12,519 research outputs found
Orbit Functions
In the paper, properties of orbit functions are reviewed and further
developed. Orbit functions on the Euclidean space are symmetrized
exponential functions. The symmetrization is fulfilled by a Weyl group
corresponding to a Coxeter-Dynkin diagram. Properties of such functions will be
described. An orbit function is the contribution to an irreducible character of
a compact semisimple Lie group of rank from one of its Weyl group
orbits. It is shown that values of orbit functions are repeated on copies of
the fundamental domain of the affine Weyl group (determined by the initial
Weyl group) in the entire Euclidean space . Orbit functions are solutions
of the corresponding Laplace equation in , satisfying the Neumann
condition on the boundary of . Orbit functions determine a symmetrized
Fourier transform and a transform on a finite set of points.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Non-Gatherable Triples for Non-Affine Root Systems
This paper contains a complete description of minimal non-gatherable triangle
triples in the lambda-sequences for the classical root systems, and
. Such sequences are associated with reduced decompositions (words) in
affine and non-affine Weyl groups. The existence of the non-gatherable triples
is a combinatorial obstacle for using the technique of intertwiners for an
explicit description of the irreducible representations of the (double) affine
Hecke algebras, complementary to their algebraic-geometric theory.Comment: This is a contribution to the Special Issue on Kac-Moody Algebras and
Applications, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
E-Orbit Functions
We review and further develop the theory of -orbit functions. They are
functions on the Euclidean space obtained from the multivariate
exponential function by symmetrization by means of an even part of a
Weyl group , corresponding to a Coxeter-Dynkin diagram. Properties of such
functions are described. They are closely related to symmetric and
antisymmetric orbit functions which are received from exponential functions by
symmetrization and antisymmetrization procedure by means of a Weyl group .
The -orbit functions, determined by integral parameters, are invariant with
respect to even part of the affine Weyl group corresponding
to . The -orbit functions determine a symmetrized Fourier transform,
where these functions serve as a kernel of the transform. They also determine a
transform on a finite set of points of the fundamental domain of the
group (the discrete -orbit function transform).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Antisymmetric Orbit Functions
In the paper, properties of antisymmetric orbit functions are reviewed and
further developed. Antisymmetric orbit functions on the Euclidean space
are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a
Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such
functions are described. These functions are closely related to irreducible
characters of a compact semisimple Lie group of rank . Up to a sign,
values of antisymmetric orbit functions are repeated on copies of the
fundamental domain of the affine Weyl group (determined by the initial Weyl
group) in the entire Euclidean space . Antisymmetric orbit functions are
solutions of the corresponding Laplace equation in , vanishing on the
boundary of the fundamental domain . Antisymmetric orbit functions determine
a so-called antisymmetrized Fourier transform which is closely related to
expansions of central functions in characters of irreducible representations of
the group . They also determine a transform on a finite set of points of
(the discrete antisymmetric orbit function transform). Symmetric and
antisymmetric multivariate exponential, sine and cosine discrete transforms are
given.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Some results on affine Deligne-Lusztig varieties
The study of affine Deligne-Lusztig varieties originally arose from
arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are
purely Lie-theoretic in nature. This survey deals with recent progress on
several important problems on affine Deligne-Lusztig varieties. The emphasis is
on the Lie-theoretic aspect, while some connections and applications to
arithmetic geometry will also be mentioned.Comment: 2018 ICM report, reference update
Maximal Newton points and the quantum Bruhat graph
We discuss a surprising relationship between the partially ordered set of
Newton points associated to an affine Schubert cell and the quantum cohomology
of the complex flag variety. The main theorem provides a combinatorial formula
for the unique maximum element in this poset in terms of paths in the quantum
Bruhat graph, whose vertices are indexed by elements in the finite Weyl group.
Key to establishing this connection is the fact that paths in the quantum
Bruhat graph encode saturated chains in the strong Bruhat order on the affine
Weyl group. This correspondence is also fundamental in the work of Lam and
Shimozono establishing Peterson's isomorphism between the quantum cohomology of
the finite flag variety and the homology of the affine Grassmannian. One
important geometric application of the present work is an inequality which
provides a necessary condition for non-emptiness of certain affine
Deligne-Lusztig varieties in the affine flag variety.Comment: 39 pages, 4 figures best viewed in color; final version to appear in
Michigan Math.
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