## Kyoto Journal of Mathematics

### Degenerate affine Grassmannians and loop quivers

#### Abstract

We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers, and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type $\operatorname{GL}_{n}$ and identify it with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional appro- ximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type $A_{1}^{(1)}$, propose a conjectural description in the symplectic case, and discuss the generalization to the case of the affine degenerate flag varieties.

#### Article information

Source
Kyoto J. Math., Volume 57, Number 2 (2017), 445-474.

Dates
Received: 30 September 2015
Revised: 26 November 2015
Accepted: 31 March 2016
First available in Project Euclid: 9 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1494295226

Digital Object Identifier
doi:10.1215/21562261-3821864

Mathematical Reviews number (MathSciNet)
MR3648057

Zentralblatt MATH identifier
06736609

#### Citation

Feigin, Evgeny; Finkelberg, Michael; Reineke, Markus. Degenerate affine Grassmannians and loop quivers. Kyoto J. Math. 57 (2017), no. 2, 445--474. doi:10.1215/21562261-3821864. https://projecteuclid.org/euclid.kjm/1494295226

#### References

• [1] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65–68.
• [2] G. Cerulli Irelli, E. Feigin, and M. Reineke, Quiver Grassmannians and degenerate flag varieties, Algebra Number Theory 6 (2012), 165–194.
• [3] G. Cerulli Irelli and M. Lanini, Degenerate flag varieties of type A and C are Schubert varieties, Int. Math. Res. Not. IMRN 2015, no. 15, 6353–6374.
• [4] B. Feigin, R. Kedem, S. Loktev, T. Miwa, and E. Mukhin, Combinatorics of the $\widehat{\mathfrak{sl}}_{2}$ spaces of coinvariants, Transform. Groups 6 (2001), 25–52.
• [5] E. Feigin, Degenerate flag varieties and the median Genocchi numbers, Math. Res. Lett. 18 (2011), 1163–1178.
• [6] E. Feigin, ${\mathbb{G}}_{a}^{M}$ degeneration of flag varieties, Selecta Math. (N.S.) 18 (2012), 513–537.
• [7] E. Feigin, Systems of correlation functions, co-invariants, and the Verlinde algebra, Funktsional. Anal. i Prilozhen. 46 (2012), 49–64; English translation in Funct. Anal. Appl. 46 (2012), 41–52.
• [8] E. Feigin and M. Finkelberg, Degenerate flag varieties of type A: Frobenius splitting and BW theorem, Math. Z. 275 (2013), 55–77.
• [9] E. Feigin, M. Finkelberg, and P. Littelmann, Symplectic degenerate flag varieties, Canad. J. Math. 66 (2014), 1250–1286.
• [10] E. Feigin, G. Fourier, and P. Littelmann, PBW filtration and bases for irreducible modules in type $A_{n}$, Transform. Groups 16 (2011), 71–89.
• [11] E. Feigin, G. Fourier, and P. Littelmann, “PBW-filtration over $\mathbb{Z}$ and compatible bases for $V_{\mathbb{Z}}(\lambda)$ in type ${\tt a}_{n}$ and ${\tt c}_{n}$,” in Symmetries, Integrable Systems and Representations, Springer Proc. Math. Stat. 40, Springer, Heidelberg, 2013, 35–63.
• [12] I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123–168.
• [13] C. Geiß, “Introduction to moduli spaces associated to quivers,” with an appendix by L. Le Bruyn and M. Reineke, in Trends in Representation Theory of Algebras and Related Topics, Contemp. Math. 406, Amer. Math. Soc., Providence, 2006, 31–50.
• [14] U. Görtz, On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann. 321 (2001), 689–727.
• [15] V. G. Kac and D. H. Peterson, “Lectures on the infinite wedge-representation and the MKP hierarchy” in Systèmes dynamiques non linéaires: intégrabilité et comportement qualitatif, Sém. Math. Sup. 102, Presses Univ. Montréal, Montréal, 1986, 141–184.
• [16] H. Kraft and C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 (1979), 227–247.
• [17] S. Kumar, Demazure character formula in arbitrary Kac-Moody setting, Invent. Math. 89 (1987), 395–423.
• [18] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169–178.
• [19] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math Monogr., Oxford Univ. Press, New York, 1995.
• [20] O. Mathieu, Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque 159-160, Soc. Math. France, Paris, 1988.
• [21] G. Pappas, M. Rapoport, and B. Smithling, “Local models of Shimura varieties, I: Geometry and combinatorics,” in Handbook of Moduli, Vol. III, Adv. Lect. Math. (ALM) 26, Int. Press, Somerville, Mass., 2013, 135–217.
• [22] J.-P. Serre, Espaces fibrés algebriques, Séminaire Claude Chevalley 3 (1958), 1–37.