Skip to main content
×
Home
    • Aa
    • Aa

Characters of equivariant ${\mathcal{D}}$ -modules on spaces of matrices

  • Claudiu Raicu (a1) (a2)
Abstract

We compute the characters of the simple $\text{GL}$ -equivariant holonomic ${\mathcal{D}}$ -modules on the vector spaces of general, symmetric, and skew-symmetric matrices. We realize some of these ${\mathcal{D}}$ -modules explicitly as subquotients in the pole order filtration associated to the $\text{determinant}/\text{Pfaffian}$ of a generic matrix, and others as local cohomology modules. We give a direct proof of a conjecture of Levasseur in the case of general and skew-symmetric matrices, and provide counterexamples in the case of symmetric matrices. The character calculations are used in subsequent work with Weyman to describe the ${\mathcal{D}}$ -module composition factors of local cohomology modules with determinantal and Pfaffian support.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

T. Braden and M. Grinberg , Perverse sheaves on rank stratifications , Duke Math. J. 96 (1999), 317362, doi:10.1215/S0012-7094-99-09609-6; MR 1666554 (2000f:14031).

S. C. Coutinho and D. Levcovitz , D-modules and étale morphisms , Comm. Algebra 29 (2001), 14871497, doi:10.1081/AGB-100002114; MR 1853107 (2002m:14014).

R. Hotta , K. Takeuchi and T. Tanisaki , D-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236 (Birkhäuser, Boston, MA, 2008); MR 2357361 (2008k:32022).

M. Kashiwara , B-functions and holonomic systems. Rationality of roots of B-functions , Invent. Math. 38 (1976–1977), 3353; MR 0430304 (55 #3309).

G. Kempf , The Grothendieck–Cousin complex of an induced representation , Adv. Math. 29 (1978), 310396, doi:10.1016/0001-8708(78)90021-X; MR 509802 (80g:14042).

R. MacPherson and K. Vilonen , Elementary construction of perverse sheaves , Invent. Math. 84 (1986), 403435, doi:10.1007/BF01388812; MR 833195 (87m:32028).

P. Nang , On a class of holonomic D-modules on M n (ℂ) related to the action of GL n (ℂ) × GL n (ℂ) , Adv. Math. 218 (2008), 635648, doi:10.1016/j.aim.2008.01.001; MR 2414315 (2009c:14028).

P. Nang , On the classification of regular holonomic D-modules on skew-symmetric matrices , J. Algebra 356 (2012), 115132, doi:10.1016/j.jalgebra.2012.01.021; MR 2891125.

C. Raicu and J. Weyman , Local cohomology with support in generic determinantal ideals , Algebra Number Theory 8 (2014), 12311257, doi:10.2140/ant.2014.8.1231; MR 3263142.

C. Raicu , J. Weyman and E. E. Witt , Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians , Adv. Math. 250 (2014), 596610, doi:10.1016/j.aim.2013.10.005; MR 3122178.

M. Van den Bergh , Local cohomology of modules of covariants , Adv. Math. 144 (1999), 161220, doi:10.1006/aima.1998.1809; MR 1695237 (2000d:14051).

K. Vilonen , Intersection homology D-module on local complete intersections with isolated singularities , Invent. Math. 81 (1985), 107114, doi:10.1007/BF01388774; MR 796193 (87f:32023).

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 38 *
Loading metrics...

Abstract views

Total abstract views: 171 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd September 2017. This data will be updated every 24 hours.