Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Spinoriality of orthogonal representations of reductive groups

Authors: Rohit Joshi and Steven Spallone
Journal: Represent. Theory 24 (2020), 435-469
MSC (2010): Primary 20G15; Secondary 22E46
Published electronically: September 16, 2020
MathSciNet review: 4150223
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a connected reductive group over a field $F$ of characteristic $0$, and $\varphi : G \to \operatorname {SO}(V)$ an orthogonal representation over $F$. We give criteria to determine when $\varphi$ lifts to the double cover $\operatorname {Spin}(V)$.

References [Enhancements On Off] (What's this?)

  • J. Frank Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0252560
  • Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629
  • Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR 2109105
  • E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates) (Russian). MR 0047629
  • E. B. Dynkin, Selected papers of E. B. Dynkin with commentary, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000. Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik. MR 1757976
  • William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
  • Roe Goodman and Nolan R. Wallach, Symmetry, representations, and invariants, Graduate Texts in Mathematics, vol. 255, Springer, Dordrecht, 2009. MR 2522486
  • Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
  • R. Joshi, Spinorial representations of Lie groups, PhD thesis, Indian Institute of Science Education and Research, Pune, 2018.
  • Alexander Kirillov Jr., An introduction to Lie groups and Lie algebras, Cambridge Studies in Advanced Mathematics, vol. 113, Cambridge University Press, Cambridge, 2008. MR 2440737
  • Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
  • Bertram Kostant, On Macdonald’s $\eta $-function formula, the Laplacian and generalized exponents, Advances in Math. 20 (1976), no. 2, 179–212. MR 485661, DOI
  • Dipendra Prasad and Dinakar Ramakrishnan, Lifting orthogonal representations to spin groups and local root numbers, Proc. Indian Acad. Sci. Math. Sci. 105 (1995), no. 3, 259–267. MR 1369731, DOI
  • Hans Samelson, Notes on Lie algebras, 2nd ed., Universitext, Springer-Verlag, New York, 1990. MR 1056083
  • Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
  • T. A. Springer, Reductive groups, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR 546587
  • T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713
  • Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112
  • Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1763974

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 20G15, 22E46

Retrieve articles in all journals with MSC (2010): 20G15, 22E46

Additional Information

Rohit Joshi
Affiliation: Bhaskaracharya Pratishthana, 56/14, Erandavane, Damle Path, off Law College Road, Pune-411004, Maharashtra, India; and Indian Institute of Science Education and Research, Pune-411021, India
ORCID: 0000-0002-2471-2737

Steven Spallone
Affiliation: Indian Institute of Science Education and Research, Pune-411021, India
MR Author ID: 824479

Keywords: Reductive groups, orthogonal representations, Dynkin index, lifting criterion, Weyl dimension formula
Received by editor(s): October 31, 2018
Received by editor(s) in revised form: February 21, 2020
Published electronically: September 16, 2020
Additional Notes: This paper comes out of the first author’s Ph.D. thesis at IISER Pune, during which he was supported by an Institute Fellowship. Afterwards he was supported by a fellowship from Bhaskaracharya Pratishthan.
Article copyright: © Copyright 2020 American Mathematical Society