski, Limits of weight spaces, Lusztig’s $q$-analogs, and fiberings of adjoint orbits, J. Amer. Math. Soc. 2 (1989), no. 3, 517–533. MR 984511, DOI 10.1090/S0894-0347-1989-0984511-X
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.
E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (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
È. 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. 5–259 (Russian).
Roger Howe, The classical groups and invariants of binary forms, The mathematical heritage of Hermann Weyl (Durham, NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 133–166. MR 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. 191–231. 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), 973–1032., 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. 3, 581–615. MR 2743105
Ivan Penkov and Gregg Zuckerman, A construction of generalized Harish-Chandra modules for locally reductive Lie algebras, Transform. Groups 13 (2008), no. 3-4, 799–817. 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. 1, 29–36. 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. 385–397. 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. 11, 3809–3814. 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. 403–429. MR 2401818, DOI 10.1142/9789812770790_{0}011