×

zbMATH — the first resource for mathematics

Algebraic families of groups and commuting involutions. (English) Zbl 1428.20046
This paper constructs real forms of one-parameter algebraic families of complex affine algebraic groups. For the notion of an algebraic family, see [“Algebraic families of Harish-Chandra pairs”, Preprint, arXiv:1610.03435; “Contractions of representations and algebraic families of Harish-Chandra modules”, Preprint, arXiv:1703.04028] by J. Bernstein, N. Higson and E. Subag.
Let \(\sigma_1\) and \(\sigma_2\) be two commuting antiholomorphic involutions of a complex affine algebraic group \(G\). Its main result states that there exist an algebraic family \(\boldsymbol{G}\) of affine algebraic groups and an antiholomorphic involution \(\boldsymbol{\sigma}\) of the family \(\boldsymbol{G}\) that interpolates between the real forms \(G^{\sigma_1}\) and \(G^{\sigma_2}\). More precisely, if \([\alpha: \beta] \in \mathbb{RP}^1\) then \[ \boldsymbol{G}^{\boldsymbol{\sigma}}|_{[\alpha:\beta]} \cong \begin{cases} G^{\sigma_1}, & \alpha\beta >0,\\ (G^{\sigma_1}\cap G^{\sigma_2}) \ltimes (\mathfrak{g}^{\sigma_1} \cap \mathfrak{g}^{-\sigma_2}), & \alpha \beta =0,\\ G^{\sigma_2}, & \alpha \beta <0. \end{cases} \]

MSC:
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
22E15 General properties and structure of real Lie groups
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Adams, J.; Barbasch, D.; Vogan, D. A. Jr., The Langlands Classification and Irreducible Characters for Real Reductive Groups, 104, (1992), Birkhäuser Boston, Boston, MA · Zbl 0756.22004
[2] A. Afgoustidis, How tempered representations of a semisimple Lie group contract to its Cartan motion group, preprint (2015), arXiv:1510.02650.
[3] Aizenbud, A.; Gourevitch, D., Schwartz functions on Nash manifolds, Int. Math. Res. Not., 2008, 5, 37, (2008) · Zbl 1161.58002
[4] J. Bernstein, Stacks in representation theory. What is a continuous representation of an algebraic group? preprint (2014), arXiv:1410.0435.
[5] J. Bernstein, N. Higson and E. M. Subag, Algebraic families of Harish-Chandra pairs, preprint (2016), arXiv:1610.03435.
[6] J. Bernstein, N. Higson and E. M. Subag, Contractions of representations and algebraic families of Harish-Chandra modules, preprint (2017), arXiv:1703.04028.
[7] Bernstein, J.; Krötz, B., Smooth Fréchet globalizations of harish-chandra modules, Israel J. Math., 199, 1, 45-111, (2014) · Zbl 1294.22010
[8] Dooley, A. H.; Rice, J. W., Contractions of rotation groups and their representations, Math. Proc. Cambridge Philos. Soc., 94, 3, 509-517, (1983) · Zbl 0532.22014
[9] Dooley, A. H.; Rice, J. W., On contractions of semisimple Lie groups, Trans. Amer. Math. Soc., 289, 1, 185-202, (1985) · Zbl 0546.22017
[10] Flensted-Jensen, M., Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal., 30, 1, 106-146, (1978) · Zbl 0419.22019
[11] Hartshorne, R., Algebraic Geometry, (1977), Springer-Verlag, New York
[12] Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, 34, (2001), American Mathematical Society, Providence, RI · Zbl 0993.53002
[13] Helminck, A. G., Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces, Adv. Math., 71, 1, 21-91, (1988) · Zbl 0685.22007
[14] Helminck, A. G.; Schwarz, G. W., Orbits and invariants associated with a pair of commuting involutions, Duke Math. J., 106, 2, 237-279, (2001) · Zbl 1015.20031
[15] Higson, N., Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey, The MacKey analogy and \(K\)-theory, 149-172, (2008), American Mathematical Society, Providence, RI · Zbl 1172.46046
[16] Higson, N., Noncommutative Geometry and Global Analysis, 546, On the analogy between complex semisimple groups and their Cartan motion groups, 137-170, (2011), American Mathematical Society, Providence, RI · Zbl 1236.22010
[17] Inonu, E.; Wigner, E. P., On the contraction of groups and their representations, Proc. Natl. Acad. Sci. USA, 39, 510-524, (1953) · Zbl 0050.02601
[18] Loos, O., Symmetric Spaces, I, II, (1969), W. A. Benjamin · Zbl 0175.48601
[19] Mackey, G. W., Lie Groups and Their Representations, On the analogy between semisimple Lie groups and certain related semi-direct product groups, 339-363, (1975), Halsted, New York
[20] Milnor, J., Singular Points of Complex Hypersurfaces, (1968), Princeton University Press, Princeton, NJ · Zbl 0184.48405
[21] Onishchik, A. L.; Vinberg, È. B., Lie Groups and Algebraic Groups, (1990), Springer-Verlag, Berlin · Zbl 0722.22004
[22] Panyushev, D. I., Commuting involutions and degenerations of isotropy representations, Transform. Groups, 18, 2, 507-537, (2013) · Zbl 1317.17022
[23] Serre, J.-P., Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble), 6, 1-42, (19551956) · Zbl 0075.30401
[24] E. M. Subag, The algebraic Mackey-Higson bijections, preprint (2017), arXiv:1706.05616.
[25] Vogan, D. A. Jr., Irreducible characters of semisimple Lie groups. IV. character-multiplicity duality, Duke Math. J., 49, 4, 943-1073, (1982) · Zbl 0536.22022
[26] Vogan, D. A. Jr., Representation Theory of Groups and Algebras, 145, The local Langlands conjecture, 305-379, (1993), American Mathematical Society, Providence, RI
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.