We formulate the Nahm transform in the context of parabolic Higgs bundles on P 1 and extend its scope in completely algebraic terms. This transform requires parabolic Higgs bundles to satisfy an admissibility condition and allows Higgs fields to have poles of arbitrary order and arbitrary behavior. Our methods are constructive in nature and examples are provided. The extended Nahm transform is established as an algebraic duality between moduli spaces of parabolic Higgs bundles. The guiding principle behind the construction is to investigate the behavior of spectral data near the poles of Higgs fields. 14H60; 14E05, 14J26
A pair (G, K) of a group and its subgroup is called a Gelfand pair if the induced trivial representation of K on G is multiplicity free. Let (a j ) be a sequence of positive integers of length n, and let (b i ) be its non-decreasing rearrangement. The sequence (a i ) is called a parking function of length n if b i ≤ i for all i = 1, . . . , n. In this paper we study certain Gelfand pairs in relation with parking functions. In particular, we find explicit descriptions of the decomposition of the associated induced trivial representations into irreducibles. We obtain and study a new q analogue of the Catalan numbers 1 n+1 2n n , n ≥ 1.
This paper studies the combinatorics of the orbit Hecke algebras associated with W × W orbits in the Renner monoid of a finite monoid of Lie type, M, where W is the Weyl group associated with M. It is shown by Putcha in [12] that the Kazhdan–Lusztig involution [6] can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using the R-polynomials, we calculate the Möbius function of the Bruhat–Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.
We define and study the rational Schubert, rational Grothendieck, rational key polynomials in an effort to understand Molev's dual Schur functions from the viewpoint of Lascoux.
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