On the principal symbols of KC-invariant differential operators on Hermitian symmetric spaces
Takashi HASHIMOTO
Source: J. Math. Soc. Japan Volume 63, Number 3
(2011), 837-869.
Abstract
Let (G,K) be one of the following Hermitian symmetric pair: (SU(p,q), S(U(p) × U(q))), (Sp(n,R), U(n)), or (SO*(2n), U(n)). Let Gc and KC be the complexifications of G and K, respectively, Q the maximal parabolic subgroup of Gc whose Levi part is KC, and V the holomorphic tangent space at the origin of G/K. It is known that the ring of KC-invariant differential operators on V has a generating system {Γk} given in terms of determinant or Pfaffian that plays an essential role in the Capelli identities. Our main result is that determinant or Pfaffian of a deformation of the twisted moment map on the holomorphic cotangent bundle of Gc/Q provides a generating function for the principal symbols of Γk's.
First Page:
Show
Hide
Keywords: Hermitian symmetric space; KC-invariant differential operator; principal symbol; Capelli identity; generating function; twisted moment map
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1312203803
Digital Object Identifier: doi:10.2969/jmsj/06330837
Mathematical Reviews number (MathSciNet): MR2836747
References
T. Hashimoto, A central element in the universal enveloping algebra of type $\mathsf{D}_n$ via minor summation formula of Pfaffians, J. Lie Theory, 18 (2008), 581–594.
Mathematical Reviews (MathSciNet): MR2493055
Zentralblatt MATH: 1210.17018
T. Hashimoto, Generating function for $\mathrm{GL}_{n}$-invariant differential operators in the skewCapelli identity, 2009, Lett. Math. Rhys., 93 (2010), 157–168, arXiv:0803.1339v2 [math.RT].
Mathematical Reviews (MathSciNet): MR2679967
Digital Object Identifier: doi:10.1007/s11005-010-0405-5
Zentralblatt MATH: 05788011
T. Hashimoto, K. Ogura, K. Okamoto and R. Sawae, Borel-Weil theory and Feynman path integrals on flag manifolds, Hiroshima Math. J., 23 (1993), 231–247.
Mathematical Reviews (MathSciNet): MR1228571
Project Euclid: euclid.hmj/1206128252
R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann., 290 (1991), 565–619.
Mathematical Reviews (MathSciNet): MR1116239
Digital Object Identifier: doi:10.1007/BF01459261
Zentralblatt MATH: 0733.20019
M. Ishikawa and M. Wakayama, Application of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities, J. Comb. Theory A, 113 (2006), 113–155.
Mathematical Reviews (MathSciNet): MR2192773
Digital Object Identifier: doi:10.1016/j.jcta.2005.05.008
M. Itoh, A Cayley-Hamilton theorem for the skew Capelli elements, J. Algebra., 242 (2001), 740–761.
Mathematical Reviews (MathSciNet): MR1848969
Zentralblatt MATH: 0981.17006
Digital Object Identifier: doi:10.1006/jabr.2001.8824
M. Itoh and T. Umeda, On central elements in the universal enveloping algebras of the orthogonal Lie algebra, Compositio Math., 127 (2001), 333–359.
Mathematical Reviews (MathSciNet): MR1845042
Digital Object Identifier: doi:10.1023/A:1017571403369
Zentralblatt MATH: 1007.17007
K. Kinoshita and M. Wakayama, Explicit Capelli identities for skew symmetric matrices, Proc. Edinburgh Math. Soc., 45 (2002), 449–465.
Mathematical Reviews (MathSciNet): MR1912651
Digital Object Identifier: doi:10.1017/S0013091500001176
Zentralblatt MATH: 1035.17020
A. W. Knapp, Representation theory of semisimple groups: An overview based on examples, Princeton Mathematical Series, 36, Princeton Univ. Press, 1986.
Mathematical Reviews (MathSciNet): MR855239
D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory (3rd enlarged edition), Springer-Verlag, 1994.
Mathematical Reviews (MathSciNet): MR1304906
W. Schmid and K. Vilonen, Characteristic cycles of constructible sheaves, Invent. Math., 124 (1996), 451–502.
Mathematical Reviews (MathSciNet): MR1369425
Zentralblatt MATH: 0851.32011
Digital Object Identifier: doi:10.1007/s002220050060
G. Shimura, On differential operators attached to certain representations of classical groups, Invent. Math., 77 (1984), 463–488.
Mathematical Reviews (MathSciNet): MR759261
Zentralblatt MATH: 0558.10023
Digital Object Identifier: doi:10.1007/BF01388834
Journal of the Mathematical Society of Japan
