Reflection positivity for unitary representations of Lie groups

Author:
Humberto Prado

Journal:
Proc. Amer. Math. Soc. **114** (1992), 723-731

MSC:
Primary 22E45; Secondary 46N99, 47A67, 53C35

MathSciNet review:
1072089

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a Lie group, and let be an involutive automorphism on . Then we establish a correspondence between unitary representations of and unitary representations of a simply connected Lie group dual to , where the duality is defined by the given involution . The correspondence is obtained from a geometric assumption that was considered earlier in connection with reflection positivity. As a consequence of this construction, we obtain unitary representations of universal covering groups.

**[FOS]**J. Fröhlich, K. Osterwalder, and E. Seiler,*On virtual representations of symmetric spaces and their analytic continuation*, Ann. of Math. (2)**118**(1983), no. 3, 461–489. MR**727701**, 10.2307/2006979**[H]**Sigurđur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[J1]**Palle E. T. Jorgensen,*Analytic continuation of local representations of Lie groups*, Pacific J. Math.**125**(1986), no. 2, 397–408. MR**863534****[J2]**Palle E. T. Jorgensen,*Analytic continuation of local representations of symmetric spaces*, J. Funct. Anal.**70**(1987), no. 2, 304–322. MR**874059**, 10.1016/0022-1236(87)90115-7**[KL]**Abel Klein and Lawrence J. Landau,*From the Euclidean group to the Poincaré group via Osterwalder-Schrader positivity*, Comm. Math. Phys.**87**(1982/83), no. 4, 469–484. MR**691039****[LM]**M. Lüscher and G. Mack,*Global conformal invariance in quantum field theory*, Comm. Math. Phys.**41**(1975), 203–234. MR**0371282****[OS]**K. Osterwalder and R. Schrader,*Axioms for Euclidean Green's functions*, Comm. Math. Phys.**31**(1973), 83-112; Part II,**42**(1975), 281-305.**[P1]**H. Prado,*Local representations of symmetric spaces and*-*positivity*, Ph.D. Thesis, University of Iowa, 1989.**[P2]**-,*Linear subspaces with analytic structure, and approximation in Hilbert space*(to appear).**[S]**Robert Schrader,*Reflection positivity for the complementary series of 𝑆𝐿(2𝑛,𝐶)*, Publ. Res. Inst. Math. Sci.**22**(1986), no. 1, 119–141. MR**834352**, 10.2977/prims/1195178376**[Se]**Erhard Seiler,*Gauge theories as a problem of constructive quantum field theory and statistical mechanics*, Lecture Notes in Physics, vol. 159, Springer-Verlag, Berlin, 1982. MR**785937**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
22E45,
46N99,
47A67,
53C35

Retrieve articles in all journals with MSC: 22E45, 46N99, 47A67, 53C35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1072089-6

Keywords:
Local representations,
symmetric spaces,
reflection positivity,
induced representations and phrases

Article copyright:
© Copyright 1992
American Mathematical Society