⃝Springer Science+Business Media New York (2017)
ENHANCED VARIETY OF HIGHER LEVEL
AND KOSTKA FUNCTIONS
ASSOCIATED TO COMPLEX REFLECTION GROUPS
Department of Mathematics
1239 Siping Road
Shanghai 200092, P.R. China
Abstract. Let V be an n-dimensional vector space over an algebraic closure of a ﬁnite
, and G = GL(V ). A variety X = G × V
is called an enhanced variety of
level r. Let X
be the unipotent variety of X . We have a partition
indexed by r-partitions λ of n. In the case where r = 1 or 2, X
single G-orbit, but if r ≥ 3, X
is, in general, a union of inﬁnitely many G-orbits. In
this paper, we prove certain orthogonality relations for the characteristic functions (over
) of the intersection cohomology IC(X
), and show some results, which suggest a
close relationship between those characteristic functions and Kostka functions associated
to the complex reﬂection group S
Let V be an n-dimensional vector space over an algebraic closure of a ﬁnite ﬁeld F
and G = GL(V ) ≃ GL
. In 1981, Lusztig showed in [L1] that Kostka polynomials
(t) have a geometric interpretation in terms of the intersection cohomology
associated to the closure of unipotent classes in G in the following sense. Let C
be the unipotent class corresponding to a partition λ of n, and K = IC(
be the intersection cohomology complex on the closure C
. He proved that
K = 0 for odd i, and that for partitions λ, µ of n,
) = t
where x ∈ C
, and n(λ) is the usual n-function.
Kostka polynomials are polynomials indexed by a pair of partitions. In [S1],
[S2], as a generalization of Kostka polynomials, Kostka functions K
ated to the complex reﬂection group S
were introduced, which are
a-priori rational functions in Q(t) indexed by r-partitions λ, µ of n (see 3.10 for
Received August 18, 2015. Accepted October 23, 2016.
Corresponding Author: T. Shoji, e-mail: email@example.com.
Vol. 22, No.
, 2017, pp.