Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Integrability of unitary representations on reproducing kernel spaces

Authors: Stéphane Merigon, Karl-Hermann Neeb and Gestur Ólafsson
Journal: Represent. Theory 19 (2015), 24-55
MSC (2010): Primary 17B15, 22E30, 22E70
Published electronically: March 10, 2015
MathSciNet review: 3319397
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathfrak{g}$ be a Banach-Lie algebra and $ \tau : \mathfrak{g} \to \mathfrak{g}$ an involution. Write $ \mathfrak{g}=\mathfrak{h}\oplus \mathfrak{q}$ for the eigenspace decomposition of $ \mathfrak{g}$ with respect to $ \tau $ and $ \mathfrak{g}^c := \mathfrak{h}\oplus i\mathfrak{q}$ for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of $ \mathfrak{g}^c$. The first class of representation is determined by a smooth positive definite kernel $ K$ on a locally convex manifold $ M$. The kernel is assumed to satisfy a natural invariance condition with respect to an infinitesimal action $ \beta \colon \mathfrak{g} \to \mathcal {V}(M)$ by locally integrable vector fields that is compatible with a smooth action of a connected Lie group $ H$ with Lie algebra $ \mathfrak{h}$. The second class is constructed from a positive definite kernel corresponding to a positive definite distribution $ K \in C^{-\infty }(M \times M)$ on a finite dimensional smooth manifold $ M$ which satisfies a similar invariance condition with respect to a homomorphism $ \beta \colon \mathfrak{g} \to \mathcal {V}(M)$. As a consequence, we get a generalization of the Lüscher-Mack Theorem which applies to a class of semigroups that need not have a polar decomposition. Our integrability results also apply naturally to local representations and representations arising in the context of reflection positivity.

