46 research outputs found
Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras
We introduce a generalization of the classical Hall-Littlewood and
Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove
that these Kostka-Foulkes polynomials coincide with the natural generalization
of Lusztig's -analog of weight multiplicities, thereby extending a theorem
of Kato. For an affine Kac-Moody algebra, we define -analogs of string
functions and use Cherednik's constant term identities to derive explicit
product expressions for them.Comment: 19 page
Poincare series of subsets of affine Weyl groups
In this note, we identify a natural class of subsets of affine Weyl groups
whose Poincare series are rational functions. This class includes the sets of
minimal coset representatives of reflection subgroups. As an application, we
construct a generalization of the classical length-descent generating function,
and prove its rationality.Comment: 7 page
On growth types of quotients of Coxeter groups by parabolic subgroups
The principal objects studied in this note are Coxeter groups that are
neither finite nor affine. A well known result of de la Harpe asserts that such
groups have exponential growth. We consider quotients of by its parabolic
subgroups and by a certain class of reflection subgroups. We show that these
quotients have exponential growth as well. To achieve this, we use a theorem of
Dyer to construct a reflection subgroup of that is isomorphic to the
universal Coxeter group on three generators. The results are all proved under
the restriction that the Coxeter diagram of is simply laced, and some
remarks made on how this restriction may be relaxed.Comment: 10 pages; The exposition has been made more concise and an additional
proposition is proved in the final sectio
A note on exponents vs root heights for complex simple Lie algebras
We give an elementary combinatorial proof of a special case of a result due
to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel.
This can be used to give yet another proof of the classical fact that for a
complex simple Lie algebra, the partition formed by its exponents is dual to
that formed by the numbers of positive roots at each height.Comment: 5 page
Stability of the Chari-Pressley-Loktev bases for local Weyl modules of
We prove stability of the Chari-Pressley-Loktev bases for natural inclusions
of local Weyl modules of the current algebra . These modules being
known to be Demazure submodules in the level 1 representations of the affine
Lie algebra , we obtain, by passage to the direct limit, bases
for the level 1 representations themselves.Comment: 20 pages; minor revision