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Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

100%
Zielenkiewicz M.
Open Mathematics
|
2014
|
tom 12
|
nr 4
574-583
EN
Using the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases can be expressed as special cases of one formula involving the Weyl group action on the characters of the natural representation of the torus.
2
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Wahl’s conjecture holds in odd characteristics for symplectic and orthogonal Grassmannians

81%
Lakshmibai V., Raghavan K., Sankaran P.
Open Mathematics
|
2009
|
tom 7
|
nr 2
214-223
EN
It is shown that the proof by Mehta and Parameswaran of Wahl’s conjecture for Grassmannians in positive odd characteristics also works for symplectic and orthogonal Grassmannians.
3
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Elliptic curves on spinor varieties

61%
Perrin N.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1393-1406
EN
We prove irreducibility of the scheme of morphisms, of degree large enough, from a smooth elliptic curve to spinor varieties. We give an explicit bound on the degree.
4
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On parabolic Whittaker functions II

52%
Oblezin S.
Open Mathematics
|
2012
|
tom 10
|
nr 2
543-558
EN
We propose a Givental-type stationary phase integral representation for the restricted Grm,N-Whittaker function, which is expected to describe the (S 1×U N)-equivariant Gromov-Witten invariants of the Grassmann variety Grm,N. Our key tool is a generalization of the Whittaker model for principal series U(gl N)-modules, and its realization in the space of functions of totally positive unipotent matrices. In particular, our construction involves a representation theoretic derivation of the Batyrev-Ciocan-Fontanine-Kim-van Straten toric degeneration of Grm,N.
5
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The geometry of the space of Cauchy data of nonlinear PDEs

42%
Moreno G.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1960-1981
EN
First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.
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