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  • 1
    Electronic Resource
    Electronic Resource
    Quantum integrable systems and differential Galois theory (1997)
    Springer
    Transformation groups 2 (1997), S. 31-56 
    Details
    ISSN: 1531-586X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01234630
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The Cohen-Macaulay property of the category of $ ({\frak g}, K) $-modules (1997)
    Springer
    Selecta mathematica 3 (1997), S. 303-314 
    Details
    ISSN: 1420-9020
    Keywords: Key words. $ ({\frak g}, K) $-modules, Cohen-Macaulay categories, Grothendieck duality.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. Let $ ({\frak g}, K) $ be a Harish-Chandra pair. In this paper we prove that if P and P' are two projective $ ({\frak g}, K) $ -modules, then Hom(P, P') is a Cohen-Macaulay module over the algebra $ {\cal Z}({\frak g}, K) $ of K-invariant elements in the center of $ U({\frak g}) $ . This fact implies that the category of $ ({\frak g}, K) $ -modules is locally equivalent to the category of modules over a Cohen-Macaulay algebra, where by a Cohen-Macaulay algebra we mean an associative algebra that is a free finitely generated module over a polynomial subalgebra of its center.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000290050012
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  • 3
    Electronic Resource
    Electronic Resource
    On the Schwartz space of the basic affine space (1999)
    Springer
    Selecta mathematica 5 (1999), S. 1-28 
    Details
    ISSN: 1420-9020
    Keywords: Key words. Representation theory, automorphic forms.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space ${\cal S}(X)$ (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of $ {\cal S}(X)^I $ , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of $ {\cal S}(X) $ , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000290050041
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