## Duke Mathematical Journal

### On the existence of $\mathfrak{g}$, $\mathfrak{k}$-modules of finite type

#### Abstract

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed field of characteristic zero, and let $\mathfrak{k}$ be a subalgebra reductive in $\mathfrak{g}$. We prove that $\mathfrak{g}$ admits an irreducible ($\mathfrak{g}$,$\mathfrak{k}$)-module M which has finite $\mathfrak{k}$-multiplicities and which is not a ($\mathfrak{g}$,$\mathfrak{k}$′)-module for any proper inclusion of reductive subalgebras $\mathfrak{k}$⊂$\mathfrak{k}$′⊂$\mathfrak{g}$ if and only if $\mathfrak{k}$ contains its centralizer in $\mathfrak{g}$. The main point of the proof is a geometric construction of ($\mathfrak{g}$,$\mathfrak{k}$)-modules which is analogous to cohomological induction. For $\mathfrak{g}=\mathfrak{g}\mathfrak{l}$(n) we show that whenever $\mathfrak{k}$ contains its centralizer, there is an irreducible ($\mathfrak{g}$,$\mathfrak{k}$)-module M of finite type over $\mathfrak{k}$ such that $\mathfrak{k}$ coincides with the subalgebra of all $g∈\mathfrak{g}$ which act locally finitely on M. Finally, for a root subalgebra $\mathfrak{k}⊂\mathfrak{g}\mathfrak{l}(n)$, we describe all possibilities for the subalgebra $\mathfrak{l}$⊃$\mathfrak{k}$ of all elements acting locally finitely on some M.

#### Article information

Source
Duke Math. J., Volume 125, Number 2 (2004), 329-349.

Dates
First available in Project Euclid: 27 October 2004

https://projecteuclid.org/euclid.dmj/1098892272

Digital Object Identifier
doi:10.1215/S0012-7094-04-12525-4

Mathematical Reviews number (MathSciNet)
MR2096676

Zentralblatt MATH identifier
1097.17007

Subjects
Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 22E46: Semisimple Lie groups and their representations

#### Citation

Penkov, Ivan; Serganova, Vera; Zuckerman, Gregg. On the existence of $\mathfrak{g}$, $\mathfrak{k}$-modules of finite type. Duke Math. J. 125 (2004), no. 2, 329--349. doi:10.1215/S0012-7094-04-12525-4. https://projecteuclid.org/euclid.dmj/1098892272

#### References

• \lccA. Beĭlinson and J. Bernstein, “A proof of Jantzen conjectures” in I. M. Gel'fand Seminar, Adv. Soviet Math. 16, Part 1, Amer. Math. Soc., Providence, 1993, 1–50.
• \lccN. Bourbaki, Éléments de mathématique, fasc. 38: Groupes et algèbres de Lie, Chapitre 7, Actualités Sci. Indust. 1364, Hermann, Paris, 1975.
• \lccE. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras (in Russian), Mat. Sb. N.S., 30(72) (1952), 349–462; English translation in Amer. Math. Soc. Transl. Ser. (2) 6 (1957), 111–244.
• \lccJ. Dixmier, Enveloping Algebras, North-Holland Math. Library 14, North Holland, Amsterdam, 1977.
• \lccS. L. Fernando, Lie algebra modules with finite-dimensional weight spaces, I, Trans. Amer. Math. Soc. 322 (1990), 757–781.
• \lccI. M. Gel'fand, “The cohomology of infinite dimensional Lie algebras: Some questions of integral geometry” in Actes du Congrès International des Mathématiciens, Tome 1 (Nice, 1970), Gauthier-Villars, Paris, 1971, 95–111.
• \lccV. G. Kac, “Constructing groups associated to infinite-dimensional Lie algebras” in Infinite-Dimensional Groups with Applications (Berkeley, 1984), Math. Sci. Res. Inst. Publ. 4, Springer, New York, 1985, 167–216.
• \lccF. I. Karpelevič, On nonsemisimple maximal subalgebras of semisimple Lie algebras (in Russian), Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 775–778.
• \lccP. Kekäläinen, On irreducible $A_2$-finite $G_2$-modules, J. Algebra 117 (1988), 72–80.
• \lccA. W. Knapp and D. A. Vogan, Cohomological Induction and Unitary Representations, Princeton Math. Ser. 45, Princeton Univ. Press, Princeton, 1995.
• \lccO. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), 537–592.
• \lccI. Penkov and V. Serganova, Generalized Harish-Chandra modules, Mosc. Math. J. 2 (2002), no. 4, 753–767.
• \lccG. Savin, Dual pair $\PGL(3) \times G_2$ and $(\mathfrak{g}_2, \SL(3))$-modules, Internat. Math. Res. Not. 1994, no. 4, 177–184; correction, Int. Math. Res. Not. 1994, no. 6, 273.
• \lccD. A. Vogan, Representations of Real Reductive Lie Groups, Progr. Math. 15, Birkhäuser, Boston, 1981. \lccV. G. Kac, “Constructing groups associated to infinite-dimensional Lie algebras” in Infinite-Dimensional Groups with Applications (Berkeley, 1984), Math. Sci. Res. Inst. Publ. 4, Springer, New York, 1985, 167–216.
• \lccF. I. Karpelevič, On nonsemisimple maximal subalgebras of semisimple Lie algebras (in Russian), Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 775–778.
• \lccP. Kekäläinen, On irreducible $A_2$-finite $G_2$-modules, J. Algebra 117 (1988), 72–80.
• \lccA. W. Knapp and D. A. Vogan, Cohomological Induction and Unitary Representations, Princeton Math. Ser. 45, Princeton Univ. Press, Princeton, 1995.
• \lccO. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), 537–592.
• \lccI. Penkov and V. Serganova, Generalized Harish-Chandra modules, Mosc. Math. J. 2 (2002), no. 4, 753–767.
• \lccG. Savin, Dual pair $\PGL(3) \times G_2$ and $(\mathfrak{g}_2, \SL(3))$-modules, Internat. Math. Res. Not. 1994, no. 4, 177–184; correction, Int. Math. Res. Not. 1994, no. 6, 273.
• \lccD. A. Vogan, Representations of Real Reductive Lie Groups, Progr. Math. 15, Birkhäuser, Boston, 1981.