Spinoriality of orthogonal representations of reductive groups

Joshi, Rohit; Spallone, Steven

doi:10.1090/ert/552

Spinoriality of orthogonal representations of reductive groups
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by Rohit Joshi and Steven Spallone PDF

Represent. Theory 24 (2020), 435-469 Request permission

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

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, DOI 10.1007/978-3-540-89394-3
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, DOI 10.1007/978-1-4612-0979-9
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
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, DOI 10.1017/CBO9780511755156
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
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 10.1016/0001-8708(76)90186-9
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 10.1007/BF02837191
Hans Samelson, Notes on Lie algebras, 2nd ed., Universitext, Springer-Verlag, New York, 1990. MR 1056083, DOI 10.1007/978-1-4613-9014-5
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, DOI 10.1007/978-0-8176-4840-4
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, DOI 10.1007/978-3-662-12622-6