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

  • [B95] Emile Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. École Norm. Sup. (3) 12 (1895), 9-55 (French). MR 1508908
  • [Fro80] J. Fröhlich, Unbounded, symmetric semigroups on a separable Hilbert space are essentially selfadjoint, Adv. in Appl. Math. 1 (1980), no. 3, 237-256. MR 603131 (82c:47044),
  • [FOS83] 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 (85j:22024),
  • [Jo86] Palle E. T. Jorgensen, Analytic continuation of local representations of Lie groups, Pacific J. Math. 125 (1986), no. 2, 397-408. MR 863534 (88m:22030)
  • [Jo87] Palle E. T. Jorgensen, Analytic continuation of local representations of symmetric spaces, J. Funct. Anal. 70 (1987), no. 2, 304-322. MR 874059 (88d:22021),
  • [JOl98] Palle E. T. Jorgensen and Gestur Ólafsson, Unitary representations of Lie groups with reflection symmetry, J. Funct. Anal. 158 (1998), no. 1, 26-88. MR 1641554 (99m:22013),
  • [JOl00] Palle E. T. Jorgensen and Gestur Ólafsson, Unitary representations and Osterwalder-Schrader duality, The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998) Proc. Sympos. Pure Math., vol. 68, Amer. Math. Soc., Providence, RI, 2000, pp. 333-401. MR 1767902 (2001f:22036),
  • [KL81] Abel Klein and Lawrence J. Landau, Construction of a unique selfadjoint generator for a symmetric local semigroup, J. Funct. Anal. 44 (1981), no. 2, 121-137. MR 642913 (83b:47051),
  • [KL82] 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 (84g:22043)
  • [LM75] M. Lüscher and G. Mack, Global conformal invariance in quantum field theory, Comm. Math. Phys. 41 (1975), 203-234. MR 0371282 (51 #7503)
  • [Mer11] Stéphane Merigon, Integrating representations of Banach-Lie algebras, J. Funct. Anal. 260 (2011), no. 5, 1463-1475. MR 2749434 (2012a:22036),
  • [MN12] Stéphane Merigon and Karl-Hermann Neeb, Analytic extension techniques for unitary representations of Banach-Lie groups, Int. Math. Res. Not. IMRN 18 (2012), 4260-4300. MR 2975382,
  • [MN14] S. Mergon and K.-H. Neeb, Semibounded unitary representations of mapping groups with values in infinite dimensional hermitian groups, in preparation
  • [Ne00] Karl-Hermann Neeb, Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, vol. 28, Walter de Gruyter & Co., Berlin, 2000. MR 1740617 (2001j:32020)
  • [Ne06] Karl-Hermann Neeb, Towards a Lie theory of locally convex groups, Jpn. J. Math. 1 (2006), no. 2, 291-468. MR 2261066 (2007k:22020),
  • [Ne10] Karl-Hermann Neeb, On differentiable vectors for representations of infinite dimensional Lie groups, J. Funct. Anal. 259 (2010), no. 11, 2814-2855. MR 2719276 (2012b:22031),
  • [Ne13] K.-H. Neeb, Unitary representations of unitary groups, ``Lie theory workshops'', Eds. G. Mason, I. Penkov, J. Wolf, ``Developments in Math.'' Vol. 37, Springer, 2014, 197-243.
  • [NO14] Karl-Hermann Neeb and Gestur Ólafsson, Reflection positivity and conformal symmetry, J. Funct. Anal. 266 (2014), no. 4, 2174-2224. MR 3150157,
  • [NO13] K.-H. Neeb and G. Ólafsson, Reflection positive one-parameter groups and dilations, Complex Analysis and Operator Theory, to appear.
  • [Nel59] Edward Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572-615. MR 0107176 (21 #5901)
  • [Ol84] G. I. Olshanskiĭ, Infinite-dimensional classical groups of finite $ R$-rank: description of representations and asymptotic theory, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 28-42 (Russian). MR 739087 (86a:22037)
  • [Ol90] G. I. Olshanskiĭ, Unitary representations of infinite-dimensional pairs $ (G,K)$ and the formalism of R. Howe, Representation of Lie groups and related topics, Adv. Stud. Contemp. Math., vol. 7, Gordon and Breach, New York, 1990, pp. 269-463. MR 1104279 (92c:22043)
  • [OS73] Konrad Osterwalder and Robert Schrader, Axioms for Euclidean Green's functions, Comm. Math. Phys. 31 (1973), 83-112. MR 0329492 (48 #7834)
  • [Pr92] Humberto Prado, Reflection positivity for unitary representations of Lie groups, Proc. Amer. Math. Soc. 114 (1992), no. 3, 723-731. MR 1072089 (92f:22019),
  • [Sch86] Robert Schrader, Reflection positivity for the complementary series of $ {\rm SL}(2n,{\bf C})$, Publ. Res. Inst. Math. Sci. 22 (1986), no. 1, 119-141. MR 834352 (87h:81111),
  • [Sh84] David S. Shucker, Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups, J. Funct. Anal. 58 (1984), no. 3, 291-309. MR 759101 (86b:47077),
  • [Si72] Jacques Simon, On the integrability of representations of infinite dimensional real Lie algebras, Comm. Math. Phys. 28 (1972), 39-46. MR 0308333 (46 #7447)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 17B15, 22E30, 22E70

Retrieve articles in all journals with MSC (2010): 17B15, 22E30, 22E70

Additional Information

Stéphane Merigon
Affiliation: Department of Mathematics, Friedrich-Alexander University, Erlangen-Nuremberg, Cauerstrasse 11, 91058 Erlangen, Germany

Karl-Hermann Neeb
Affiliation: Department of Mathematics, Friedrich-Alexander University, Erlangen-Nuremberg, Cauerstrasse 11, 91058 Erlangen, Germany

Gestur Ólafsson
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Keywords: Unitary representations, finite and infinite dimensional Lie groups, manifolds, reproducing kernel
Received by editor(s): June 10, 2014
Received by editor(s) in revised form: June 11, 2014, July 11, 2014, and October 1, 2014
Published electronically: March 10, 2015
Additional Notes: The first author would like to thank Louisiana State University for their hospitality during his visit in 2014 when most of the work on this article was done
The research of the second author was supported by DFG-grant NE 413/7-2, SPP “Representation Theory”.
The research of the third author was supported by NSF grant DMS-1101337, “Representation Theory and Harmonic Analysis on Homogeneous Spaces”
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society